Algebra questions with solutions for grade 8

Grade 8 algebra questions are presented with solutions. Questions include solving equations, simplifying expressions, and expressions with fractions.

1. Simplify the following algebraic expressions.

   A) \(-2x + 5 + 10x - 9\)

   B) \(3(x + 7) + 2(-x + 4) + 5x\)

2. Simplify the expressions.

   A) \(\dfrac{2x - 6}{2}\)

   B) \(\dfrac{-x - 2}{x + 2}\)

   C) \(\dfrac{5x - 5}{10}\)

3. Solve for x the following equations.

   A) \(-x = 6\)

   B) \(2x - 8 = -x + 4\)

   C) \(2x + \dfrac{1}{2} = \dfrac{2}{3}\)

   D) \(\dfrac{x}{3} + 2 = 5\)

   E) \(\dfrac{-5}{x} = 2\)

4. Evaluate for the given values of \(x\) and \( y \).

   A) \(x^2 - y^2\), for \(x = 4\) and \(y = 5\)

   B) \(|4x - 2y|\), for \(x = -2\) and \(y = 3\)

   C) \(3x^3 - 4y^4\), for \(x = -1\) and \(y = -2\)

5. Solve the following inequalities.

   A) \(x + 6 < 0\)

   B) \(x + 1 > 5\)

   C) \(2(x - 2) < 12\)

6. What is the reciprocal of each of the following numbers?

   A) \(-1\)

   B) \(0\)

   C) \(\dfrac{3}{4}\)

   D) \(2\dfrac{5}{7}\)

   E) \(0.02\)

7. Evaluate the following expressions involving mixed numbers.

   A) \(3\dfrac{3}{4} + 6\dfrac{1}{7}\)

   B) \((1\dfrac{3}{5}) \times (3\dfrac{1}{3}) - 2\dfrac{1}{2}\)

   C) \((5\dfrac{2}{3}) \div (4\dfrac{1}{5})\)

   D) \((3\dfrac{4}{7} - 1\dfrac{1}{2}) \div (2\dfrac{3}{8} + 2\dfrac{1}{4})\)

8. Evaluate the following exponential expressions.

   A) \(-4^2\)

   B) \((-2)^3\)

   C) \((-2)^4\)

   D) \(1000^0\)

   E) \(566^1\)

9. Convert to fractions and write in simplest form.

   A) \(0.02\)

   B) \(12\%\)

   C) \(0.5\%\)

   D) \(1.12\)

10. Convert to decimals.

    A) \(\dfrac{1}{5}\)

    B) \(120\%\)

    C) \(0.2\%\)

    D) \(4\dfrac{8}{5}\)

11. Convert to percent.

    A) \(\dfrac{3}{10}\)

    B) \(1.4\)

    C) \(123.45\)

    D) \(2\dfrac{4}{5}\)

12. Which of these numbers is divisible by 3?

    A) \(156312\)

    B) \(176314\)

13. Which of these numbers is divisible by 4?

    A) \(3432\)

    B) \(1257\)

14. Which of these numbers is divisible by 6?

    A) \(1233\)

    B) \(3432\)

15. Which of these numbers is divisible by 9?

    A) \(2538\)

    B) \(1451\)

16. Evaluate \(8x + 7\) given that \(x - 3 = 10\).

Solutions to the Above Questions

1. A) \(-2x + 5 + 10x - 9\)

   \(= (10x - 2x) + (5 - 9)\)

   put like terms together

   \(= 8x - 4\)

   group

   B) \(3(x + 7) + 2(-x + 4) + 5x\)

   \(= 3x + 21 - 2x + 8 + 5x\)

   expand

   \(= (3x - 2x + 5x) + (21 + 8)\)

   put like terms together

   \(= 6x + 29\)

   group

2. A) \(\dfrac{2x - 6}{2}\)

   \(= \dfrac{2(x - 3)}{2}\)

   factor 2 in numerator

   \(= x - 3\)

   divide numerator and denominator by 2 to simplify

   B) \(\dfrac{-x - 2}{x + 2}\)

