Grades 7 / 8 Tests and Answer Keys

Find and simplify

Write your answer as a simple fraction where the numerator and denominator have no common factors.

  (2 pts) 1.

A class has 22 students. Every student in the class has a dog, a cat, or both a cat and a dog. If 18 students have a dog (with or without a cat), and 9 students have a cat (with or without a dog), how many students have both a cat and a dog?

  (3 pts) 2.

A right triangle has one leg exactly twice as long as the other leg. If the area of the triangle is 25 square centimeters, what is the length of the hypotenuse (in centimeters)? Write your answer to two decimal places.

  (3 pts) 3.

A recipe calls for $1⁤\frac{1}{2}$ cups of sugar and 4 cups of flour. If we increase the recipe to use $2⁤\frac{1}{2}$ cups of sugar, how many cups of flour should we use? Write your answer as a mixed number where the numerator and the denominator of the fraction have no common factors.

  (3 pts) 4.

What is the largest prime factor of $2023$?

  (3 pts) 5.

Four students take a quiz. The highest score is 10, and the lowest is 2. If the mean (average) score is 7, what is the median score?

  (3 pts) 6.

A rectangular prism is 8 cm long, 4 cm wide, and 3 cm high. What is the total area (in square cm) of all of the faces of the prism?

  (3 pts) 7.

Find and simplify

Write your answer as a simple fraction where the numerator and denominator have no common factors.

  (2 pts) 1. ${\displaystyle \frac{13}{60}}$

A class has 22 students. Every student in the class has a dog, a cat, or both a cat and a dog. If 18 students have a dog (with or without a cat), and 9 students have a cat (with or without a dog), how many students have both a cat and a dog?

  (3 pts) 2. $5$

A right triangle has one leg exactly twice as long as the other leg. If the area of the triangle is 25 square centimeters, what is the length of the hypotenuse (in centimeters)? Write your answer to two decimal places.

  (3 pts) 3. $11.18$

A recipe calls for $1⁤\frac{1}{2}$ cups of sugar and 4 cups of flour. If we increase the recipe to use $2⁤\frac{1}{2}$ cups of sugar, how many cups of flour should we use? Write your answer as a mixed number where the numerator and the denominator of the fraction have no common factors.

  (3 pts) 4. $6⁤\frac{2}{3}$

What is the largest prime factor of $2023$?

  (3 pts) 5. $17$

Four students take a quiz. The highest score is 10, and the lowest is 2. If the mean (average) score is 7, what is the median score?

  (3 pts) 6. $8$

A rectangular prism is 8 cm long, 4 cm wide, and 3 cm high. What is the total area (in square cm) of all of the faces of the prism?

  (3 pts) 7. $136$

Find and simplify

Write your answer as a simple fraction where the numerator and denominator have no common factors.

  (2 pts) 1. ${\displaystyle \frac{13}{60}}$

Solution:

A class has 22 students. Every student in the class has a dog, a cat, or both a cat and a dog. If 18 students have a dog (with or without a cat), and 9 students have a cat (with or without a dog), how many students have both a cat and a dog?

  (3 pts) 2. $5$

Solution: If 22 students have a pet, and 18 have a dog, 4 must only have a cat. If 9 have a cat at all, 5 must have both.

A right triangle has one leg exactly twice as long as the other leg. If the area of the triangle is 25 square centimeters, what is the length of the hypotenuse (in centimeters)? Write your answer to two decimal places.

  (3 pts) 3. $11.18$

Solution: Let the legs have lengths $a$ and $2a$. Then the area must be $\frac{1}{2}a\cdot 2a=a^2$, so that $a=5$. So the length of the hypotenuse must be $\sqrt{5^2+{10}^2}=\sqrt{125}\approx 11.18$.

A recipe calls for $1⁤\frac{1}{2}$ cups of sugar and 4 cups of flour. If we increase the recipe to use $2⁤\frac{1}{2}$ cups of sugar, how many cups of flour should we use? Write your answer as a mixed number where the numerator and the denominator of the fraction have no common factors.

  (3 pts) 4. $6⁤\frac{2}{3}$

Solution:

What is the largest prime factor of $2023$?

