violetxi/omni-math-difficulty-1_2 · Datasets at Hugging Face

domain list	difficulty float64	problem string	solution string	answer string	source string
[ "Mathematics -> Geometry -> Plane Geometry -> Angles" ]	1.5	In triangle $BCD$, $\angle CBD=\angle CDB$ because $BC=CD$. If $\angle BCD=80+50+30=160$, find $\angle CBD=\angle CDB$.	$\angle CBD=\angle CDB=10$	10	HMMT_11
[ "Mathematics -> Algebra -> Linear Algebra -> Linear Transformations" ]	1.5	What is the $y$-intercept of the line $y = x + 4$ after it is translated down 6 units?	The line with equation $y = x + 4$ has a $y$-intercept of 4. When the line is translated 6 units downwards, all points on the line are translated 6 units down. This moves the $y$-intercept from 4 to $4 - 6 = -2$.	-2	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1.5	If $(pq)(qr)(rp) = 16$, what is a possible value for $pqr$?	Since $(pq)(qr)(rp) = 16$, then $pqqrrp = 16$ or $p^2q^2r^2 = 16$. Thus, $(pqr)^2 = 16$ and so $pqr = \pm 4$. Using the given answers, $pqr$ is positive and so $pqr = 4$.	4	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]	1.5	What is the value of \((-1)^{3}+(-1)^{2}+(-1)\)?	Since -1 raised to an even exponent equals 1 and -1 raised to an odd exponent equals -1, then \((-1)^{3}+(-1)^{2}+(-1)=-1+1-1=-1\).	-1	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1.5	What is the value of $x$ if the three numbers $2, x$, and 10 have an average of $x$?	Since the average of $2, x$ and 10 is $x$, then $\frac{2 + x + 10}{3} = x$. Multiplying by 3, we obtain $2 + x + 10 = 3x$. Re-arranging, we obtain $x + 12 = 3x$ and then $2x = 12$ which gives $x = 6$.	6	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1.5	The three numbers $5, a, b$ have an average (mean) of 33. What is the average of $a$ and $b$?	Since $5, a, b$ have an average of 33, then $\frac{5+a+b}{3}=33$. Multiplying by 3, we obtain $5+a+b=3 \times 33=99$, which means that $a+b=94$. The average of $a$ and $b$ is thus equal to $\frac{a+b}{2}=\frac{94}{2}=47$.	47	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1.5	If $x=2018$, what is the value of the expression $x^{2}+2x-x(x+1)$?	For any value of $x$, we have $x^{2}+2x-x(x+1)=x^{2}+2x-x^{2}-x=x$. When $x=2018$, the value of this expression is thus 2018.	2018	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1.5	If \( \sqrt{100-x}=9 \), what is the value of \( x \)?	Since \( \sqrt{100-x}=9 \), then \( 100-x=9^{2}=81 \), and so \( x=100-81=19 \).	19	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1	If $x+\sqrt{81}=25$, what is the value of $x$?	If $x+\sqrt{81}=25$, then $x+9=25$ or $x=16$.	16	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Fractions" ]	1	The symbol $\diamond$ is defined so that $a \diamond b=\frac{a+b}{a \times b}$. What is the value of $3 \diamond 6$?	Using the definition of the symbol, $3 \diamond 6=\frac{3+6}{3 \times 6}=\frac{9}{18}=\frac{1}{2}$.	\frac{1}{2}	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1.5	If $a(x+b)=3 x+12$ for all values of $x$, what is the value of $a+b$?	Since $a(x+b)=3 x+12$ for all $x$, then $a x+a b=3 x+12$ for all $x$. Since the equation is true for all $x$, then the coefficients on the left side must match the coefficients on the right side. Therefore, $a=3$ and $a b=12$, which gives $3 b=12$ or $b=4$. Finally, $a+b=3+4=7$.	7	cayley
[ "Mathematics -> Applied Mathematics -> Math Word Problems" ]	1.5	A class of 30 students was asked what they did on their winter holiday. 20 students said that they went skating. 9 students said that they went skiing. Exactly 5 students said that they went skating and went skiing. How many students did not go skating and did not go skiing?	Since 20 students went skating and 5 students went both skating and skiing, then \( 20-5=15 \) students went skating only. Since 9 students went skiing and 5 students went both skating and skiing, then \( 9-5=4 \) students went skiing only. The number of students who went skating or skiing or both equals the sum of the number who went skating only, the number who went skiing only, and the number who went both skating and skiing, or \( 15+4+5=24 \). Therefore, \( 30-24=6 \) students did not go skating or skiing.	6	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Fractions" ]	1	At Wednesday's basketball game, the Cayley Comets scored 90 points. At Friday's game, they scored $80\%$ as many points as they scored on Wednesday. How many points did they score on Friday?	On Friday, the Cayley Comets scored $80\%$ of 90 points. This is equal to $\frac{80}{100} \times 90 = \frac{8}{10} \times 90 = 8 \times 9 = 72$ points. Alternatively, since $80\%$ is equivalent to 0.8, then $80\%$ of 90 is equal to $0.8 \times 90 = 72$.	72	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1.5	The Cayley Corner Store sells three types of toys: Exes, Wyes and Zeds. All Exes are identical, all Wyes are identical, and all Zeds are identical. The mass of 2 Exes equals the mass of 29 Wyes. The mass of 1 Zed equals the mass of 16 Exes. The mass of 1 Zed equals the mass of how many Wyes?	Since the mass of 2 Exes equals the mass of 29 Wyes, then the mass of $8 \times 2$ Exes equals the mass of $8 \times 29$ Wyes. In other words, the mass of 16 Exes equals the mass of 232 Wyes. Since the mass of 1 Zed equals the mass of 16 Exes, then the mass of 1 Zed equals the mass of 232 Wyes.	232	cayley
[ "Mathematics -> Geometry -> Plane Geometry -> Triangulations" ]	1.5	Square $P Q R S$ has an area of 900. $M$ is the midpoint of $P Q$ and $N$ is the midpoint of $P S$. What is the area of triangle $P M N$?	Since square $P Q R S$ has an area of 900, then its side length is $\sqrt{900}=30$. Thus, $P Q=P S=30$. Since $M$ and $N$ are the midpoints of $P Q$ and $P S$, respectively, then $P N=P M=\frac{1}{2}(30)=15$. Since $P Q R S$ is a square, then the angle at $P$ is $90^{\circ}$, so $\triangle P M N$ is right-angled. Therefore, the area of $\triangle P M N$ is $\frac{1}{2}(P M)(P N)=\frac{1}{2}(15)(15)=\frac{225}{2}=112.5$.	112.5	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1.5	If \( x=2 \) and \( v=3x \), what is the value of \((2v-5)-(2x-5)\)?	Since \( v=3x \) and \( x=2 \), then \( v=3 \cdot 2=6 \). Therefore, \((2v-5)-(2x-5)=(2 \cdot 6-5)-(2 \cdot 2-5)=7-(-1)=8\).	8	fermat
[ "Mathematics -> Algebra -> Prealgebra -> Fractions" ]	1	Calculate the value of the expression $\left(2 \times \frac{1}{3}\right) \times \left(3 \times \frac{1}{2}\right)$.	Re-arranging the order of the numbers being multiplied, $\left(2 \times \frac{1}{3}\right) \times \left(3 \times \frac{1}{2}\right) = 2 \times \frac{1}{2} \times 3 \times \frac{1}{3} = \left(2 \times \frac{1}{2}\right) \times \left(3 \times \frac{1}{3}\right) = 1 \times 1 = 1$.	1	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Fractions", "Mathematics -> Geometry -> Plane Geometry -> Area" ]	1.5	A rectangle has a length of $\frac{3}{5}$ and an area of $\frac{1}{3}$. What is the width of the rectangle?	In a rectangle, length times width equals area, so width equals area divided by length. Therefore, the width is $\frac{1}{3} \div \frac{3}{5}=\frac{1}{3} \times \frac{5}{3}=\frac{5}{9}$.	\\frac{5}{9}	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Fractions" ]	1.5	What is the value of the expression $\frac{20+16 \times 20}{20 \times 16}$?	Evaluating, $\frac{20+16 \times 20}{20 \times 16}=\frac{20+320}{320}=\frac{340}{320}=\frac{17}{16}$. Alternatively, we could notice that each of the numerator and denominator is a multiple of 20, and so $\frac{20+16 \times 20}{20 \times 16}=\frac{20(1+16)}{20 \times 16}=\frac{1+16}{16}=\frac{17}{16}$.	\frac{17}{16}	fermat
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1	If $x=3$, $y=2x$, and $z=3y$, what is the value of $z$?	Since $x=3$ and $y=2x$, then $y=2 \cdot 3=6$. Since $y=6$ and $z=3y$, then $z=3 \cdot 6=18$.	18	fermat
[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]	1	If \( (2^{a})(2^{b})=64 \), what is the mean (average) of \( a \) and \( b \)?	Since \( (2^{a})(2^{b})=64 \), then \( 2^{a+b}=64 \), using an exponent law. Since \( 64=2^{6} \), then \( 2^{a+b}=2^{6} \) and so \( a+b=6 \). Therefore, the average of \( a \) and \( b \) is \( \frac{1}{2}(a+b)=3 \).	3	fermat
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1	If $x+\sqrt{25}=\sqrt{36}$, what is the value of $x$?	Since $x+\sqrt{25}=\sqrt{36}$, then $x+5=6$ or $x=1$.	1	cayley
[ "Mathematics -> Algebra -> Algebra -> Algebraic Expressions" ]	1.5	The line with equation $y = 3x + 5$ is translated 2 units to the right. What is the equation of the resulting line?	The line with equation $y = 3x + 5$ has slope 3 and $y$-intercept 5. Since the line has $y$-intercept 5, it passes through $(0, 5)$. When the line is translated 2 units to the right, its slope does not change and the new line passes through $(2, 5)$. A line with slope $m$ that passes through the point $(x_1, y_1)$ has equation $y - y_1 = m(x - x_1)$. Therefore, the line with slope 3 that passes through $(2, 5)$ has equation $y - 5 = 3(x - 2)$ or $y - 5 = 3x - 6$, which gives $y = 3x - 1$. Alternatively, we could note that when the graph of $y = 3x + 5$ is translated 2 units to the right, the equation of the new graph is $y = 3(x - 2) + 5$ or $y = 3x - 1$.	y = 3x - 1	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]	1.5	Calculate the value of $\frac{2 \times 3 + 4}{2 + 3}$.	Evaluating, $\frac{2 \times 3+4}{2+3}=\frac{6+4}{5}=\frac{10}{5}=2$.	2	fermat
[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]	1.5	The mean (average) of 5 consecutive integers is 9. What is the smallest of these 5 integers?	Since the mean of five consecutive integers is 9, then the middle of these five integers is 9. Therefore, the integers are $7,8,9,10,11$, and so the smallest of the five integers is 7.	7	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]	1	If $x = -3$, what is the value of $(x-3)^{2}$?	Evaluating, $(x-3)^{2}=(-3-3)^{2}=(-6)^{2}=36$.	36	fermat
[ "Mathematics -> Geometry -> Plane Geometry -> Polygons" ]	1.5	A rectangle has width $x$ and length $y$. The rectangle is cut along the horizontal and vertical dotted lines to produce four smaller rectangles. The sum of the perimeters of these four rectangles is 24. What is the value of $x+y$?	The sum of the lengths of the horizontal line segments is $4x$, because the tops of the four small rectangles contribute a total of $2x$ to their combined perimeter and the bottoms contribute a total of $2x$. Similarly, the sum of the lengths of the vertical line segments is $4y$. In other words, the sum of the perimeters of the four rectangles is $4x + 4y$. Since the sum of the perimeters also equals 24, then $4x + 4y = 24$ and so $x + y = 6$.	6	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]	1	If \( 3-5+7=6-x \), what is the value of \( x \)?	Simplifying the left side of the equation, we obtain \( 5=6-x \). Therefore, \( x=6-5=1 \).	1	fermat