   \(= \dfrac{-1(x + 2)}{x + 2}\)

   factor -1 in numerator

   \(= -1\)

   divide numerator and denominator by x + 2 to simplify

   C) \(\dfrac{5x - 5}{10}\)

   \(= \dfrac{5(x - 1)}{10}\)

   factor 5 in numerator

   \(= \dfrac{x - 1}{2}\)

   divide numerator and denominator by 5 to simplify

3. A) \(-x = 6\)

   \(x = -6\)

   multiply both sides of the equation by -1

   B) \(2x - 8 = -x + 4\)

   \(2x - 8 + 8 = -x + 4 + 8\)

   add +8 to both sides of the equation

   \(2x = -x + 12\)

   group like terms

   \(2x + x = -x + 12 + x\)

   add +x to both sides

   \(3x = 12\)

   group like terms

   \(x = 4\)

   multiply both sides by 1/3

   C) \(2x + \dfrac{1}{2} = \dfrac{2}{3}\)

   \(2x + \dfrac{1}{2} - \dfrac{1}{2} = \dfrac{2}{3} - \dfrac{1}{2}\)

   subtract 1/2 from both sides

   \(2x = \dfrac{1}{6}\)

   group like terms

   \(x = \dfrac{1}{12}\)

   multiply both sides by 1/2

   D) \(\dfrac{x}{3} + 2 = 5\)

   \(\dfrac{x}{3} + 2 - 2 = 5 - 2\)

   subtract 2 from both sides

   \(\dfrac{x}{3} = 3\)

   group like terms

   \(x = 9\)

   multiply both sides by 3

   E) \(\dfrac{-5}{x} = 2\)

   \(-5 = 2x\)

   multiply both sides by x and simplify

   \(x = -\dfrac{5}{2}\)

   multiply both sides by 1/2

4. A) \(x^2 - y^2\), for \(x = 4\) and \(y = 5\)

   \(4^2 - 5^2\)

   substitute x and y by the given values

   \(= 16 - 25 = -9\)

   evaluate

   B) \(|4x - 2y|\), for \(x = -2\) and \(y = 3\)

   \(|4(-2) - 2(3)|\)

   substitute x and y by the given values

   \(= |-8 - 6| = |-14| = 14\)

   evaluate

   C) \(3x^3 - 4y^4\), for \(x = -1\) and \(y = -2\)

   \(3(-1)^3 - 4(-2)^4\)

   substitute x and y by the given values

   \(= 3(-1) - 4(16) = -3 - 64 = -67\)

   evaluate

5. A) \(x + 6 < 0\)

   \(x + 6 - 6 < 0 - 6\)

   subtract 6 from both sides

   \(x < -6\)

   group like terms

   B) \(x + 1 > 5\)

   \(x + 1 - 1 > 5 - 1\)

   subtract 1 from both sides

   \(x > 4\)

   group like terms

   C) \(2(x - 2) < 12\)

   \(x - 2 < 6\)

   multiply both sides by 1/2

   \(x - 2 + 2 < 6 + 2\)

   add 2 to both sides

   \(x < 8\)

   group like terms

6. A) \(-1\)

   \((-1) \cdot a = 1\)

   definition: a is the reciprocal of -1

   \(a = \dfrac{1}{-1} = -1\)

   solve for a; -1 is the reciprocal of -1

   B) \(0\)

   \((0) \cdot b = 1\)

   definition: b is the reciprocal of 0

   \(b = \text{undefined}\)

   no value of b satisfies the above equation

   C) \(\dfrac{3}{4}\)

   \(\dfrac{3}{4} \cdot c = 1\)

   definition: c is the reciprocal of 3/4

   \(c = \dfrac{4}{3}\)

   solve for c; c = 4/3 is the reciprocal of 3/4

   D) \(2\dfrac{5}{7}\)

   \(2\dfrac{5}{7} \cdot d = 1\)

   definition: d is the reciprocal of 2 5/7

   \(\dfrac{19}{7} \cdot d = 1\)

   convert the mixed number 2 5/7 into a fraction

   \(d = \dfrac{7}{19}\)

   solve for d; d = 7/19 is the reciprocal of 2(5/7)