  (3 pts) 5. $17$

Solution: $2023=7\cdot 17\cdot 17$

Four students take a quiz. The highest score is 10, and the lowest is 2. If the mean (average) score is 7, what is the median score?

  (3 pts) 6. $8$

Solution: Since the mean is 7, the total must be $7\cdot 4=28$. Subtracting 10 and 2, the middle two values must sum to 16. They could be 10 and 6, 9 and 7, or 8 and 8, but no matter what, their average, and the median for the whole set, must be 8.

A rectangular prism is 8 cm long, 4 cm wide, and 3 cm high. What is the total area (in square cm) of all of the faces of the prism?

  (3 pts) 7. $136$

Solution: There are 2 faces which are rectangles with sides of lengths equal to each pair of lengths. That is, 2 that are 8 cm by 4 cm, 2 that are 8 cm by 3 cm, and 2 that are 4 cm by 3 cm. So the total surface area is $2\cdot 8\cdot 4+2\cdot 8\cdot 3+2\cdot 4\cdot 3=136c{m}^2$.

Tom dug a circular quarry in his sandbox that is 8 inches in diameter and 4 inches deep. Callie also dug a circular quarry that is 7 inches in diameter and 5 inches deep. Whose quarry will hold the most water? Tom or Callie?

  (2 pts) 1.

Evaluate the expression: $3+3^2-3\sqrt{9}+9÷3$

  (3 pts) 2.

Find all solutions for $x$:

  (3 pts) 3.

Given values $a$ and $b$ on a real number line, which expression will always give the distance between $a$ and $b$ ?

A. $a-b$ B. $b-a$ C. $|a-b|$ D. $|a+b|$

  (3 pts) 4.

Find the solution to the system of equations and put them in $(x,y)$ form. Simplify answers if possible.

  (3 pts) 5.

Factor completely: $x^4-169x^2$

  (3 pts) 6.

Simplify the expression with no negative exponents.

  (3 pts) 7.

Tom dug a circular quarry in his sandbox that is 8 inches in diameter and 4 inches deep. Callie also dug a circular quarry that is 7 inches in diameter and 5 inches deep. Whose quarry will hold the most water? Tom or Callie?

  (2 pts) 1. Tom

Evaluate the expression: $3+3^2-3\sqrt{9}+9÷3$

  (3 pts) 2. 6

Find all solutions for $x$:

  (3 pts) 3. $x=1,2$

Given values $a$ and $b$ on a real number line, which expression will always give the distance between $a$ and $b$ ?

A. $a-b$ B. $b-a$ C. $|a-b|$ D. $|a+b|$

  (3 pts) 4. C.

Find the solution to the system of equations and put them in $(x,y)$ form. Simplify answers if possible.

  (3 pts) 5. $({\displaystyle \frac{3}{4}},-1)$

Factor completely: $x^4-169x^2$

  (3 pts) 6. $x^2(x+13)(x-13)$

Simplify the expression with no negative exponents.

  (3 pts) 7. ${\displaystyle \frac{-a^6}{6b^{13}{c}^4}}$

Evaluate the following expression as a decimal to the nearest thousandth.

  (2 pts) 1.

Jack and Jill start running laps of the school track (in the same direction) when the school clock shows the time as 3:00:00 PM. Jack takes 2 minute and 45 seconds to run each lap and Jill takes 3 minutes and 15 seconds. They run until the first time that they reach the starting line at exactly the same time. What time does the school clock show when they stop?

  (3 pts) 2.

The rectangle in the figure below has an area of 32 cm². What is the area in cm² of the shaded region inside the rectangle and exterior to the circles?

  (3 pts) 3.

Two six-sided dice are rolled. What is the probability that the product of the two numbers is a perfect square?

  (3 pts) 4.

If you add all the dates of the Sundays in a month and the total is 75, what day of the week is the 17th day of the month?

  (3 pts) 5.

Find the set of all $x$ for which ${(x+1)}^4-5{(x+1)}^2+4=0$.

  (3 pts) 6.

Jay wants to buy a phone which has a regular price of $449. If Jay lives in a city with 5% sales tax and has saved up $420, what is the smallest whole number percentage discount on the phone which will allow Jay to buy the phone?