[ "Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Other" ]	1.5	Two standard six-sided dice are rolled. What is the probability that the product of the two numbers rolled is 12?	When two standard six-sided dice are rolled, there are $6 \times 6 = 36$ possibilities for the pair of numbers that are rolled. Of these, the pairs $2 \times 6, 3 \times 4, 4 \times 3$, and $6 \times 2$ each give 12. (If one of the numbers rolled is 1 or 5, the product cannot be 12.) Since there are 4 pairs of possible rolls whose product is 12, the probability that the product is 12 is $\frac{4}{36}$.	\frac{4}{36}	cayley
[ "Mathematics -> Applied Mathematics -> Math Word Problems", "Mathematics -> Algebra -> Prealgebra -> Ratios -> Other" ]	1.5	There are 400 students at Pascal H.S., where the ratio of boys to girls is $3: 2$. There are 600 students at Fermat C.I., where the ratio of boys to girls is $2: 3$. What is the ratio of boys to girls when considering all students from both schools?	Since the ratio of boys to girls at Pascal H.S. is $3: 2$, then $rac{3}{3+2}=rac{3}{5}$ of the students at Pascal H.S. are boys. Thus, there are $rac{3}{5}(400)=rac{1200}{5}=240$ boys at Pascal H.S. Since the ratio of boys to girls at Fermat C.I. is $2: 3$, then $rac{2}{2+3}=rac{2}{5}$ of the students at Fermat C.I. are boys. Thus, there are $rac{2}{5}(600)=rac{1200}{5}=240$ boys at Fermat C.I. There are $400+600=1000$ students in total at the two schools. Of these, $240+240=480$ are boys, and so the remaining $1000-480=520$ students are girls. Therefore, the overall ratio of boys to girls is $480: 520=48: 52=12: 13$.	12:13	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1.5	John ate a total of 120 peanuts over four consecutive nights. Each night he ate 6 more peanuts than the night before. How many peanuts did he eat on the fourth night?	Suppose that John ate \( x \) peanuts on the fourth night. Since he ate 6 more peanuts each night than on the previous night, then he ate \( x-6 \) peanuts on the third night, \((x-6)-6=x-12\) peanuts on the second night, and \((x-12)-6=x-18\) peanuts on the first night. Since John ate 120 peanuts in total, then \( x+(x-6)+(x-12)+(x-18)=120 \), and so \( 4x-36=120 \) or \( 4x=156 \) or \( x=39 \). Therefore, John ate 39 peanuts on the fourth night.	39	fermat
[ "Mathematics -> Algebra -> Prealgebra -> Fractions" ]	1.5	If $\frac{1}{6} + \frac{1}{3} = \frac{1}{x}$, what is the value of $x$?	Simplifying, $\frac{1}{6} + \frac{1}{3} = \frac{1}{6} + \frac{2}{6} = \frac{3}{6} = \frac{1}{2}$. Thus, $\frac{1}{x} = \frac{1}{2}$ and so $x = 2$.	2	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1.5	If $3n=9+9+9$, what is the value of $n$?	Since $3n=9+9+9=3 imes 9$, then $n=9$. Alternatively, we could note that $9+9+9=27$ and so $3n=27$ which gives $n=rac{27}{3}=9$.	9	cayley
[ "Mathematics -> Applied Mathematics -> Math Word Problems" ]	1.5	Arturo has an equal number of $\$5$ bills, of $\$10$ bills, and of $\$20$ bills. The total value of these bills is $\$700$. How many $\$5$ bills does Arturo have?	Since Arturo has an equal number of $\$5$ bills, of $\$10$ bills, and of $\$20$ bills, then we can divide Arturo's bills into groups, each of which contains one $\$5$ bill, one $\$10$ bill, and one $\$20$ bill. The value of the bills in each group is $\$5 + \$10 + \$20 = \$35$. Since the total value of Arturo's bills is $\$700$, then there are $\frac{\$700}{\$35} = 20$ groups. Thus, Arturo has $20 \$5$ bills.	20	cayley
[ "Mathematics -> Geometry -> Solid Geometry -> Volume" ]	1.5	The volume of a prism is equal to the area of its base times its depth. If the prism has identical bases with area $400 \mathrm{~cm}^{2}$ and depth 8 cm, what is its volume?	The volume of a prism is equal to the area of its base times its depth. Here, the prism has identical bases with area $400 \mathrm{~cm}^{2}$ and depth 8 cm, and so its volume is $400 \mathrm{~cm}^{2} \times 8 \mathrm{~cm} = 3200 \mathrm{~cm}^{3}$.	3200 \mathrm{~cm}^{3}	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]	1	What is the value of $(-2)^{3}-(-3)^{2}$?	Evaluating, $(-2)^{3}-(-3)^{2}=-8-9=-17$.	-17	fermat
[ "Mathematics -> Algebra -> Prealgebra -> Fractions" ]	1	Evaluate the expression $8-rac{6}{4-2}$.	Evaluating, $8-rac{6}{4-2}=8-rac{6}{2}=8-3=5$.	5	fermat
[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]	1	Calculate the value of the expression $2 \times 0 + 2 \times 4$.	Calculating, $2 \times 0 + 2 \times 4 = 0 + 8 = 8$.	8	cayley
[ "Mathematics -> Applied Mathematics -> Math Word Problems" ]	1.5	A 6 m by 8 m rectangular field has a fence around it. There is a post at each of the four corners of the field. Starting at each corner, there is a post every 2 m along each side of the fence. How many posts are there?	A rectangle that is 6 m by 8 m has perimeter $2 \times(6 \mathrm{~m}+8 \mathrm{~m})=28 \mathrm{~m}$. If posts are put in every 2 m around the perimeter starting at a corner, then we would guess that it will take $\frac{28 \mathrm{~m}}{2 \mathrm{~m}}=14$ posts.	14	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1.5	In a cafeteria line, the number of people ahead of Kaukab is equal to two times the number of people behind her. There are $n$ people in the line. What is a possible value of $n$?	Suppose that there are $p$ people behind Kaukab. This means that there are $2p$ people ahead of her. Including Kaukab, the total number of people in line is $n = p + 2p + 1 = 3p + 1$, which is one more than a multiple of 3. Of the given choices $(23, 20, 24, 21, 25)$, the only one that is one more than a multiple of 3 is 25, which equals $3 \times 8 + 1$. Therefore, a possible value for $n$ is 25.	25	cayley
[ "Mathematics -> Geometry -> Plane Geometry -> Area" ]	1.5	A rectangular field has a length of 20 metres and a width of 5 metres. If its length is increased by 10 m, by how many square metres will its area be increased?	Since the field originally has length 20 m and width 5 m, then its area is $20 \times 5=100 \mathrm{~m}^{2}$. The new length of the field is $20+10=30 \mathrm{~m}$, so the new area is $30 \times 5=150 \mathrm{~m}^{2}$. The increase in area is $150-100=50 \mathrm{~m}^{2}$. (Alternatively, we could note that since the length increases by 10 m and the width stays constant at 5 m, then the increase in area is $10 \times 5=50 \mathrm{~m}^{2}$.)	50	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1.5	The average of 1, 3, and \( x \) is 3. What is the value of \( x \)?	Since the average of three numbers equals 3, then their sum is \( 3 \times 3 = 9 \). Therefore, \( 1+3+x=9 \) and so \( x=9-4=5 \).	5	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1	What is the value of $(3x + 2y) - (3x - 2y)$ when $x = -2$ and $y = -1$?	The expression $(3x + 2y) - (3x - 2y)$ is equal to $3x + 2y - 3x + 2y$ which equals $4y$. When $x = -2$ and $y = -1$, this equals $4(-1)$ or $-4$.	-4	fermat
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1	If $x=3$, what is the value of $-(5x - 6x)$?	When $x=3$, we have $-(5x - 6x) = -(-x) = x = 3$. Alternatively, when $x=3$, we have $-(5x - 6x) = -(15 - 18) = -(-3) = 3$.	3	cayley
[ "Mathematics -> Applied Mathematics -> Math Word Problems", "Mathematics -> Discrete Mathematics -> Algorithms" ]	1.5	A robotic grasshopper jumps 1 cm to the east, then 2 cm to the north, then 3 cm to the west, then 4 cm to the south. After every fourth jump, the grasshopper restarts the sequence of jumps: 1 cm to the east, then 2 cm to the north, then 3 cm to the west, then 4 cm to the south. After a total of $n$ jumps, the position of the grasshopper is 162 cm to the west and 158 cm to the south of its original position. What is the sum of the squares of the digits of $n$?	Each group of four jumps takes the grasshopper 1 cm to the east and 3 cm to the west, which is a net movement of 2 cm to the west, and 2 cm to the north and 4 cm to the south, which is a net movement of 2 cm to the south. In other words, we can consider each group of four jumps, starting with the first, as resulting in a net movement of 2 cm to the west and 2 cm to the south. We note that $158=2 \times 79$. Thus, after 79 groups of four jumps, the grasshopper is $79 \times 2=158 \mathrm{~cm}$ to the west and 158 cm to the south of its original position. (We need at least 79 groups of these because the grasshopper cannot be 158 cm to the south of its original position before the end of 79 such groups.) The grasshopper has made $4 \times 79=316$ jumps so far. After the 317th jump (1 cm to the east), the grasshopper is 157 cm west and 158 cm south of its original position. After the 318th jump (2 cm to the north), the grasshopper is 157 cm west and 156 cm south of its original position. After the 319th jump (3 cm to the west), the grasshopper is 160 cm west and 156 cm south of its original position. After the 320th jump (4 cm to the south), the grasshopper is 160 cm west and 160 cm south of its original position. After the 321st jump (1 cm to the east), the grasshopper is 159 cm west and 160 cm south of its original position. After the 322nd jump (2 cm to the north), the grasshopper is 159 cm west and 158 cm south of its original position. After the 323rd jump (3 cm to the west), the grasshopper is 162 cm west and 158 cm south of its original position, which is the desired position. As the grasshopper continues jumping, each of its positions will always be at least 160 cm south of its original position, so this is the only time that it is at this position. Therefore, $n=323$. The sum of the squares of the digits of $n$ is $3^{2}+2^{2}+3^{2}=9+4+9=22$.	22	cayley
[ "Mathematics -> Geometry -> Plane Geometry -> Angles" ]	1.5	What is the measure of the largest angle in $\triangle P Q R$?	Since the sum of the angles in a triangle is $180^{\circ}$, then $3 x^{\circ}+x^{\circ}+6 x^{\circ}=180^{\circ}$ or $10 x=180$ or $x=18$. The largest angle in the triangle is $6 x^{\circ}=6(18^{\circ})=108^{\circ}$.	108^{\\circ}	cayley
[ "Mathematics -> Number Theory -> Congruences" ]	1.5	How many of the four integers $222, 2222, 22222$, and $222222$ are multiples of 3?	We could use a calculator to divide each of the four given numbers by 3 to see which calculations give an integer answer. Alternatively, we could use the fact that a positive integer is divisible by 3 if and only if the sum of its digits is divisible by 3. The sums of the digits of $222, 2222, 22222$, and $222222$ are $6, 8, 10$, and 12, respectively. Two of these sums are divisible by 3 (namely, 6 and 12) so two of the four integers (namely, 222 and 222222) are divisible by 3.	2	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]	1.5	When three consecutive integers are added, the total is 27. What is the result when the same three integers are multiplied?	If the sum of three consecutive integers is 27, then the numbers must be 8, 9, and 10. Their product is $8 imes 9 imes 10=720$.	720	fermat