   E) \(0.02\)

   \(0.02 \cdot d = 1\)

   definition: d is the reciprocal of 0.02

   \(d = \dfrac{1}{0.02} = 50\)

   solve for d; d = 50 is the reciprocal of 0.02

7. A) \(3\dfrac{3}{4} + 6\dfrac{1}{7}\)

   \(= (3 + 6) + \left(\dfrac{3}{4} + \dfrac{1}{7}\right)\)

   put the whole parts together and the fractional parts together

   \(= 9 + \left(\dfrac{21}{28} + \dfrac{4}{28}\right)\)

   use common denominator 28

   \(= 9 + \dfrac{25}{28}\)

   add fractions

   \(= 9\dfrac{25}{28}\)

   write as a mixed number

   B) \((1\dfrac{3}{5}) \times (3\dfrac{1}{3}) - 2\dfrac{1}{2}\)

   \(= \dfrac{8}{5} \times \dfrac{10}{3} - \dfrac{5}{2}\)

   convert mixed numbers to improper fractions

   \(= \dfrac{80}{15} - \dfrac{5}{2}\)

   multiply

   \(= \dfrac{16}{3} - \dfrac{5}{2}\)

   simplify the fraction

   \(= \dfrac{32}{6} - \dfrac{15}{6}\)

   use common denominator 6

   \(= \dfrac{17}{6}\)

   subtract

   \(= 2\dfrac{5}{6}\)

   convert improper fractions to mixed number

   C) \((5\dfrac{2}{3}) \div (4\dfrac{1}{5})\)

   \(= \dfrac{17}{3} \div \dfrac{21}{5}\)

   convert mixed numbers to improper fractions

   \(= \dfrac{17}{3} \times \dfrac{5}{21}\)

   multiply by the reciprocal

   \(= \dfrac{85}{63}\)

   multiply numerators and denominators

   \(= 1\dfrac{22}{63}\)

   convert improper fraction to mixed number

   D) \((3\dfrac{4}{7} - 1\dfrac{1}{2}) \div (2\dfrac{3}{8} + 2\dfrac{1}{4})\)

   \(= \left(\dfrac{25}{7} - \dfrac{3}{2}\right) \div \left(\dfrac{19}{8} + \dfrac{9}{4}\right)\)

   convert mixed numbers to improper fractions

   \(= \left(\dfrac{50}{14} - \dfrac{21}{14}\right) \div \left(\dfrac{19}{8} + \dfrac{18}{8}\right)\)

   use common denominators

   \(= \dfrac{29}{14} \div \dfrac{37}{8}\)

   subtract and add

   \(= \dfrac{29}{14} \times \dfrac{8}{37}\)

   multiply by the reciprocal

   \(= \dfrac{232}{518}\)

   multiply numerators and denominators

   \(= \dfrac{116}{259}\)

   divide numerator and denominator by 2 to reduce fraction

8. A) \(-4^2\)

   \(= -(4 \times 4) = -16\)

   expand and calculate

   B) \((-2)^3\)

   \(= (-2) \times (-2) \times (-2) = -8\)

   expand and calculate

   C) \((-2)^4\)

   \(= (-2) \times (-2) \times (-2) \times (-2) = 16\)

   expand and calculate

   D) \(1000^0\)

   \(= 1\)

   definition: any nonzero number to the power zero gives 1

   E) \(566^1\)

   \(= 566\)

   any number to the power 1 is the number itself

9. A) \(0.02 = \dfrac{2}{100} = \dfrac{1}{50}\)
   Explanation: The decimal 0.02 is read as 2 hundredths, which is the fraction 2/100. Dividing both numerator and denominator by 2 simplifies it to 1/50.

   B) \(12\% = \dfrac{12}{100} = \dfrac{3}{25}\)
   Explanation: The percent symbol (%) means "per hundred," so 12% is 12 per 100, or 12/100. Dividing both numerator and denominator by 4 gives the simplified fraction 3/25.