  (3 pts) 7.

Evaluate the following expression as a decimal to the nearest thousandth.

  (2 pts) 1. $0.875$

Jack and Jill start running laps of the school track (in the same direction) when the school clock shows the time as 3:00:00 PM. Jack takes 2 minute and 45 seconds to run each lap and Jill takes 3 minutes and 15 seconds. They run until the first time that they reach the starting line at exactly the same time. What time does the school clock show when they stop?

  (3 pts) 2. 3:35:45 PM

The rectangle in the figure below has an area of 32 cm². What is the area in cm² of the shaded region inside the rectangle and exterior to the circles?

  (3 pts) 3. $32-8\pi \approx 6.87$

Two six-sided dice are rolled. What is the probability that the product of the two numbers is a perfect square?

  (3 pts) 4. ${\displaystyle \frac{2}{9}}$ or $0.22$ or $22$%

If you add all the dates of the Sundays in a month and the total is 75, what day of the week is the 17th day of the month?

  (3 pts) 5. Tuesday

Find the set of all $x$ for which ${(x+1)}^4-5{(x+1)}^2+4=0$.

  (3 pts) 6. $\{-3,-2,0,1\}$

Jay wants to buy a phone which has a regular price of $449. If Jay lives in a city with 5% sales tax and has saved up $420, what is the smallest whole number percentage discount on the phone which will allow Jay to buy the phone?

  (3 pts) 7. $11$

Evaluate the following expression as a decimal to the nearest thousandth.

  (2 pts) 1. $0.875$

Jack and Jill start running laps of the school track (in the same direction) when the school clock shows the time as 3:00:00 PM. Jack takes 2 minute and 45 seconds to run each lap and Jill takes 3 minutes and 15 seconds. They run until the first time that they reach the starting line at exactly the same time. What time does the school clock show when they stop?

  (3 pts) 2. 3:35:45 PM

The rectangle in the figure below has an area of 32 cm². What is the area in cm² of the shaded region inside the rectangle and exterior to the circles?

  (3 pts) 3. $32-8\pi \approx 6.87$

Two six-sided dice are rolled. What is the probability that the product of the two numbers is a perfect square?

  (3 pts) 4. ${\displaystyle \frac{2}{9}}$ or $0.22$ or $22$%

If you add all the dates of the Sundays in a month and the total is 75, what day of the week is the 17th day of the month?

  (3 pts) 5. Tuesday

Find the set of all $x$ for which ${(x+1)}^4-5{(x+1)}^2+4=0$.

  (3 pts) 6. $\{-3,-2,0,1\}$

Jay wants to buy a phone which has a regular price of $449. If Jay lives in a city with 5% sales tax and has saved up $420, what is the smallest whole number percentage discount on the phone which will allow Jay to buy the phone?

  (3 pts) 7. $11$

Evaluate

  (2 pts) 1.

A math club at Oxford High School consists of only ninth-grade and tenth-grade students. If ${\displaystyle \frac{3}{10}}$ of the students in the math club are from ninth grade and there are 60 more tenth-grade students than ninth-grade students, how many tenth-grade students are there in the math club?

  (3 pts) 2.

Two standard six-sided dice are thrown, and the dice are considered fair. What is the probability that the sum of the two numbers facing up is at least 11?

  (3 pts) 3.

How many gallons of paint would be needed to paint the sides of an uncovered tank that is 100 feet long, 25 feet wide and 15 feet high if one gallon of paint will cover 200 square feet?

  (3 pts) 4.

A factory makes pizza boxes using 8 cutting machines. Each machine cuts 12 boxes every ${\displaystyle \frac{3}{4}}$ minute. How many boxes can all 8 machines cut in one minute?

  (3 pts) 5.

Find the $x$-intercept(s) for

  (3 pts) 6.

Simplify the expression

  (3 pts) 7.

Evaluate

  (2 pts) 1. $81$

A math club at Oxford High School consists of only ninth-grade and tenth-grade students. If ${\displaystyle \frac{3}{10}}$ of the students in the math club are from ninth grade and there are 60 more tenth-grade students than ninth-grade students, how many tenth-grade students are there in the math club?