[ "Mathematics -> Algebra -> Prealgebra -> Fractions" ]	1	If $\frac{1}{9}+\frac{1}{18}=\frac{1}{\square}$, what is the number that replaces the $\square$ to make the equation true?	We simplify the left side and express it as a fraction with numerator 1: $\frac{1}{9}+\frac{1}{18}=\frac{2}{18}+\frac{1}{18}=\frac{3}{18}=\frac{1}{6}$. Therefore, the number that replaces the $\square$ is 6.	6	pascal
[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]	1.5	What is the expression $2^{3}+2^{2}+2^{1}$ equal to?	Since $2^{1}=2$ and $2^{2}=2 imes 2=4$ and $2^{3}=2 imes 2 imes 2=8$, then $2^{3}+2^{2}+2^{1}=8+4+2=14$.	14	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1	If $3 imes n=6 imes 2$, what is the value of $n$?	Since $3 imes n=6 imes 2$, then $3n=12$ or $n=\frac{12}{3}=4$.	4	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]	1	What is the value of $rac{(20-16) imes (12+8)}{4}$?	Using the correct order of operations, $rac{(20-16) imes (12+8)}{4} = rac{4 imes 20}{4} = rac{80}{4} = 20$.	20	pascal
[ "Mathematics -> Geometry -> Solid Geometry -> 3D Shapes" ]	1.5	How many solid $1 imes 1 imes 1$ cubes are required to make a solid $2 imes 2 imes 2$ cube?	The volume of a $1 imes 1 imes 1$ cube is 1 . The volume of a $2 imes 2 imes 2$ cube is 8 . Thus, 8 of the smaller cubes are needed to make the larger cube.	8	fermat
[ "Mathematics -> Applied Mathematics -> Math Word Problems" ]	1.5	Charlie is making a necklace with yellow beads and green beads. She has already used 4 green beads and 0 yellow beads. How many yellow beads will she have to add so that $rac{4}{5}$ of the total number of beads are yellow?	If $rac{4}{5}$ of the beads are yellow, then $rac{1}{5}$ are green. Since there are 4 green beads, the total number of beads must be $4 imes 5=20$. Thus, Charlie needs to add $20-4=16$ yellow beads.	16	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1	If \( x=2 \), what is the value of \( (x+2-x)(2-x-2) \)?	When \( x=2 \), we have \( (x+2-x)(2-x-2)=(2+2-2)(2-2-2)=(2)(-2)=-4 \). Alternatively, we could simplify \( (x+2-x)(2-x-2) \) to obtain \( (2)(-x) \) or \( -2x \) and then substitute \( x=2 \) to obtain a result of \( -2(2) \) or -4.	-4	fermat
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1.5	Ten numbers have an average (mean) of 87. Two of those numbers are 51 and 99. What is the average of the other eight numbers?	Since 10 numbers have an average of 87, their sum is $10 \times 87 = 870$. When the numbers 51 and 99 are removed, the sum of the remaining 8 numbers is $870 - 51 - 99$ or 720. The average of these 8 numbers is $\frac{720}{8} = 90$.	90	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1.5	If $x \%$ of 60 is 12, what is $15 \%$ of $x$?	Since $x \%$ of 60 is 12, then $\frac{x}{100} \cdot 60=12$ or $x=\frac{12 \cdot 100}{60}=20$. Therefore, $15 \%$ of $x$ is $15 \%$ of 20, or $0.15 \cdot 20=3$.	3	fermat
[ "Mathematics -> Applied Mathematics -> Math Word Problems" ]	1.5	Ava's machine takes four-digit positive integers as input. When the four-digit integer $ABCD$ is input, the machine outputs the integer $A imes B + C imes D$. What is the output when the input is 2023?	Using the given rule, the output of the machine is $2 imes 0 + 2 imes 3 = 0 + 6 = 6$.	6	fermat
[ "Mathematics -> Algebra -> Prealgebra -> Fractions" ]	1	What is the value of \( \frac{5-2}{2+1} \)?	Simplifying, \( \frac{5-2}{2+1}=\frac{3}{3}=1 \).	1	cayley
[ "Mathematics -> Algebra -> Algebra -> Algebraic Expressions" ]	1	Evaluate the expression $2x^{2}+3x^{2}$ when $x=2$.	When $x=2$, we obtain $2x^{2}+3x^{2}=5x^{2}=5 \cdot 2^{2}=5 \cdot 4=20$.	20	fermat
[ "Mathematics -> Algebra -> Prealgebra -> Fractions" ]	1	If $\frac{1}{9}+\frac{1}{18}=\frac{1}{\square}$, what is the number that replaces the $\square$ to make the equation true?	We simplify the left side and express it as a fraction with numerator 1: $\frac{1}{9}+\frac{1}{18}=\frac{2}{18}+\frac{1}{18}=\frac{3}{18}=\frac{1}{6}$. Therefore, the number that replaces the $\square$ is 6.	6	pascal
[ "Mathematics -> Applied Mathematics -> Math Word Problems" ]	1.5	A bicycle trip is 30 km long. Ari rides at an average speed of 20 km/h. Bri rides at an average speed of 15 km/h. If Ari and Bri begin at the same time, how many minutes after Ari finishes the trip will Bri finish?	Riding at 15 km/h, Bri finishes the 30 km in $\frac{30 \text{ km}}{15 \text{ km/h}} = 2 \text{ h}$. Riding at 20 km/h, Ari finishes the 30 km in $\frac{30 \text{ km}}{20 \text{ km/h}} = 1.5 \text{ h}$. Therefore, Bri finishes 0.5 h after Ari, which is 30 minutes.	30	fermat
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1.5	If $\frac{x-y}{x+y}=5$, what is the value of $\frac{2x+3y}{3x-2y}$?	Since $\frac{x-y}{x+y}=5$, then $x-y=5(x+y)$. This means that $x-y=5x+5y$ and so $0=4x+6y$ or $2x+3y=0$. Therefore, $\frac{2x+3y}{3x-2y}=\frac{0}{3x-2y}=0$.	0	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1	Calculate the value of the expression $(8 \times 6)-(4 \div 2)$.	We evaluate the expression by first evaluating the expressions in brackets: $(8 \times 6)-(4 \div 2)=48-2=46$.	46	pascal
[ "Mathematics -> Geometry -> Plane Geometry -> Polygons" ]	1	What is the perimeter of the figure shown if $x=3$?	Since $x=3$, the side lengths of the figure are $4,3,6$, and 10. Thus, the perimeter of the figure is $4+3+6+10=23$. (Alternatively, the perimeter is $x+6+10+(x+1)=2x+17$. When $x=3$, this equals $2(3)+17$ or 23.)	23	pascal
[ "Mathematics -> Applied Mathematics -> Math Word Problems" ]	1	Carrie sends five text messages to her brother each Saturday and Sunday, and two messages on other days. Over four weeks, how many text messages does Carrie send?	Each week, Carrie sends 5 messages to her brother on each of 2 days, for a total of 10 messages. Each week, Carrie sends 2 messages to her brother on each of the remaining 5 days, for a total of 10 messages. Therefore, Carrie sends $10+10=20$ messages per week. In four weeks, Carrie sends $4 \cdot 20=80$ messages.	80	fermat
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1	If $x=2y$ and $y \neq 0$, what is the value of $(x+2y)-(2x+y)$?	We simplify first, then substitute $x=2y$: $(x+2y)-(2x+y)=x+2y-2x-y=y-x=y-2y=-y$. Alternatively, we could substitute first, then simplify: $(x+2y)-(2x+y)=(2y+2y)-(2(2y)+y)=4y-5y=-y$.	-y	pascal
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1.5	What is the value of $2^{4}-2^{3}$?	We note that $2^{2}=2 \times 2=4,2^{3}=2^{2} \times 2=4 \times 2=8$, and $2^{4}=2^{2} \times 2^{2}=4 \times 4=16$. Therefore, $2^{4}-2^{3}=16-8=8=2^{3}$.	2^{3}	pascal
[ "Mathematics -> Applied Mathematics -> Math Word Problems" ]	1.5	Calculate the number of minutes in a week.	There are 60 minutes in an hour and 24 hours in a day. Thus, there are $60 \cdot 24=1440$ minutes in a day. Since there are 7 days in a week, the number of minutes in a week is $7 \cdot 1440=10080$. Of the given choices, this is closest to 10000.	10000	fermat
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1	When $x=-2$, what is the value of $(x+1)^{3}$?	When $x=-2$, we have $(x+1)^{3}=(-2+1)^{3}=(-1)^{3}=-1$.	-1	pascal
[ "Mathematics -> Applied Mathematics -> Math Word Problems" ]	1	How many points does a sports team earn for 9 wins, 3 losses, and 4 ties, if they earn 2 points for each win, 0 points for each loss, and 1 point for each tie?	The team earns 2 points for each win, so 9 wins earn $2 \times 9=18$ points. The team earns 0 points for each loss, so 3 losses earn 0 points. The team earns 1 point for each tie, so 4 ties earn 4 points. In total, the team earns $18+0+4=22$ points.	22	pascal
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations", "Mathematics -> Number Theory -> Integer Factorization -> Other" ]	1.5	How many pairs of positive integers $(x, y)$ have the property that the ratio $x: 4$ equals the ratio $9: y$?	The equality of the ratios $x: 4$ and $9: y$ is equivalent to the equation $\frac{x}{4}=\frac{9}{y}$. This equation is equivalent to the equation $xy=4(9)=36$. The positive divisors of 36 are $1,2,3,4,6,9,12,18,36$, so the desired pairs are $(x, y)=(1,36),(2,18),(3,12),(4,9),(6,6),(9,4),(12,3),(18,2),(36,1)$. There are 9 such pairs.	9	pascal
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1.5	Jitka hiked a trail. After hiking 60% of the length of the trail, she had 8 km left to go. What is the length of the trail?	After Jitka hiked 60% of the trail, 40% of the trail was left, which corresponds to 8 km. This means that 10% of the trail corresponds to 2 km. Therefore, the total length of the trail is \( 10 \times 2 = 20 \text{ km} \).	20 \text{ km}	pascal
[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]	1.5	In the addition problem shown, $m, n, p$, and $q$ represent positive digits. What is the value of $m+n+p+q$?	From the ones column, we see that $3 + 2 + q$ must have a ones digit of 2. Since $q$ is between 1 and 9, inclusive, then $3 + 2 + q$ is between 6 and 14. Since its ones digit is 2, then $3 + 2 + q = 12$ and so $q = 7$. This also means that there is a carry of 1 into the tens column. From the tens column, we see that $1 + 6 + p + 8$ must have a ones digit of 4. Since $p$ is between 1 and 9, inclusive, then $1 + 6 + p + 8$ is between 16 and 24. Since its ones digit is 4, then $1 + 6 + p + 8 = 24$ and so $p = 9$. This also means that there is a carry of 2 into the hundreds column. From the hundreds column, we see that $2 + n + 7 + 5$ must have a ones digit of 0. Since $n$ is between 1 and 9, inclusive, then $2 + n + 7 + 5$ is between 15 and 23. Since its ones digit is 0, then $2 + n + 7 + 5 = 20$ and so $n = 6$. This also means that there is a carry of 2 into the thousands column. This means that $m = 2$. Thus, we have $m + n + p + q = 2 + 6 + 9 + 7 = 24$.	24	cayley
[ "Mathematics -> Algebra -> Prealgebra -> Other" ]	1.5	In a magic square, what is the sum \( a+b+c \)?	Using the properties of a magic square, \( a+b+c = 14+18+15 = 47 \).	47	pascal
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1	If $10x+y=75$ and $10y+x=57$ for some positive integers $x$ and $y$, what is the value of $x+y$?	Since $10x+y=75$ and $10y+x=57$, then $(10x+y)+(10y+x)=75+57$ and so $11x+11y=132$. Dividing by 11, we get $x+y=12$. (We could have noticed initially that $(x, y)=(7,5)$ is a pair that satisfies the two equations, thence concluding that $x+y=12$.)	12	fermat
[ "Mathematics -> Algebra -> Prealgebra -> Fractions" ]	1.5	What is the number halfway between $\frac{1}{12}$ and $\frac{1}{10}$?	The number halfway between two numbers is their average. Therefore, the number halfway between $\frac{1}{10}$ and $\frac{1}{12}$ is $\frac{1}{2}\left(\frac{1}{10}+\frac{1}{12}\right)=\frac{1}{2}\left(\frac{12}{120}+\frac{10}{120}\right)=\frac{1}{2}\left(\frac{22}{120}\right)=\frac{11}{120}$.	\frac{11}{120}	fermat
