# Grade 8 Math Practice Test

Online grade 8 math practice test questions are presented along with their detailed solutions .

## 1 - Numbers

1. Evaluate the following expressions.

2. Which of the following is NOT a rational number?

3. What is the value of the digit 5 in the number 34.6597 ?

## 2 - Sequences

4. What is the next term in the sequence?

5. What is the next term in the sequence?

6. A sequence of numbers is defined by

a) Find the first five terms of the sequence starting with n = 1.
b) Is the given sequence arithmetic, geometric or neither?

7. A sequence of numbers is defined by

a) Find the first five terms of the sequence starting with n = 1.
b) Is the given sequence arithmetic, geometric or neither?

 

## 3 - Sets

8. Sets $$S_1$$ and $$S_2$$ are defined as follows?
$$S_1 = \{0, 2, 9, 12\} \quad$$ and $$\quad S_2 = \{2, 9, 10, 11, 12\}$$
Find the elements of the sets
a) The intersection of $$S_1$$ and $$S_2$$ written as $$S_1 \cap S_2$$
and
b) The union of $$S_1$$ and $$S_2$$ written as $$S_1 \cup S_2$$

9. Let the following sets be defined as follows
$$Q = \text{All Rational Numbers}$$
$$P = \text{All Irrational Numbers}$$
$$R = \text{All Real Numbers}$$
Which of the following is true?
a)   $$Q \cap P = R$$      b)   $$Q \cup P = R$$       c)   $$Q \cup P = \text{Empty Set}$$       d)   $$Q \cap P = \text{Empty Set}$$

## 4 - Factors, Multiples and Divisibility of Numbers

10. Express the numbers as a product of prime factors
a)   $$345$$        b)   $$150$$        c)   $$210$$

11. What is the Greatest Common Factor (GCF) of $$100$$ and $$180$$?

12. What is the Least Common Multiple (LCM) of $$100$$ and $$15$$?

13.   Which of the following numbers is divisible by $$3$$?
a)   $$101899$$        b)   $$900234$$        c)   $$134567280$$

14.   Which of the following numbers is divisible by $$4$$?
a)   $$189001$$        b)   $$1005612$$        c)   $$1003456024$$

15.   Which of the following numbers is divisible by $$6$$?
a)   $$234$$        b)   $$12345$$        c)   $$12114290910$$

## 5 - Fractions and Mixed Numbers

16. Which of the following pairs of fractions are equivalent?
a)   $$\displaystyle \frac{10}{15} \; , \; \frac{7}{3}$$        b)   $$\displaystyle \frac{8}{12} \; , \; \frac{2}{3}$$        c)   $$\displaystyle \frac{7}{12} \; , \; \frac{21}{36}$$

17. Evaluate the following expressions
a)   $$\displaystyle \frac{2}{5} + \frac{3}{10} - \frac{1}{15}$$        b)   $$\displaystyle \frac{7}{16} \times \frac{4}{14}$$        c)   $$\displaystyle \frac{11}{2} \div 4$$

d)   $$\displaystyle 4 \frac{3}{4} - 1 \frac{1}{2} + 1 \frac{1}{8}$$        e)   $$\displaystyle 1 \frac{3}{4} \div \left(3 + \frac{1}{3} \right)$$

18. Write the following expressions as fractions or mixed numbers in reduced form
a)   $$0.2 \div 0.6$$        b)   $$1 \div 0.4$$

19. Dalia spends the quarter of her salary on food and beverages. She spends the fifth of her money for food and beverages on soft drinks and the sixth of her money for food and beverages on cookies.
What fraction of Dalia's salary is spent on soft drinks and cookies?

20. James spends two hours a day on homework during weekdays (Monday to Friday inclusive). During the same period of time, Ben spends 3/4 as much time as James on homework and Linda spend 5/4 as much time as James on homework.
How many hours are spent by Ben and Linda on homework during weekdays (Monday to Friday inclusive)?

21. Sara made one liter and a half of juice. She has glasses that can hold one sixth of a liter .
How many glasses will Sara be able to fill?

## 6 - Exponents and Scientific Notation

22. Evaluate the expressions
a)   $$(-2)^3 - 5^3 + (-3)^4$$        b)   $$\quad (-1)^{-3} - 5^0 + \frac{4^2}{(-2)^4}$$        c)   $$\quad \left( \frac{3}{4} \right)^2 + \left( \frac{4}{3} \right)^{-2}$$

23. Write in exponential form with base ten
a)   $$10000$$        b)   $$0.0000001$$        c)   $$\frac{1}{100000}$$

24. Write in Scientific Notation
a)   $$12.4 \times 10^3$$        b)   $$0.0023 \times 10^{-2}$$        c)   $$\frac{12}{100000}$$

## 7 - Roots

25. Simplify
a)   $$\sqrt{16}$$        b)   $$\quad \sqrt{9}$$        b)   $$\quad \sqrt[3]{8}$$

26. Reduce to simplest form
a)   $$\sqrt{3 \times 25}$$        b)   $$\quad \sqrt{36 \times 5}$$        b)   $$\quad \sqrt[3]{8 \times 7}$$

## 8 - Proportionality and Related Problems

27.   Leila went for a 5 hours walk and the graph below shows the distance $$d$$ (in km) walked after time $$t$$ (in hours).
a) Write an equation of the relationship between the distance $$d$$ and the $$time$$ of the form $d = k \times t$ b) If she started walking at 8 am, at what time was she 10 km away from her starting point?