  (3 pts) 2. $105$

Two standard six-sided dice are thrown, and the dice are considered fair. What is the probability that the sum of the two numbers facing up is at least 11?

  (3 pts) 3. ${\displaystyle \frac{1}{12}}$

How many gallons of paint would be needed to paint the sides of an uncovered tank that is 100 feet long, 25 feet wide and 15 feet high if one gallon of paint will cover 200 square feet?

  (3 pts) 4. $18.75$ gallons

A factory makes pizza boxes using 8 cutting machines. Each machine cuts 12 boxes every ${\displaystyle \frac{3}{4}}$ minute. How many boxes can all 8 machines cut in one minute?

  (3 pts) 5. $128$ boxes

Find the $x$-intercept(s) for

  (3 pts) 6. $0$ and $6$

Simplify the expression

  (3 pts) 7. ${\displaystyle \frac{1}{8}}$

A square is inscribed in a circle of radius 5. A smaller circle is inscribed in the square. Find the area of the smaller circle.

  (20 pts) 1.

The area of a triangle is $10$. If the height of the triangle is twice the base, what is the base?

  (20 pts) 2.

Find positive numbers $a$ and $b$ with $a^{2}b=2023$ and $a{b}^2=325$.

  (20 pts) 3.

In a class your grade is determined by the average of five exam scores. If the average of your first three exams is 75%, what must the average of your last two exams be to get an 83% in the class?

  (20 pts) 4.

A bacterial culture doubles every $20$ minutes. There are $1500$ bacteria at noon. How many bacteria will there be at 4 pm later that same day?

  (20 pts) 5.

What is the greatest common divisor of $360$ and $525$?

  (20 pts) 6.

Find the positive solution to $2^{x^2-x}=2\cdot 8^x$.

  (20 pts) 7.

Suppose the price of an item is increased by $15$% and then later the new price is decreased by $10$%. Compared to the original price, what percentage increase does the final price represent?

  (20 pts) 8.

Solve the system of equations

  (20 pts) 9.

Suppose $a$ and $b$ are positive real numbers such that $a^2+b^2=53$ and $ab=2$. What is $|a-b|$?

  (20 pts) 10.

A square is inscribed in a circle of radius 5. A smaller circle is inscribed in the square. Find the area of the smaller circle.

  (20 pts) 1. $25\pi /2\approx 39.27$

The area of a triangle is $10$. If the height of the triangle is twice the base, what is the base?

  (20 pts) 2. $\sqrt{10}\approx 3.16$

Find positive numbers $a$ and $b$ with $a^{2}b=2023$ and $a{b}^2=325$.

  (20 pts) 3. $b\approx 3.74$, $a\approx 23.26$

In a class your grade is determined by the average of five exam scores. If the average of your first three exams is 75%, what must the average of your last two exams be to get an 83% in the class?

  (20 pts) 4. $95$%

A bacterial culture doubles every $20$ minutes. There are $1500$ bacteria at noon. How many bacteria will there be at 4 pm later that same day?

  (20 pts) 5. $6144000$

What is the greatest common divisor of $360$ and $525$?

  (20 pts) 6. $15$

Find the positive solution to $2^{x^2-x}=2\cdot 8^x$.

  (20 pts) 7. $2+\sqrt{5}\approx 4.24$

Suppose the price of an item is increased by $15$% and then later the new price is decreased by $10$%. Compared to the original price, what percentage increase does the final price represent?

  (20 pts) 8. $3.5$%

Solve the system of equations

  (20 pts) 9. $x=6074$, $y=-8099$

Suppose $a$ and $b$ are positive real numbers such that $a^2+b^2=53$ and $ab=2$. What is $|a-b|$?

  (20 pts) 10. $7$

The average weight of five turkeys on a scale is 13 pounds. If a 7-pound turkey is removed from the scale, what is the average weight of the four remaining turkeys?

  (20 pts) 1.

You used $\frac{1}{2}$ can of paint to paint $\frac{3}{5}$ of a wall. How many cans of paint are needed to paint the whole wall?

  (20 pts) 2.

What is the area of a triangle with sides twice as long as those of a triangle with an area of 50 square inches?