[ "Mathematics -> Algebra -> Prealgebra -> Fractions" ]	1.5	If \( 50\% \) of \( N \) is 16, what is \( 75\% \) of \( N \)?	The percentage \( 50\% \) is equivalent to the fraction \( \frac{1}{2} \), while \( 75\% \) is equivalent to \( \frac{3}{4} \). Since \( 50\% \) of \( N \) is 16, then \( \frac{1}{2}N=16 \) or \( N=32 \). Therefore, \( 75\% \) of \( N \) is \( \frac{3}{4}N \) or \( \frac{3}{4}(32) \), which equals 24.	24	fermat
[ "Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations" ]	1.5	In how many different ways can André form exactly \( \$10 \) using \( \$1 \) coins, \( \$2 \) coins, and \( \$5 \) bills?	Using combinations of \( \$5 \) bills, \( \$2 \) coins, and \( \$1 \) coins, there are 10 ways to form \( \$10 \).	10	pascal
[ "Mathematics -> Algebra -> Prealgebra -> Fractions" ]	1.5	What is the value of $rac{8+4}{8-4}$?	Simplifying, $rac{8+4}{8-4}=rac{12}{4}=3$.	3	cayley
[ "Mathematics -> Precalculus -> Functions" ]	1.5	Numbers $m$ and $n$ are on the number line. What is the value of $n-m$?	On a number line, the markings are evenly spaced. Since there are 6 spaces between 0 and 30, each space represents a change of $\frac{30}{6}=5$. Since $n$ is 2 spaces to the right of 60, then $n=60+2 \times 5=70$. Since $m$ is 3 spaces to the left of 30, then $m=30-3 \times 5=15$. Therefore, $n-m=70-15=55$.	55	pascal
[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]	1.5	What is the value of \( \sqrt{16 \times \sqrt{16}} \)?	Evaluating, \( \sqrt{16 \times \sqrt{16}} = \sqrt{16 \times 4} = \sqrt{64} = 8 \). Since \( 8 = 2^3 \), then \( \sqrt{16 \times \sqrt{16}} = 2^3 \).	2^3	pascal
[ "Mathematics -> Algebra -> Prealgebra -> Fractions" ]	1	The operation \( \otimes \) is defined by \( a \otimes b = \frac{a}{b} + \frac{b}{a} \). What is the value of \( 4 \otimes 8 \)?	From the given definition, \( 4 \otimes 8 = \frac{4}{8} + \frac{8}{4} = \frac{1}{2} + 2 = \frac{5}{2} \).	\frac{5}{2}	pascal
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1	If \( 8 + 6 = n + 8 \), what is the value of \( n \)?	Since \( 8+6=n+8 \), then subtracting 8 from both sides, we obtain \( 6=n \) and so \( n \) equals 6.	6	pascal
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1	If $x=3$, $y=2x$, and $z=3y$, what is the average of $x$, $y$, and $z$?	Since $x=3$ and $y=2x$, then $y=2 \times 3=6$. Since $y=6$ and $z=3y$, then $z=3 \times 6=18$. Therefore, the average of $x, y$ and $z$ is $\frac{x+y+z}{3}=\frac{3+6+18}{3}=9$.	9	pascal
[ "Mathematics -> Algebra -> Prealgebra -> Decimals" ]	1	What is 30% of 200?	$30\%$ of 200 equals $\frac{30}{100} \times 200=60$. Alternatively, we could note that $30\%$ of 100 is 30 and $200=2 \times 100$, so $30\%$ of 200 is $30 \times 2$ which equals 60.	60	pascal
[ "Mathematics -> Applied Mathematics -> Math Word Problems" ]	1	What is the value of \( z \) in the carpet installation cost chart?	Using the cost per square metre, \( z = 1261.40 \).	1261.40	pascal
[ "Mathematics -> Applied Mathematics -> Math Word Problems" ]	1.5	It takes Pearl 7 days to dig 4 holes. It takes Miguel 3 days to dig 2 holes. If they work together and each continues digging at these same rates, how many holes in total will they dig in 21 days?	Since Pearl digs 4 holes in 7 days and $\frac{21}{7}=3$, then in 21 days, Pearl digs $3 \cdot 4=12$ holes. Since Miguel digs 2 holes in 3 days and $\frac{21}{3}=7$, then in 21 days, Miguel digs $7 \cdot 2=14$ holes. In total, they dig $12+14=26$ holes in 21 days.	26	fermat
[ "Mathematics -> Algebra -> Prealgebra -> Integers", "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1	If 7:30 a.m. was 16 minutes ago, how many minutes will it be until 8:00 a.m.?	If 7:30 a.m. was 16 minutes ago, then it is currently $30+16=46$ minutes after 7:00 a.m., or 7:46 a.m. Since 8:00 a.m. is 60 minutes after 7:00 a.m., then it will be 8:00 a.m. in $60-46=14$ minutes.	14	pascal
[ "Mathematics -> Algebra -> Prealgebra -> Integers", "Mathematics -> Applied Mathematics -> Math Word Problems" ]	1.5	At the Lacsap Hospital, Emily is a doctor and Robert is a nurse. Not including Emily, there are five doctors and three nurses at the hospital. Not including Robert, there are $d$ doctors and $n$ nurses at the hospital. What is the product of $d$ and $n$?	Since Emily is a doctor and there are 5 doctors and 3 nurses aside from Emily at the hospital, then there are 6 doctors and 3 nurses in total. Since Robert is a nurse, then aside from Robert, there are 6 doctors and 2 nurses. Therefore, $d=6$ and $n=2$, so $dn=12$.	12	pascal
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1	If $y=1$ and $4x-2y+3=3x+3y$, what is the value of $x$?	Substituting $y=1$ into the second equation, we obtain $4x-2(1)+3=3x+3(1)$. Simplifying, we obtain $4x-2+3=3x+3$ or $4x+1=3x+3$. Therefore, $4x-3x=3-1$ or $x=2$.	2	pascal
[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]	1	Determine which of the following expressions has the largest value: $4^2$, $4 \times 2$, $4 - 2$, $\frac{4}{2}$, or $4 + 2$.	We evaluate each of the five choices: $4^{2}=16$, $4 \times 2=8$, $4-2=2$, $\frac{4}{2}=2$, $4+2=6$. Of these, the largest is $4^{2}=16$.	16	pascal
[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]	1	Calculate the expression $8 \times 10^{5}+4 \times 10^{3}+9 \times 10+5$.	First, we write out the powers of 10 in full to obtain $8 \times 100000+4 \times 1000+9 \times 10+5$. Simplifying, we obtain $800000+4000+90+5$ or 804095.	804095	pascal
[ "Mathematics -> Geometry -> Plane Geometry -> Triangulations" ]	1.5	In $\triangle PQR, \angle RPQ=90^{\circ}$ and $S$ is on $PQ$. If $SQ=14, SP=18$, and $SR=30$, what is the area of $\triangle QRS$?	Since $\triangle RPS$ is right-angled at $P$, then by the Pythagorean Theorem, $PR^{2}+PS^{2}=RS^{2}$ or $PR^{2}+18^{2}=30^{2}$. This gives $PR^{2}=900-324=576$, from which $PR=24$. Since $P, S$ and $Q$ lie on a straight line and $RP$ is perpendicular to this line, then $RP$ is actually a height for $\triangle QRS$ corresponding to base $SQ$. Thus, the area of $\triangle QRS$ is $\frac{1}{2}(24)(14)=168$.	168	pascal
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1	The average (mean) of two numbers is 7. One of the numbers is 5. What is the other number?	Since the average of two numbers is 7, their sum is $2 imes 7=14$. Since one of the numbers is 5, the other is $14-5=9$.	9	fermat
[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]	1.5	Jim wrote a sequence of symbols a total of 50 times. How many more of one symbol than another did he write?	The sequence of symbols includes 5 of one symbol and 2 of another. This means that, each time the sequence is written, there are 3 more of one symbol written than the other. When the sequence is written 50 times, in total there are \( 50 \times 3 = 150 \) more of one symbol written than the other.	150	pascal
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1.5	Chris received a mark of $50 \%$ on a recent test. Chris answered 13 of the first 20 questions correctly. Chris also answered $25 \%$ of the remaining questions on the test correctly. If each question on the test was worth one mark, how many questions in total were on the test?	Suppose that there were $n$ questions on the test. Since Chris received a mark of $50 \%$ on the test, then he answered $\frac{1}{2} n$ of the questions correctly. We know that Chris answered 13 of the first 20 questions correctly and then $25 \%$ of the remaining questions. Since the test has $n$ questions, then after the first 20 questions, there are $n-20$ questions. Since Chris answered $25 \%$ of these $n-20$ questions correctly, then Chris answered $\frac{1}{4}(n-20)$ of these questions correctly. The total number of questions that Chris answered correctly can be expressed as $\frac{1}{2} n$ and also as $13+\frac{1}{4}(n-20)$. Therefore, $\frac{1}{2} n=13+\frac{1}{4}(n-20)$ and so $2 n=52+(n-20)$, which gives $n=32$. (We can check that if $n=32$, then Chris answers 13 of the first 20 and 3 of the remaining 12 questions correctly, for a total of 16 correct out of 32.)	32	pascal
[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]	1.5	If $2 \times 2 \times 3 \times 3 \times 5 \times 6=5 \times 6 \times n \times n$, what is a possible value of $n$?	We rewrite the left side of the given equation as $5 \times 6 \times(2 \times 3) \times(2 \times 3)$. Since $5 \times 6 \times(2 \times 3) \times(2 \times 3)=5 \times 6 \times n \times n$, then a possible value of $n$ is $2 \times 3$ or 6.	6	pascal
[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]	1	If $x=2$, what is the value of $4x^2 - 3x^2$?	Simplifying, $4 x^{2}-3 x^{2}=x^{2}$. When $x=2$, this expression equals 4 . Alternatively, when $x=2$, we have $4 x^{2}-3 x^{2}=4 \cdot 2^{2}-3 \cdot 2^{2}=16-12=4$.	4	fermat
[ "Mathematics -> Algebra -> Prealgebra -> Integers", "Mathematics -> Discrete Mathematics -> Combinatorics" ]	1.5	The smallest of nine consecutive integers is 2012. These nine integers are placed in the circles to the right. The sum of the three integers along each of the four lines is the same. If this sum is as small as possible, what is the value of $u$?	If we have a configuration of the numbers that has the required property, then we can add or subtract the same number from each of the numbers in the circles and maintain the property. (This is because there are the same number of circles in each line.) Therefore, we can subtract 2012 from all of the numbers and try to complete the diagram using the integers from 0 to 8. We label the circles as shown in the diagram, and call $S$ the sum of the three integers along any one of the lines. Since $p, q, r, t, u, w, x, y, z$ are 0 through 8 in some order, then $p+q+r+t+u+w+x+y+z=0+1+2+3+4+5+6+7+8=36$. From the desired property, we want $S=p+q+r=r+t+u=u+w+x=x+y+z$. Therefore, $(p+q+r)+(r+t+u)+(u+w+x)+(x+y+z)=4S$. From this, $(p+q+r+t+u+w+x+y+z)+r+u+x=4S$ or $r+u+x=4S-36=4(S-9)$. We note that the right side is an integer that is divisible by 4. Also, we want $S$ to be as small as possible so we want the sum $r+u+x$ to be as small as possible. Since $r+u+x$ is a positive integer that is divisible by 4, then the smallest that it can be is $r+u+x=4$. If $r+u+x=4$, then $r, u$ and $x$ must be 0,1 and 3 in some order since each of $r, u$ and $x$ is a different integer between 0 and 8. In this case, $4=4S-36$ and so $S=10$. Since $S=10$, then we cannot have $r$ and $u$ or $u$ and $x$ equal to 0 and 1 in some order, or else the third number in the line would have to be 9, which is not possible. This tells us that $u$ must be 3, and $r$ and $x$ are 0 and 1 in some order. Therefore, the value of $u$ in the original configuration is $3+2012=2015$.	2015	pascal