28.   Does the table below indicate that $$y$$ is proportional to $$x$$?

a) If the answer to part a) is yes , find the coefficient of proportionality $$k$$ such that $y = k \; x$ b) Assuming the above equation is valid for values of $$x$$ higher than $$6$$, find $$y$$ for $$x = 10.2$$.

29.   A water tap fills a tank of 10 Liters in 2 minutes. The same tap fills a tank of 20 liters in 4 minutes and a tank of 30 liters in 6 minutes.
a) Let $$t$$ be the time it takes to fill a tank and $$V$$ the volume of the tank filled. Use the information given above to graph points with $$V$$ on the vertical axis and $$t$$ on the horizontal axis.
b) Is there a proportionality relationship between $$V$$ and $$t$$?
c) If the answer to part b) is yes, find the coefficient $$k$$ such that $$V = k \; t$$?
d) Assuming that the above equation is valid for values of $$t$$ higher than $$6$$ minutes, find the time it takes to fill a tank of 100 Liters.

## 9 - Percent and Related Problems

30. The original price of an item was $$120$$ and is increased by 12%. What is the price of this item after the increase?

31. Jimmy spends $$50\%$$ of his salary on rent and bills and $$15\%$$ of the amount spent on rent and bills is spent on bills.
What percent of Jimmy's salary is spent on bills?

32. A meal in a restaurant costs $40 before taxes. You pay 15% taxes and a tip of 5% of the cost including the taxes. How much do you pay for this meal? 33. Kamelea has a monthly salary of $$5000$$. She spends $$400$$ on clothes, $$1200$$ on rent, $$200$$ on bills, $$1200$$ on food, beverages and other spending when going out, $$600$$ on gas and insurance for her car per month, and the rest is saved. What percent of Kamelea's monthly salary is saved? 34. $$10\%$$ of the third of a number is equal to 3. What is the number? 35. From one year to the next, the average price of gas increased from \$3 per gallon to \\$4 per gallon in the US and from 1.5 Euros to 2 Euros per liter in France. Which country saw a large percentage of gas increase?

## 10 - Convert Units of Measurement

36. How many many meters (m) are in $$10.5 \text{ ft}$$ knowing that $$1 \text{ m} = 3.28084 \text{ ft}$$?

37. How many yards (yd) are in 1.3 kilometers knowing that $$1 \text{ km} = 1093.61 \text{ yd}$$?

38. How many square yards $$yd^2$$ are in $$1.2$$ square meters ( $$m^2$$ ) knowing that $$1 \; m = 1.09361 \; yd$$?

39. Convert the rate of $$100 \; km / hr$$ to hour into meters per second?

## 11 - Evaluate Expressions

40. Evaluate the expression $$\; \frac{1}{x+2} - \frac{1}{x-2} \;$$ for $$x = 1$$

41. Evaluate the expression $$\; | \frac{-x+1}{-6} | + x^2 - 1\;$$ for $$x = -5$$

42. Evaluate the expression $$\; 2^a - \sqrt{b^2} \;$$ for $$a = 2$$ and $$b = -2$$

## 12 - Algebra

Review
The distributive property in algebra may be used to expand as follows $a (x + y ) = \color{red}a \times x + \color{red}a \times y$ The distributive property may also be used in reverse to factor as follows $\color{red}a \times x + \color{red}a \times y = \color{red}a (x + y )$

43. Simplify the expressions
a)   $$3 (x + 2) + x - 12$$        b)   $$\displaystyle \frac{1}{5}( 15 x + 20) + 2x + 4$$        c)   $$0.2 ( 5 x + 10) + 3x - 4$$

44. Simplify the expressions
a)   $$2x \times 3 x$$        b)   $$\displaystyle \frac{1}{2}x \times \frac{4}{5} x$$        c)   $$3x^2 \times 5 x^3$$

45.  Factor the expressions
a)   $$21 x + 7$$        b)   $$24 - 20 x$$        c)   $$8 b - 4 a + 32$$

## 13 - Equation with One Variable and Related Problems

46.  Solve the equations
a)   $$3(x - 2 ) = 3$$        b)   $$2(9 - x) = - (x + 5)$$

c)   $$\displaystyle \frac{x+1}{3} = 6$$        d)   $$4 \left(x + \displaystyle \frac{1}{4} \right) = -15$$        e)   $$x - \displaystyle \frac{x}{2} = 3$$

47. A rectangular garden has a length of 12 meters and a width of 8 meters (Diagram below not drawn to scales). A path of width $$x$$ is to be constructed along the perimeter of the garden such that the outer perimeter is equal to twice the perimeter of the garden.

a) Write an equation in $$x$$.
b) Solve for $$x$$ the equation obtained in part a)
c) Find the length and width of the outer part of the garden
d) Find the total area of the garden and the path
e) Find the area of the garden.
f) Find the area of the path.