  (20 pts) 3.

If $S_{66}$ represents the sum of all the factors of the number 66 and $S_{70}$ represents the sum of all of the factors of the number 70, then find the value of $S_{70}-S_{66}$.

  (20 pts) 4.

The prize money at a raffle was given out as follows:

20 people received $5
10 people received $10
5 people received $20
2 people received $50
1 person received $100
1 person received $500
1 person received $1000

What is the difference between the mean cash prize and median cash prize?

  (20 pts) 5.

The surface area of a cube is 150 cm². What is the volume of the cube?

  (20 pts) 6.

If you use the eight digits $1,2,3,4,5,6,7,$ and, $9$ each once and only once to form 4 two-digit prime numbers, what will be the sum of the four prime numbers you created?

1. A.

   190

2. B.

   253

3. C.

   172

4. D.

   235

5. E.

   None of these

  (20 pts) 7.

If the ratio of the number of widgets X has to the number Z has is two to seven, and their total number of widgets is 1908, how many more widgets does Z have than X?

1. A.

   1050

2. B.

   1060

3. C.

   1484

4. D.

   1030

5. E.

   None of these

  (20 pts) 8.

A rectangular solid (shoebox) has dimensions 4 high, 6 wide, and 7 deep. How long is the diagonal through the interior of the solid?

1. A.

   $\sqrt{98}$

2. B.

   $\sqrt{102}$

3. C.

   $\sqrt{85}$

4. D.

   $\sqrt{101}$

5. E.

   None of these

  (20 pts) 9.

Two cars leave at the same time from the same starting point traveling “down and back.” Car A goes 60 mph for 2 hours then returns on the same road going 40 mph. At what constant speed, in mph, would car B need to travel to finish the course at the same time as car A?

  (20 pts) 10.

The average weight of five turkeys on a scale is 13 pounds. If a 7-pound turkey is removed from the scale, what is the average weight of the four remaining turkeys?

  (20 pts) 1. $14.5$ pounds

You used $\frac{1}{2}$ can of paint to paint $\frac{3}{5}$ of a wall. How many cans of paint are needed to paint the whole wall?

  (20 pts) 2. ${\displaystyle \frac{5}{6}}$

What is the area of a triangle with sides twice as long as those of a triangle with an area of 50 square inches?

  (20 pts) 3. $200$ in²

If $S_{66}$ represents the sum of all the factors of the number 66 and $S_{70}$ represents the sum of all of the factors of the number 70, then find the value of $S_{70}-S_{66}$.

  (20 pts) 4. $0$

The prize money at a raffle was given out as follows:

20 people received $5
10 people received $10
5 people received $20
2 people received $50
1 person received $100
1 person received $500
1 person received $1000

What is the difference between the mean cash prize and median cash prize?

  (20 pts) 5. $42.50

The surface area of a cube is 150 cm². What is the volume of the cube?

  (20 pts) 6. 125 cm²

If you use the eight digits $1,2,3,4,5,6,7,$ and, $9$ each once and only once to form 4 two-digit prime numbers, what will be the sum of the four prime numbers you created?

1. A.

   190

2. B.

   253

3. C.

   172

4. D.

   235

5. E.

   None of these

  (20 pts) 7. A) 190

If the ratio of the number of widgets X has to the number Z has is two to seven, and their total number of widgets is 1908, how many more widgets does Z have than X?

1. A.

   1050

2. B.

   1060

3. C.

   1484

4. D.

   1030

5. E.

   None of these

  (20 pts) 8. B) 1060

A rectangular solid (shoebox) has dimensions 4 high, 6 wide, and 7 deep. How long is the diagonal through the interior of the solid?

1. A.

   $\sqrt{98}$

2. B.

   $\sqrt{102}$

3. C.

   $\sqrt{85}$

4. D.

   $\sqrt{101}$

5. E.

   None of these

  (20 pts) 9. D) $\sqrt{101}$

Two cars leave at the same time from the same starting point traveling “down and back.” Car A goes 60 mph for 2 hours then returns on the same road going 40 mph. At what constant speed, in mph, would car B need to travel to finish the course at the same time as car A?

  (20 pts) 10. 48 mph