[ "Mathematics -> Geometry -> Plane Geometry -> Angles" ]

1.5

In triangle $BCD$, $\angle CBD=\angle CDB$ because $BC=CD$. If $\angle BCD=80+50+30=160$, find $\angle CBD=\angle CDB$.

$\angle CBD=\angle CDB=10$

10

HMMT_11

[ "Mathematics -> Algebra -> Linear Algebra -> Linear Transformations" ]

1.5

What is the $y$-intercept of the line $y = x + 4$ after it is translated down 6 units?

The line with equation $y = x + 4$ has a $y$-intercept of 4. When the line is translated 6 units downwards, all points on the line are translated 6 units down. This moves the $y$-intercept from 4 to $4 - 6 = -2$.

-2

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1.5

If $(pq)(qr)(rp) = 16$, what is a possible value for $pqr$?

Since $(pq)(qr)(rp) = 16$, then $pqqrrp = 16$ or $p^2q^2r^2 = 16$. Thus, $(pqr)^2 = 16$ and so $pqr = \pm 4$. Using the given answers, $pqr$ is positive and so $pqr = 4$.

4

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]

1.5

What is the value of $(-1)^{3}+(-1)^{2}+(-1)$?

Since -1 raised to an even exponent equals 1 and -1 raised to an odd exponent equals -1, then $(-1)^{3}+(-1)^{2}+(-1)=-1+1-1=-1$.

-1

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1.5

What is the value of $x$ if the three numbers $2, x$, and 10 have an average of $x$?

Since the average of $2, x$ and 10 is $x$, then $\frac{2 + x + 10}{3} = x$. Multiplying by 3, we obtain $2 + x + 10 = 3x$. Re-arranging, we obtain $x + 12 = 3x$ and then $2x = 12$ which gives $x = 6$.

6

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1.5

The three numbers $5, a, b$ have an average (mean) of 33. What is the average of $a$ and $b$?

Since $5, a, b$ have an average of 33, then $\frac{5+a+b}{3}=33$. Multiplying by 3, we obtain $5+a+b=3 \times 33=99$, which means that $a+b=94$. The average of $a$ and $b$ is thus equal to $\frac{a+b}{2}=\frac{94}{2}=47$.

47

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1.5

If $x=2018$, what is the value of the expression $x^{2}+2x-x(x+1)$?

For any value of $x$, we have $x^{2}+2x-x(x+1)=x^{2}+2x-x^{2}-x=x$. When $x=2018$, the value of this expression is thus 2018.

2018

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1.5

If $ \sqrt{100-x}=9 $, what is the value of $ x $?

Since $ \sqrt{100-x}=9 $, then $ 100-x=9^{2}=81 $, and so $ x=100-81=19 $.

19

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1

If $x+\sqrt{81}=25$, what is the value of $x$?

If $x+\sqrt{81}=25$, then $x+9=25$ or $x=16$.

16

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Fractions" ]

1

The symbol $\diamond$ is defined so that $a \diamond b=\frac{a+b}{a \times b}$. What is the value of $3 \diamond 6$?

Using the definition of the symbol, $3 \diamond 6=\frac{3+6}{3 \times 6}=\frac{9}{18}=\frac{1}{2}$.

\frac{1}{2}

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1.5

If $a(x+b)=3 x+12$ for all values of $x$, what is the value of $a+b$?

Since $a(x+b)=3 x+12$ for all $x$, then $a x+a b=3 x+12$ for all $x$. Since the equation is true for all $x$, then the coefficients on the left side must match the coefficients on the right side. Therefore, $a=3$ and $a b=12$, which gives $3 b=12$ or $b=4$. Finally, $a+b=3+4=7$.

7

cayley

[ "Mathematics -> Applied Mathematics -> Math Word Problems" ]

1.5

A class of 30 students was asked what they did on their winter holiday. 20 students said that they went skating. 9 students said that they went skiing. Exactly 5 students said that they went skating and went skiing. How many students did not go skating and did not go skiing?

Since 20 students went skating and 5 students went both skating and skiing, then $ 20-5=15 $ students went skating only. Since 9 students went skiing and 5 students went both skating and skiing, then $ 9-5=4 $ students went skiing only. The number of students who went skating or skiing or both equals the sum of the number who went skating only, the number who went skiing only, and the number who went both skating and skiing, or $ 15+4+5=24 $. Therefore, $ 30-24=6 $ students did not go skating or skiing.

6

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Fractions" ]

1

At Wednesday's basketball game, the Cayley Comets scored 90 points. At Friday's game, they scored $80\%$ as many points as they scored on Wednesday. How many points did they score on Friday?

On Friday, the Cayley Comets scored $80\%$ of 90 points. This is equal to $\frac{80}{100} \times 90 = \frac{8}{10} \times 90 = 8 \times 9 = 72$ points. Alternatively, since $80\%$ is equivalent to 0.8, then $80\%$ of 90 is equal to $0.8 \times 90 = 72$.

72

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1.5

The Cayley Corner Store sells three types of toys: Exes, Wyes and Zeds. All Exes are identical, all Wyes are identical, and all Zeds are identical. The mass of 2 Exes equals the mass of 29 Wyes. The mass of 1 Zed equals the mass of 16 Exes. The mass of 1 Zed equals the mass of how many Wyes?

Since the mass of 2 Exes equals the mass of 29 Wyes, then the mass of $8 \times 2$ Exes equals the mass of $8 \times 29$ Wyes. In other words, the mass of 16 Exes equals the mass of 232 Wyes. Since the mass of 1 Zed equals the mass of 16 Exes, then the mass of 1 Zed equals the mass of 232 Wyes.

232

cayley

[ "Mathematics -> Geometry -> Plane Geometry -> Triangulations" ]

1.5

Square $P Q R S$ has an area of 900. $M$ is the midpoint of $P Q$ and $N$ is the midpoint of $P S$. What is the area of triangle $P M N$?

Since square $P Q R S$ has an area of 900, then its side length is $\sqrt{900}=30$. Thus, $P Q=P S=30$. Since $M$ and $N$ are the midpoints of $P Q$ and $P S$, respectively, then $P N=P M=\frac{1}{2}(30)=15$. Since $P Q R S$ is a square, then the angle at $P$ is $90^{\circ}$, so $\triangle P M N$ is right-angled. Therefore, the area of $\triangle P M N$ is $\frac{1}{2}(P M)(P N)=\frac{1}{2}(15)(15)=\frac{225}{2}=112.5$.

112.5

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1.5

If $ x=2 $ and $ v=3x $, what is the value of $(2v-5)-(2x-5)$?

Since $ v=3x $ and $ x=2 $, then $ v=3 \cdot 2=6 $. Therefore, $(2v-5)-(2x-5)=(2 \cdot 6-5)-(2 \cdot 2-5)=7-(-1)=8$.

8

fermat

[ "Mathematics -> Algebra -> Prealgebra -> Fractions" ]

1

Calculate the value of the expression $\left(2 \times \frac{1}{3}\right) \times \left(3 \times \frac{1}{2}\right)$.

Re-arranging the order of the numbers being multiplied, $\left(2 \times \frac{1}{3}\right) \times \left(3 \times \frac{1}{2}\right) = 2 \times \frac{1}{2} \times 3 \times \frac{1}{3} = \left(2 \times \frac{1}{2}\right) \times \left(3 \times \frac{1}{3}\right) = 1 \times 1 = 1$.

1

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Fractions", "Mathematics -> Geometry -> Plane Geometry -> Area" ]

1.5

A rectangle has a length of $\frac{3}{5}$ and an area of $\frac{1}{3}$. What is the width of the rectangle?

In a rectangle, length times width equals area, so width equals area divided by length. Therefore, the width is $\frac{1}{3} \div \frac{3}{5}=\frac{1}{3} \times \frac{5}{3}=\frac{5}{9}$.

\\frac{5}{9}

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Fractions" ]

1.5

What is the value of the expression $\frac{20+16 \times 20}{20 \times 16}$?

Evaluating, $\frac{20+16 \times 20}{20 \times 16}=\frac{20+320}{320}=\frac{340}{320}=\frac{17}{16}$. Alternatively, we could notice that each of the numerator and denominator is a multiple of 20, and so $\frac{20+16 \times 20}{20 \times 16}=\frac{20(1+16)}{20 \times 16}=\frac{1+16}{16}=\frac{17}{16}$.

\frac{17}{16}

fermat

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1

If $x=3$, $y=2x$, and $z=3y$, what is the value of $z$?

Since $x=3$ and $y=2x$, then $y=2 \cdot 3=6$. Since $y=6$ and $z=3y$, then $z=3 \cdot 6=18$.

18

fermat

[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]

1

If $ (2^{a})(2^{b})=64 $, what is the mean (average) of $ a $ and $ b $?

Since $ (2^{a})(2^{b})=64 $, then $ 2^{a+b}=64 $, using an exponent law. Since $ 64=2^{6} $, then $ 2^{a+b}=2^{6} $ and so $ a+b=6 $. Therefore, the average of $ a $ and $ b $ is $ \frac{1}{2}(a+b)=3 $.

3

fermat

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1

If $x+\sqrt{25}=\sqrt{36}$, what is the value of $x$?

Since $x+\sqrt{25}=\sqrt{36}$, then $x+5=6$ or $x=1$.

1

cayley

[ "Mathematics -> Algebra -> Algebra -> Algebraic Expressions" ]

1.5

The line with equation $y = 3x + 5$ is translated 2 units to the right. What is the equation of the resulting line?

The line with equation $y = 3x + 5$ has slope 3 and $y$-intercept 5. Since the line has $y$-intercept 5, it passes through $(0, 5)$. When the line is translated 2 units to the right, its slope does not change and the new line passes through $(2, 5)$. A line with slope $m$ that passes through the point $(x_1, y_1)$ has equation $y - y_1 = m(x - x_1)$. Therefore, the line with slope 3 that passes through $(2, 5)$ has equation $y - 5 = 3(x - 2)$ or $y - 5 = 3x - 6$, which gives $y = 3x - 1$. Alternatively, we could note that when the graph of $y = 3x + 5$ is translated 2 units to the right, the equation of the new graph is $y = 3(x - 2) + 5$ or $y = 3x - 1$.

y = 3x - 1

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]

1.5

Calculate the value of $\frac{2 \times 3 + 4}{2 + 3}$.

Evaluating, $\frac{2 \times 3+4}{2+3}=\frac{6+4}{5}=\frac{10}{5}=2$.

2

fermat

[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]

1.5

The mean (average) of 5 consecutive integers is 9. What is the smallest of these 5 integers?

Since the mean of five consecutive integers is 9, then the middle of these five integers is 9. Therefore, the integers are $7,8,9,10,11$, and so the smallest of the five integers is 7.

7

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]

1

If $x = -3$, what is the value of $(x-3)^{2}$?

Evaluating, $(x-3)^{2}=(-3-3)^{2}=(-6)^{2}=36$.

36

fermat

[ "Mathematics -> Geometry -> Plane Geometry -> Polygons" ]

1.5

A rectangle has width $x$ and length $y$. The rectangle is cut along the horizontal and vertical dotted lines to produce four smaller rectangles. The sum of the perimeters of these four rectangles is 24. What is the value of $x+y$?

The sum of the lengths of the horizontal line segments is $4x$, because the tops of the four small rectangles contribute a total of $2x$ to their combined perimeter and the bottoms contribute a total of $2x$. Similarly, the sum of the lengths of the vertical line segments is $4y$. In other words, the sum of the perimeters of the four rectangles is $4x + 4y$. Since the sum of the perimeters also equals 24, then $4x + 4y = 24$ and so $x + y = 6$.

6

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]

1

If $ 3-5+7=6-x $, what is the value of $ x $?

Simplifying the left side of the equation, we obtain $ 5=6-x $. Therefore, $ x=6-5=1 $.

1

fermat

[ "Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Other" ]

1.5

Two standard six-sided dice are rolled. What is the probability that the product of the two numbers rolled is 12?