48. When 10 is subtracted from twice a number and the result is multiplied by half the answer is 5. What is the original number?

## 14 - Inequality with One Variable

49.  Solve the inequalities
a)   $$x+2 \lt 4$$        b)   $$2(x + 3)\ge 2$$      c)   $$-3x+2 \le 11$$      d)   $$\frac{4x+1}{2} \ge x+3$$

## 15 - Functions

50.   Which of the following ordered pairs represents a function?
a)   $$\{ (1,2) , (3,4) , (5,7) , (5,9) \}$$        b)   $$\{ (-1,-2) , (3,4) , (5,7) , (7,9) \}$$        c)   $$\{ (3,3) , (9,4) , (5,7) , (9,0) \}$$

51.   Which of the graphs below may be the graph of a linear function? Find its equation.

52.   Given the function $$y = 2 x + 1$$,
a) find the value of $$y$$ for $$x = 0$$ and $$x = 1$$.
b) Use the values of $$y$$ calculated in part a) to graph the function $$y = 2 x + 1$$

53.   The graph of two functions is shown below.

a) Without doing any calculations, which function has a higher rate of change?
b) Find the coordinates of two points in each graph.
c) Use the coordinates of the points above to find the rate of change of each function
d) Compare the rate of change and confirm your answer to part a) above.

## 16 - Two-Dimensional Figures

54.   The legs of a right triangles have lengths equal to 6 cm and 8 cm. What is the length of the hypotenuse of the triangle?

55.   In the figure below, three straight lines intersect at point $$O$$ (Figure not drawn to scales). Sizes of some angles are given by $$\angle AOB = 31^{\circ}$$ and $$\angle AOC = 79^{\circ}$$
Find the size of angle $$\angle EOF$$.

56.   How many lines of symmetry does a square have?

57.   Use the figure below to find the measures of all 8 angles $$\quad m\angle 1$$ , $$m\angle 2$$, ....., $$m\angle 8 \quad$$ given that $m \angle 1 = 40^{\circ}$

## 17 - Perimeter and Area of Planar Figures

58.   Calculate the area of a circle whose diameter is $$20$$ cm.
59.   Calculate the area of a right triangle with one leg of length $$16$$ cm and the hypotenuse of length $$20$$ cm.
60.   In the figure below ABDE is a square and FC is a line of symmetry of the arrow in blue . Find the area of the arrow (shaded in blue).
61.   Find the area of the shape (shaded in blue) bounded by a semicircle of diameter DE on the right and the isosceles triangle ABG on the left. Round the answer to two decimal places.

## 18 - Volumes and Surface Area

62. Find the volume and surface area (excluding the bottom) of the silo made up of cylinder with height h = 10 m and half a sphere on top with radius r = 6 m?

63.   Below is a rectangular prism split by a plane through the diagonals (red) into two congruent triangular prisms.
Find the volume and surface area of one of the triangular prism obtained.

## 19 - Data and Graphs

64. The line plot below shows the average high temperature in degree Celsius over 12 months of the year in Ottawa, Canada.
a) What is the average high temperature of the coldest month of the year?
b) What is the average high temperature of the hottest month of the year?
c) What is the difference in the average high temperature between the coldest and hottest months?
d) What is the smallest increase in the average high temperature between any two months of the year?
e) What is the smallest decrease in the average high temperature between any two months of the year?

65. The scores in a test of a group of 25 students are given below
$$31, 44, 54, 69, 45, 55, 91, 76, 76, 77, 78, 70, 79, 67, 60, 85, 84, 89, 56, 67, 86, 64, 97, 88, 92$$
a) Order the data from the smallest to the largest values.
b) Find the range of the given data.
c) Create a frequency table classes starting from the class $$30 - 39$$ to cover all data values and make a table with the count (frequency) in each class.
d) Make a histogram of the given data.
e) If anyone who scored below 60 has failed, what percent of the students failed in this group?

## 20 - Statistics

66. Find the median, the lower quartile and the upper quartile of the data set: $$\{ 9 , 4 , 3 , 2 , 3 , 0 , 2 , 3 , 1 , 9 , 10 \}$$

67. Mark scored $$83, 94, 97$$ and $$93$$ on his first 4 quizzes. What should be the score on his fifth quiz so that the average of the 5 quizzes is at least 90?

## 21 - Probabilities

68.   Dalida is rolling a fair die with 6 sides labeled 1 to 6.
What is the probability of getting an even number?

69.   Katya is tossing a fair coin and rolling a fair die with 6 sides labeled 1 to 6.
a) What is the probability of getting a tail (coin) and a 4 (die)?
b) What is the probability of getting head (coin) and an odd number (die)?

70.   Ben selects one of three different cards, labeled 1 to 3, at random, read the number on the card and then returns the card back. He selects, at random, a card for the second time.
a) What is the probability of selecting 3 in both random selections?
b) What is the probability of selecting the same number in both random selections?

71.   Linda surveyed 20 students in her school about their favorite color and 5 said blue was their favorite color, 6 said brown was their favorite number. What is the probability that the next student surveyed will pick a color that is neither blue nor brown?