When two standard six-sided dice are rolled, there are $6 \times 6 = 36$ possibilities for the pair of numbers that are rolled. Of these, the pairs $2 \times 6, 3 \times 4, 4 \times 3$, and $6 \times 2$ each give 12. (If one of the numbers rolled is 1 or 5, the product cannot be 12.) Since there are 4 pairs of possible rolls whose product is 12, the probability that the product is 12 is $\frac{4}{36}$.

\frac{4}{36}

cayley

[ "Mathematics -> Applied Mathematics -> Math Word Problems", "Mathematics -> Algebra -> Prealgebra -> Ratios -> Other" ]

1.5

There are 400 students at Pascal H.S., where the ratio of boys to girls is $3: 2$. There are 600 students at Fermat C.I., where the ratio of boys to girls is $2: 3$. What is the ratio of boys to girls when considering all students from both schools?

Since the ratio of boys to girls at Pascal H.S. is $3: 2$, then $rac{3}{3+2}=rac{3}{5}$ of the students at Pascal H.S. are boys. Thus, there are $rac{3}{5}(400)=rac{1200}{5}=240$ boys at Pascal H.S. Since the ratio of boys to girls at Fermat C.I. is $2: 3$, then $rac{2}{2+3}=rac{2}{5}$ of the students at Fermat C.I. are boys. Thus, there are $rac{2}{5}(600)=rac{1200}{5}=240$ boys at Fermat C.I. There are $400+600=1000$ students in total at the two schools. Of these, $240+240=480$ are boys, and so the remaining $1000-480=520$ students are girls. Therefore, the overall ratio of boys to girls is $480: 520=48: 52=12: 13$.

12:13

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1.5

John ate a total of 120 peanuts over four consecutive nights. Each night he ate 6 more peanuts than the night before. How many peanuts did he eat on the fourth night?

Suppose that John ate $ x $ peanuts on the fourth night. Since he ate 6 more peanuts each night than on the previous night, then he ate $ x-6 $ peanuts on the third night, $(x-6)-6=x-12$ peanuts on the second night, and $(x-12)-6=x-18$ peanuts on the first night. Since John ate 120 peanuts in total, then $ x+(x-6)+(x-12)+(x-18)=120 $, and so $ 4x-36=120 $ or $ 4x=156 $ or $ x=39 $. Therefore, John ate 39 peanuts on the fourth night.

39

fermat

[ "Mathematics -> Algebra -> Prealgebra -> Fractions" ]

1.5

If $\frac{1}{6} + \frac{1}{3} = \frac{1}{x}$, what is the value of $x$?

Simplifying, $\frac{1}{6} + \frac{1}{3} = \frac{1}{6} + \frac{2}{6} = \frac{3}{6} = \frac{1}{2}$. Thus, $\frac{1}{x} = \frac{1}{2}$ and so $x = 2$.

2

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1.5

If $3n=9+9+9$, what is the value of $n$?

Since $3n=9+9+9=3 imes 9$, then $n=9$. Alternatively, we could note that $9+9+9=27$ and so $3n=27$ which gives $n=rac{27}{3}=9$.

9

cayley

[ "Mathematics -> Applied Mathematics -> Math Word Problems" ]

1.5

Arturo has an equal number of $\$5$ bills, of $\$10$ bills, and of $\$20$ bills. The total value of these bills is $\$700$. How many $\$5$ bills does Arturo have?

Since Arturo has an equal number of $\$5$ bills, of $\$10$ bills, and of $\$20$ bills, then we can divide Arturo's bills into groups, each of which contains one $\$5$ bill, one $\$10$ bill, and one $\$20$ bill. The value of the bills in each group is $\$5 + \$10 + \$20 = \$35$. Since the total value of Arturo's bills is $\$700$, then there are $\frac{\$700}{\$35} = 20$ groups. Thus, Arturo has $20 \$5$ bills.

20

cayley

[ "Mathematics -> Geometry -> Solid Geometry -> Volume" ]

1.5

The volume of a prism is equal to the area of its base times its depth. If the prism has identical bases with area $400 \mathrm{~cm}^{2}$ and depth 8 cm, what is its volume?

The volume of a prism is equal to the area of its base times its depth. Here, the prism has identical bases with area $400 \mathrm{~cm}^{2}$ and depth 8 cm, and so its volume is $400 \mathrm{~cm}^{2} \times 8 \mathrm{~cm} = 3200 \mathrm{~cm}^{3}$.

3200 \mathrm{~cm}^{3}

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]

1

What is the value of $(-2)^{3}-(-3)^{2}$?

Evaluating, $(-2)^{3}-(-3)^{2}=-8-9=-17$.

-17

fermat

[ "Mathematics -> Algebra -> Prealgebra -> Fractions" ]

1

Evaluate the expression $8-rac{6}{4-2}$.

Evaluating, $8-rac{6}{4-2}=8-rac{6}{2}=8-3=5$.

5

fermat

[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]

1

Calculate the value of the expression $2 \times 0 + 2 \times 4$.

Calculating, $2 \times 0 + 2 \times 4 = 0 + 8 = 8$.

8

cayley

[ "Mathematics -> Applied Mathematics -> Math Word Problems" ]

1.5

A 6 m by 8 m rectangular field has a fence around it. There is a post at each of the four corners of the field. Starting at each corner, there is a post every 2 m along each side of the fence. How many posts are there?

A rectangle that is 6 m by 8 m has perimeter $2 \times(6 \mathrm{~m}+8 \mathrm{~m})=28 \mathrm{~m}$. If posts are put in every 2 m around the perimeter starting at a corner, then we would guess that it will take $\frac{28 \mathrm{~m}}{2 \mathrm{~m}}=14$ posts.

14

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1.5

In a cafeteria line, the number of people ahead of Kaukab is equal to two times the number of people behind her. There are $n$ people in the line. What is a possible value of $n$?

Suppose that there are $p$ people behind Kaukab. This means that there are $2p$ people ahead of her. Including Kaukab, the total number of people in line is $n = p + 2p + 1 = 3p + 1$, which is one more than a multiple of 3. Of the given choices $(23, 20, 24, 21, 25)$, the only one that is one more than a multiple of 3 is 25, which equals $3 \times 8 + 1$. Therefore, a possible value for $n$ is 25.

25

cayley

[ "Mathematics -> Geometry -> Plane Geometry -> Area" ]

1.5

A rectangular field has a length of 20 metres and a width of 5 metres. If its length is increased by 10 m, by how many square metres will its area be increased?

Since the field originally has length 20 m and width 5 m, then its area is $20 \times 5=100 \mathrm{~m}^{2}$. The new length of the field is $20+10=30 \mathrm{~m}$, so the new area is $30 \times 5=150 \mathrm{~m}^{2}$. The increase in area is $150-100=50 \mathrm{~m}^{2}$. (Alternatively, we could note that since the length increases by 10 m and the width stays constant at 5 m, then the increase in area is $10 \times 5=50 \mathrm{~m}^{2}$.)

50

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1.5

The average of 1, 3, and $ x $ is 3. What is the value of $ x $?

Since the average of three numbers equals 3, then their sum is $ 3 \times 3 = 9 $. Therefore, $ 1+3+x=9 $ and so $ x=9-4=5 $.

5

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1

What is the value of $(3x + 2y) - (3x - 2y)$ when $x = -2$ and $y = -1$?

The expression $(3x + 2y) - (3x - 2y)$ is equal to $3x + 2y - 3x + 2y$ which equals $4y$. When $x = -2$ and $y = -1$, this equals $4(-1)$ or $-4$.

-4

fermat

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1

If $x=3$, what is the value of $-(5x - 6x)$?

When $x=3$, we have $-(5x - 6x) = -(-x) = x = 3$. Alternatively, when $x=3$, we have $-(5x - 6x) = -(15 - 18) = -(-3) = 3$.

3

cayley

[ "Mathematics -> Applied Mathematics -> Math Word Problems", "Mathematics -> Discrete Mathematics -> Algorithms" ]

1.5

A robotic grasshopper jumps 1 cm to the east, then 2 cm to the north, then 3 cm to the west, then 4 cm to the south. After every fourth jump, the grasshopper restarts the sequence of jumps: 1 cm to the east, then 2 cm to the north, then 3 cm to the west, then 4 cm to the south. After a total of $n$ jumps, the position of the grasshopper is 162 cm to the west and 158 cm to the south of its original position. What is the sum of the squares of the digits of $n$?

Each group of four jumps takes the grasshopper 1 cm to the east and 3 cm to the west, which is a net movement of 2 cm to the west, and 2 cm to the north and 4 cm to the south, which is a net movement of 2 cm to the south. In other words, we can consider each group of four jumps, starting with the first, as resulting in a net movement of 2 cm to the west and 2 cm to the south. We note that $158=2 \times 79$. Thus, after 79 groups of four jumps, the grasshopper is $79 \times 2=158 \mathrm{~cm}$ to the west and 158 cm to the south of its original position. (We need at least 79 groups of these because the grasshopper cannot be 158 cm to the south of its original position before the end of 79 such groups.) The grasshopper has made $4 \times 79=316$ jumps so far. After the 317th jump (1 cm to the east), the grasshopper is 157 cm west and 158 cm south of its original position. After the 318th jump (2 cm to the north), the grasshopper is 157 cm west and 156 cm south of its original position. After the 319th jump (3 cm to the west), the grasshopper is 160 cm west and 156 cm south of its original position. After the 320th jump (4 cm to the south), the grasshopper is 160 cm west and 160 cm south of its original position. After the 321st jump (1 cm to the east), the grasshopper is 159 cm west and 160 cm south of its original position. After the 322nd jump (2 cm to the north), the grasshopper is 159 cm west and 158 cm south of its original position. After the 323rd jump (3 cm to the west), the grasshopper is 162 cm west and 158 cm south of its original position, which is the desired position. As the grasshopper continues jumping, each of its positions will always be at least 160 cm south of its original position, so this is the only time that it is at this position. Therefore, $n=323$. The sum of the squares of the digits of $n$ is $3^{2}+2^{2}+3^{2}=9+4+9=22$.

22

cayley

[ "Mathematics -> Geometry -> Plane Geometry -> Angles" ]

1.5

What is the measure of the largest angle in $\triangle P Q R$?

Since the sum of the angles in a triangle is $180^{\circ}$, then $3 x^{\circ}+x^{\circ}+6 x^{\circ}=180^{\circ}$ or $10 x=180$ or $x=18$. The largest angle in the triangle is $6 x^{\circ}=6(18^{\circ})=108^{\circ}$.

108^{\\circ}

cayley

[ "Mathematics -> Number Theory -> Congruences" ]

1.5

How many of the four integers $222, 2222, 22222$, and $222222$ are multiples of 3?

We could use a calculator to divide each of the four given numbers by 3 to see which calculations give an integer answer. Alternatively, we could use the fact that a positive integer is divisible by 3 if and only if the sum of its digits is divisible by 3. The sums of the digits of $222, 2222, 22222$, and $222222$ are $6, 8, 10$, and 12, respectively. Two of these sums are divisible by 3 (namely, 6 and 12) so two of the four integers (namely, 222 and 222222) are divisible by 3.

2

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]

1.5

When three consecutive integers are added, the total is 27. What is the result when the same three integers are multiplied?

If the sum of three consecutive integers is 27, then the numbers must be 8, 9, and 10. Their product is $8 imes 9 imes 10=720$.

720

fermat

[ "Mathematics -> Algebra -> Prealgebra -> Fractions" ]

1

If $\frac{1}{9}+\frac{1}{18}=\frac{1}{\square}$, what is the number that replaces the $\square$ to make the equation true?

We simplify the left side and express it as a fraction with numerator 1: $\frac{1}{9}+\frac{1}{18}=\frac{2}{18}+\frac{1}{18}=\frac{3}{18}=\frac{1}{6}$. Therefore, the number that replaces the $\square$ is 6.

6

pascal

[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]

1.5

What is the expression $2^{3}+2^{2}+2^{1}$ equal to?

Since $2^{1}=2$ and $2^{2}=2 imes 2=4$ and $2^{3}=2 imes 2 imes 2=8$, then $2^{3}+2^{2}+2^{1}=8+4+2=14$.

14

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1

If $3 imes n=6 imes 2$, what is the value of $n$?

Since $3 imes n=6 imes 2$, then $3n=12$ or $n=\frac{12}{3}=4$.

4

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]

1

What is the value of $rac{(20-16) imes (12+8)}{4}$?

Using the correct order of operations, $rac{(20-16) imes (12+8)}{4} = rac{4 imes 20}{4} = rac{80}{4} = 20$.

20

pascal

[ "Mathematics -> Geometry -> Solid Geometry -> 3D Shapes" ]

1.5

How many solid $1 imes 1 imes 1$ cubes are required to make a solid $2 imes 2 imes 2$ cube?

The volume of a $1 imes 1 imes 1$ cube is 1 . The volume of a $2 imes 2 imes 2$ cube is 8 . Thus, 8 of the smaller cubes are needed to make the larger cube.

8

fermat

[ "Mathematics -> Applied Mathematics -> Math Word Problems" ]

1.5

Charlie is making a necklace with yellow beads and green beads. She has already used 4 green beads and 0 yellow beads. How many yellow beads will she have to add so that $rac{4}{5}$ of the total number of beads are yellow?

If $rac{4}{5}$ of the beads are yellow, then $rac{1}{5}$ are green. Since there are 4 green beads, the total number of beads must be $4 imes 5=20$. Thus, Charlie needs to add $20-4=16$ yellow beads.

16

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1

If $ x=2 $, what is the value of $ (x+2-x)(2-x-2) $?

When $ x=2 $, we have $ (x+2-x)(2-x-2)=(2+2-2)(2-2-2)=(2)(-2)=-4 $. Alternatively, we could simplify $ (x+2-x)(2-x-2) $ to obtain $ (2)(-x) $ or $ -2x $ and then substitute $ x=2 $ to obtain a result of $ -2(2) $ or -4.

-4

fermat

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1.5

Ten numbers have an average (mean) of 87. Two of those numbers are 51 and 99. What is the average of the other eight numbers?

Since 10 numbers have an average of 87, their sum is $10 \times 87 = 870$. When the numbers 51 and 99 are removed, the sum of the remaining 8 numbers is $870 - 51 - 99$ or 720. The average of these 8 numbers is $\frac{720}{8} = 90$.

90

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1.5

If $x \%$ of 60 is 12, what is $15 \%$ of $x$?

Since $x \%$ of 60 is 12, then $\frac{x}{100} \cdot 60=12$ or $x=\frac{12 \cdot 100}{60}=20$. Therefore, $15 \%$ of $x$ is $15 \%$ of 20, or $0.15 \cdot 20=3$.

3

fermat

[ "Mathematics -> Applied Mathematics -> Math Word Problems" ]

1.5

Ava's machine takes four-digit positive integers as input. When the four-digit integer $ABCD$ is input, the machine outputs the integer $A imes B + C imes D$. What is the output when the input is 2023?

Using the given rule, the output of the machine is $2 imes 0 + 2 imes 3 = 0 + 6 = 6$.

6

fermat

[ "Mathematics -> Algebra -> Prealgebra -> Fractions" ]

1

What is the value of $ \frac{5-2}{2+1} $?

Simplifying, $ \frac{5-2}{2+1}=\frac{3}{3}=1 $.

1

cayley

[ "Mathematics -> Algebra -> Algebra -> Algebraic Expressions" ]

1

Evaluate the expression $2x^{2}+3x^{2}$ when $x=2$.

When $x=2$, we obtain $2x^{2}+3x^{2}=5x^{2}=5 \cdot 2^{2}=5 \cdot 4=20$.

20

fermat

[ "Mathematics -> Algebra -> Prealgebra -> Fractions" ]

1

If $\frac{1}{9}+\frac{1}{18}=\frac{1}{\square}$, what is the number that replaces the $\square$ to make the equation true?

We simplify the left side and express it as a fraction with numerator 1: $\frac{1}{9}+\frac{1}{18}=\frac{2}{18}+\frac{1}{18}=\frac{3}{18}=\frac{1}{6}$. Therefore, the number that replaces the $\square$ is 6.

6

pascal

[ "Mathematics -> Applied Mathematics -> Math Word Problems" ]

1.5

A bicycle trip is 30 km long. Ari rides at an average speed of 20 km/h. Bri rides at an average speed of 15 km/h. If Ari and Bri begin at the same time, how many minutes after Ari finishes the trip will Bri finish?

Riding at 15 km/h, Bri finishes the 30 km in $\frac{30 \text{ km}}{15 \text{ km/h}} = 2 \text{ h}$. Riding at 20 km/h, Ari finishes the 30 km in $\frac{30 \text{ km}}{20 \text{ km/h}} = 1.5 \text{ h}$. Therefore, Bri finishes 0.5 h after Ari, which is 30 minutes.

30

fermat

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1.5

If $\frac{x-y}{x+y}=5$, what is the value of $\frac{2x+3y}{3x-2y}$?

Since $\frac{x-y}{x+y}=5$, then $x-y=5(x+y)$. This means that $x-y=5x+5y$ and so $0=4x+6y$ or $2x+3y=0$. Therefore, $\frac{2x+3y}{3x-2y}=\frac{0}{3x-2y}=0$.

0

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1

Calculate the value of the expression $(8 \times 6)-(4 \div 2)$.

We evaluate the expression by first evaluating the expressions in brackets: $(8 \times 6)-(4 \div 2)=48-2=46$.

46

pascal

[ "Mathematics -> Geometry -> Plane Geometry -> Polygons" ]

1

What is the perimeter of the figure shown if $x=3$?

Since $x=3$, the side lengths of the figure are $4,3,6$, and 10. Thus, the perimeter of the figure is $4+3+6+10=23$. (Alternatively, the perimeter is $x+6+10+(x+1)=2x+17$. When $x=3$, this equals $2(3)+17$ or 23.)

23

pascal

[ "Mathematics -> Applied Mathematics -> Math Word Problems" ]

1

Carrie sends five text messages to her brother each Saturday and Sunday, and two messages on other days. Over four weeks, how many text messages does Carrie send?

Each week, Carrie sends 5 messages to her brother on each of 2 days, for a total of 10 messages. Each week, Carrie sends 2 messages to her brother on each of the remaining 5 days, for a total of 10 messages. Therefore, Carrie sends $10+10=20$ messages per week. In four weeks, Carrie sends $4 \cdot 20=80$ messages.

80

fermat

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1

If $x=2y$ and $y \neq 0$, what is the value of $(x+2y)-(2x+y)$?

We simplify first, then substitute $x=2y$: $(x+2y)-(2x+y)=x+2y-2x-y=y-x=y-2y=-y$. Alternatively, we could substitute first, then simplify: $(x+2y)-(2x+y)=(2y+2y)-(2(2y)+y)=4y-5y=-y$.

-y

pascal

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1.5

What is the value of $2^{4}-2^{3}$?

We note that $2^{2}=2 \times 2=4,2^{3}=2^{2} \times 2=4 \times 2=8$, and $2^{4}=2^{2} \times 2^{2}=4 \times 4=16$. Therefore, $2^{4}-2^{3}=16-8=8=2^{3}$.

2^{3}

pascal

[ "Mathematics -> Applied Mathematics -> Math Word Problems" ]

1.5

Calculate the number of minutes in a week.

There are 60 minutes in an hour and 24 hours in a day. Thus, there are $60 \cdot 24=1440$ minutes in a day. Since there are 7 days in a week, the number of minutes in a week is $7 \cdot 1440=10080$. Of the given choices, this is closest to 10000.

10000

fermat

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1

When $x=-2$, what is the value of $(x+1)^{3}$?

When $x=-2$, we have $(x+1)^{3}=(-2+1)^{3}=(-1)^{3}=-1$.

-1

pascal

[ "Mathematics -> Applied Mathematics -> Math Word Problems" ]

1

How many points does a sports team earn for 9 wins, 3 losses, and 4 ties, if they earn 2 points for each win, 0 points for each loss, and 1 point for each tie?

The team earns 2 points for each win, so 9 wins earn $2 \times 9=18$ points. The team earns 0 points for each loss, so 3 losses earn 0 points. The team earns 1 point for each tie, so 4 ties earn 4 points. In total, the team earns $18+0+4=22$ points.

22

pascal

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations", "Mathematics -> Number Theory -> Integer Factorization -> Other" ]

1.5

How many pairs of positive integers $(x, y)$ have the property that the ratio $x: 4$ equals the ratio $9: y$?

The equality of the ratios $x: 4$ and $9: y$ is equivalent to the equation $\frac{x}{4}=\frac{9}{y}$. This equation is equivalent to the equation $xy=4(9)=36$. The positive divisors of 36 are $1,2,3,4,6,9,12,18,36$, so the desired pairs are $(x, y)=(1,36),(2,18),(3,12),(4,9),(6,6),(9,4),(12,3),(18,2),(36,1)$. There are 9 such pairs.

9

pascal

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1.5

Jitka hiked a trail. After hiking 60% of the length of the trail, she had 8 km left to go. What is the length of the trail?

After Jitka hiked 60% of the trail, 40% of the trail was left, which corresponds to 8 km. This means that 10% of the trail corresponds to 2 km. Therefore, the total length of the trail is $ 10 \times 2 = 20 \text{ km} $.

20 \text{ km}

pascal

[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]

1.5

In the addition problem shown, $m, n, p$, and $q$ represent positive digits. What is the value of $m+n+p+q$?

From the ones column, we see that $3 + 2 + q$ must have a ones digit of 2. Since $q$ is between 1 and 9, inclusive, then $3 + 2 + q$ is between 6 and 14. Since its ones digit is 2, then $3 + 2 + q = 12$ and so $q = 7$. This also means that there is a carry of 1 into the tens column. From the tens column, we see that $1 + 6 + p + 8$ must have a ones digit of 4. Since $p$ is between 1 and 9, inclusive, then $1 + 6 + p + 8$ is between 16 and 24. Since its ones digit is 4, then $1 + 6 + p + 8 = 24$ and so $p = 9$. This also means that there is a carry of 2 into the hundreds column. From the hundreds column, we see that $2 + n + 7 + 5$ must have a ones digit of 0. Since $n$ is between 1 and 9, inclusive, then $2 + n + 7 + 5$ is between 15 and 23. Since its ones digit is 0, then $2 + n + 7 + 5 = 20$ and so $n = 6$. This also means that there is a carry of 2 into the thousands column. This means that $m = 2$. Thus, we have $m + n + p + q = 2 + 6 + 9 + 7 = 24$.

24

cayley

[ "Mathematics -> Algebra -> Prealgebra -> Other" ]

1.5

In a magic square, what is the sum $ a+b+c $?

Using the properties of a magic square, $ a+b+c = 14+18+15 = 47 $.

47

pascal

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1

If $10x+y=75$ and $10y+x=57$ for some positive integers $x$ and $y$, what is the value of $x+y$?

Since $10x+y=75$ and $10y+x=57$, then $(10x+y)+(10y+x)=75+57$ and so $11x+11y=132$. Dividing by 11, we get $x+y=12$. (We could have noticed initially that $(x, y)=(7,5)$ is a pair that satisfies the two equations, thence concluding that $x+y=12$.)

12

fermat

[ "Mathematics -> Algebra -> Prealgebra -> Fractions" ]

1.5

What is the number halfway between $\frac{1}{12}$ and $\frac{1}{10}$?

The number halfway between two numbers is their average. Therefore, the number halfway between $\frac{1}{10}$ and $\frac{1}{12}$ is $\frac{1}{2}\left(\frac{1}{10}+\frac{1}{12}\right)=\frac{1}{2}\left(\frac{12}{120}+\frac{10}{120}\right)=\frac{1}{2}\left(\frac{22}{120}\right)=\frac{11}{120}$.

\frac{11}{120}

fermat

[ "Mathematics -> Algebra -> Prealgebra -> Fractions" ]

1.5

If $ 50\% $ of $ N $ is 16, what is $ 75\% $ of $ N $?

The percentage $ 50\% $ is equivalent to the fraction $ \frac{1}{2} $, while $ 75\% $ is equivalent to $ \frac{3}{4} $. Since $ 50\% $ of $ N $ is 16, then $ \frac{1}{2}N=16 $ or $ N=32 $. Therefore, $ 75\% $ of $ N $ is $ \frac{3}{4}N $ or $ \frac{3}{4}(32) $, which equals 24.

24

fermat

[ "Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations" ]

1.5

In how many different ways can André form exactly $ \$10 $ using $ \$1 $ coins, $ \$2 $ coins, and $ \$5 $ bills?

Using combinations of $ \$5 $ bills, $ \$2 $ coins, and $ \$1 $ coins, there are 10 ways to form $ \$10 $.

10

pascal

[ "Mathematics -> Algebra -> Prealgebra -> Fractions" ]

1.5

What is the value of $rac{8+4}{8-4}$?

Simplifying, $rac{8+4}{8-4}=rac{12}{4}=3$.

3

cayley

[ "Mathematics -> Precalculus -> Functions" ]

1.5

Numbers $m$ and $n$ are on the number line. What is the value of $n-m$?

On a number line, the markings are evenly spaced. Since there are 6 spaces between 0 and 30, each space represents a change of $\frac{30}{6}=5$. Since $n$ is 2 spaces to the right of 60, then $n=60+2 \times 5=70$. Since $m$ is 3 spaces to the left of 30, then $m=30-3 \times 5=15$. Therefore, $n-m=70-15=55$.

55

pascal

[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]

1.5

What is the value of $ \sqrt{16 \times \sqrt{16}} $?

Evaluating, $ \sqrt{16 \times \sqrt{16}} = \sqrt{16 \times 4} = \sqrt{64} = 8 $. Since $ 8 = 2^3 $, then $ \sqrt{16 \times \sqrt{16}} = 2^3 $.

2^3

pascal

[ "Mathematics -> Algebra -> Prealgebra -> Fractions" ]

1

The operation $ \otimes $ is defined by $ a \otimes b = \frac{a}{b} + \frac{b}{a} $. What is the value of $ 4 \otimes 8 $?

From the given definition, $ 4 \otimes 8 = \frac{4}{8} + \frac{8}{4} = \frac{1}{2} + 2 = \frac{5}{2} $.

\frac{5}{2}

pascal

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1

If $ 8 + 6 = n + 8 $, what is the value of $ n $?

Since $ 8+6=n+8 $, then subtracting 8 from both sides, we obtain $ 6=n $ and so $ n $ equals 6.

6

pascal

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1

If $x=3$, $y=2x$, and $z=3y$, what is the average of $x$, $y$, and $z$?

Since $x=3$ and $y=2x$, then $y=2 \times 3=6$. Since $y=6$ and $z=3y$, then $z=3 \times 6=18$. Therefore, the average of $x, y$ and $z$ is $\frac{x+y+z}{3}=\frac{3+6+18}{3}=9$.

9

pascal

[ "Mathematics -> Algebra -> Prealgebra -> Decimals" ]

1

What is 30% of 200?

$30\%$ of 200 equals $\frac{30}{100} \times 200=60$. Alternatively, we could note that $30\%$ of 100 is 30 and $200=2 \times 100$, so $30\%$ of 200 is $30 \times 2$ which equals 60.

60

pascal

[ "Mathematics -> Applied Mathematics -> Math Word Problems" ]

1

What is the value of $ z $ in the carpet installation cost chart?

Using the cost per square metre, $ z = 1261.40 $.

1261.40

pascal

[ "Mathematics -> Applied Mathematics -> Math Word Problems" ]

1.5

It takes Pearl 7 days to dig 4 holes. It takes Miguel 3 days to dig 2 holes. If they work together and each continues digging at these same rates, how many holes in total will they dig in 21 days?

Since Pearl digs 4 holes in 7 days and $\frac{21}{7}=3$, then in 21 days, Pearl digs $3 \cdot 4=12$ holes. Since Miguel digs 2 holes in 3 days and $\frac{21}{3}=7$, then in 21 days, Miguel digs $7 \cdot 2=14$ holes. In total, they dig $12+14=26$ holes in 21 days.

26

fermat

[ "Mathematics -> Algebra -> Prealgebra -> Integers", "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1

If 7:30 a.m. was 16 minutes ago, how many minutes will it be until 8:00 a.m.?

If 7:30 a.m. was 16 minutes ago, then it is currently $30+16=46$ minutes after 7:00 a.m., or 7:46 a.m. Since 8:00 a.m. is 60 minutes after 7:00 a.m., then it will be 8:00 a.m. in $60-46=14$ minutes.

14

pascal

[ "Mathematics -> Algebra -> Prealgebra -> Integers", "Mathematics -> Applied Mathematics -> Math Word Problems" ]

1.5

At the Lacsap Hospital, Emily is a doctor and Robert is a nurse. Not including Emily, there are five doctors and three nurses at the hospital. Not including Robert, there are $d$ doctors and $n$ nurses at the hospital. What is the product of $d$ and $n$?

Since Emily is a doctor and there are 5 doctors and 3 nurses aside from Emily at the hospital, then there are 6 doctors and 3 nurses in total. Since Robert is a nurse, then aside from Robert, there are 6 doctors and 2 nurses. Therefore, $d=6$ and $n=2$, so $dn=12$.

12

pascal

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1

If $y=1$ and $4x-2y+3=3x+3y$, what is the value of $x$?

Substituting $y=1$ into the second equation, we obtain $4x-2(1)+3=3x+3(1)$. Simplifying, we obtain $4x-2+3=3x+3$ or $4x+1=3x+3$. Therefore, $4x-3x=3-1$ or $x=2$.

2

pascal

[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]

1

Determine which of the following expressions has the largest value: $4^2$, $4 \times 2$, $4 - 2$, $\frac{4}{2}$, or $4 + 2$.

We evaluate each of the five choices: $4^{2}=16$, $4 \times 2=8$, $4-2=2$, $\frac{4}{2}=2$, $4+2=6$. Of these, the largest is $4^{2}=16$.

16

pascal

[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]

1

Calculate the expression $8 \times 10^{5}+4 \times 10^{3}+9 \times 10+5$.

First, we write out the powers of 10 in full to obtain $8 \times 100000+4 \times 1000+9 \times 10+5$. Simplifying, we obtain $800000+4000+90+5$ or 804095.

804095

pascal

[ "Mathematics -> Geometry -> Plane Geometry -> Triangulations" ]

1.5

In $\triangle PQR, \angle RPQ=90^{\circ}$ and $S$ is on $PQ$. If $SQ=14, SP=18$, and $SR=30$, what is the area of $\triangle QRS$?

Since $\triangle RPS$ is right-angled at $P$, then by the Pythagorean Theorem, $PR^{2}+PS^{2}=RS^{2}$ or $PR^{2}+18^{2}=30^{2}$. This gives $PR^{2}=900-324=576$, from which $PR=24$. Since $P, S$ and $Q$ lie on a straight line and $RP$ is perpendicular to this line, then $RP$ is actually a height for $\triangle QRS$ corresponding to base $SQ$. Thus, the area of $\triangle QRS$ is $\frac{1}{2}(24)(14)=168$.

168

pascal

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1

The average (mean) of two numbers is 7. One of the numbers is 5. What is the other number?

Since the average of two numbers is 7, their sum is $2 imes 7=14$. Since one of the numbers is 5, the other is $14-5=9$.

9

fermat

[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]

1.5

Jim wrote a sequence of symbols a total of 50 times. How many more of one symbol than another did he write?

The sequence of symbols includes 5 of one symbol and 2 of another. This means that, each time the sequence is written, there are 3 more of one symbol written than the other. When the sequence is written 50 times, in total there are $ 50 \times 3 = 150 $ more of one symbol written than the other.

150

pascal

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1.5

Chris received a mark of $50 \%$ on a recent test. Chris answered 13 of the first 20 questions correctly. Chris also answered $25 \%$ of the remaining questions on the test correctly. If each question on the test was worth one mark, how many questions in total were on the test?

Suppose that there were $n$ questions on the test. Since Chris received a mark of $50 \%$ on the test, then he answered $\frac{1}{2} n$ of the questions correctly. We know that Chris answered 13 of the first 20 questions correctly and then $25 \%$ of the remaining questions. Since the test has $n$ questions, then after the first 20 questions, there are $n-20$ questions. Since Chris answered $25 \%$ of these $n-20$ questions correctly, then Chris answered $\frac{1}{4}(n-20)$ of these questions correctly. The total number of questions that Chris answered correctly can be expressed as $\frac{1}{2} n$ and also as $13+\frac{1}{4}(n-20)$. Therefore, $\frac{1}{2} n=13+\frac{1}{4}(n-20)$ and so $2 n=52+(n-20)$, which gives $n=32$. (We can check that if $n=32$, then Chris answers 13 of the first 20 and 3 of the remaining 12 questions correctly, for a total of 16 correct out of 32.)

32

pascal

[ "Mathematics -> Algebra -> Prealgebra -> Integers" ]

1.5

If $2 \times 2 \times 3 \times 3 \times 5 \times 6=5 \times 6 \times n \times n$, what is a possible value of $n$?

We rewrite the left side of the given equation as $5 \times 6 \times(2 \times 3) \times(2 \times 3)$. Since $5 \times 6 \times(2 \times 3) \times(2 \times 3)=5 \times 6 \times n \times n$, then a possible value of $n$ is $2 \times 3$ or 6.

6

pascal

[ "Mathematics -> Algebra -> Prealgebra -> Simple Equations" ]

1

If $x=2$, what is the value of $4x^2 - 3x^2$?

Simplifying, $4 x^{2}-3 x^{2}=x^{2}$. When $x=2$, this expression equals 4 . Alternatively, when $x=2$, we have $4 x^{2}-3 x^{2}=4 \cdot 2^{2}-3 \cdot 2^{2}=16-12=4$.

4

fermat

[ "Mathematics -> Algebra -> Prealgebra -> Integers", "Mathematics -> Discrete Mathematics -> Combinatorics" ]

1.5

The smallest of nine consecutive integers is 2012. These nine integers are placed in the circles to the right. The sum of the three integers along each of the four lines is the same. If this sum is as small as possible, what is the value of $u$?

If we have a configuration of the numbers that has the required property, then we can add or subtract the same number from each of the numbers in the circles and maintain the property. (This is because there are the same number of circles in each line.) Therefore, we can subtract 2012 from all of the numbers and try to complete the diagram using the integers from 0 to 8. We label the circles as shown in the diagram, and call $S$ the sum of the three integers along any one of the lines. Since $p, q, r, t, u, w, x, y, z$ are 0 through 8 in some order, then $p+q+r+t+u+w+x+y+z=0+1+2+3+4+5+6+7+8=36$. From the desired property, we want $S=p+q+r=r+t+u=u+w+x=x+y+z$. Therefore, $(p+q+r)+(r+t+u)+(u+w+x)+(x+y+z)=4S$. From this, $(p+q+r+t+u+w+x+y+z)+r+u+x=4S$ or $r+u+x=4S-36=4(S-9)$. We note that the right side is an integer that is divisible by 4. Also, we want $S$ to be as small as possible so we want the sum $r+u+x$ to be as small as possible. Since $r+u+x$ is a positive integer that is divisible by 4, then the smallest that it can be is $r+u+x=4$. If $r+u+x=4$, then $r, u$ and $x$ must be 0,1 and 3 in some order since each of $r, u$ and $x$ is a different integer between 0 and 8. In this case, $4=4S-36$ and so $S=10$. Since $S=10$, then we cannot have $r$ and $u$ or $u$ and $x$ equal to 0 and 1 in some order, or else the third number in the line would have to be 9, which is not possible. This tells us that $u$ must be 3, and $r$ and $x$ are 0 and 1 in some order. Therefore, the value of $u$ in the original configuration is $3+2012=2015$.

2015

pascal