   C) \(0.5\% = \dfrac{0.5}{100} = \dfrac{1}{200}\)
   Explanation: 0.5% means 0.5 per 100, or 0.5/100. To eliminate the decimal in the numerator, multiply numerator and denominator by 2 to get 1/200.

   D) \(1.12 = \dfrac{112}{100} = \dfrac{28}{25}\)
   Explanation: The decimal 1.12 is read as 112 hundredths, so it is 112/100. Dividing both numerator and denominator by 4 simplifies it to 28/25.

10. A) \(\dfrac{1}{5} = 0.2\)
    Explanation: The fraction 1/5 means 1 divided by 5. Performing the division gives the decimal 0.2.

    B) \(120\% = \dfrac{120}{100} = 1.2\)
    Explanation: Percent means "per hundred," so 120% is 120/100. Dividing 120 by 100 results in the decimal 1.2.

    C) \(0.2\% = \dfrac{0.2}{100} = 0.002\)
    Explanation: 0.2% means 0.2 per 100, or 0.2/100. Dividing 0.2 by 100 moves the decimal point two places to the left, giving 0.002.

    D) \(4\dfrac{8}{5} = 4 + 1.6 = 5.6\)
    Explanation: The mixed number consists of a whole number 4 and a fraction 8/5. Since 8/5 is an improper fraction, it equals 1.6. Adding 4 and 1.6 gives the decimal 5.6.

11. A) \(\dfrac{3}{10} = \dfrac{30}{100} = 30\%\)
    Explanation: To convert a fraction to a percentage, first rewrite it with a denominator of 100. Multiplying the numerator and denominator of 3/10 by 10 gives 30/100, which is 30 percent.

    B) \(1.4 = \dfrac{140}{100} = 140\%\)
    Explanation: The decimal 1.4 is equivalent to 140/100 because moving the decimal point two places to the right converts it to hundredths. A fraction with a denominator of 100 is read as a percentage, so it equals 140%.

    C) \(123.45 = \dfrac{12345}{100} = 12345\%\)
    Explanation: To express the decimal 123.45 as a percentage, multiply it by 100, which gives 12345, and add the percent symbol. This corresponds to the fraction 12345/100.

    D) \(2\dfrac{4}{5} = 2 + \dfrac{4}{5} = 2 + 0.8 = 2.8 = \dfrac{280}{100} = 280\%\)
    Explanation: The mixed number 2 and 4/5 is converted to a decimal by adding the whole number 2 to the fraction 4/5 (which equals 0.8), resulting in 2.8. Writing 2.8 as a fraction with denominator 100 gives 280/100, which is 280%.

12. A) \(156312\)

    Sum of digits: \(1+5+6+3+1+2 = 18\)

    18 is divisible by 3 → YES \(156312\) is divisible by 3

    B) \(176314\)

    Sum of digits: \(1+7+6+3+1+4 = 22\)

    22 is not divisible by 3 → NO, \(176314\) is NOT divisible by 3

13. A) \(3432\)

    Last two digits: 32

    32 is divisible by 4 → YES, \(3432\) is divisible by 4

    B) \(1257\)

    Last two digits: 57

    57 is not divisible by 4 → NO, \(1257\) is not divisible by 4

14. A) \(1233\)

    Sum of digits = 9 (divisible by 3), hence \(1233\) is divisible by 3

    \(1233\) is divisible by 3 but is Not divisible by 2 → \(1233\) is NOT divisible by 6

    B) \(3432\)

    Sum of digits = 12 , hence \(3432\) is divisible by 3

    \(3432\) is divisible by 3 and 2 → YES \(3432\) is divisble by 6

15. A) \(2538\)

    Sume of digits: \(2+5+3+8 = 18\)

    18 is divisible by 9 → YES \(2538\) is divisible by 9

    B) \(1451\)

    Sume of digits: \(1+4+5+1 = 11\)

    11 is not divisible by 9 → \(1451\) is NOT divisible by 9

16. Evaluate \(8x + 7\) given that \(x - 3 = 10\).

    \(x - 3 = 10\)

    given equation

    \(x = 10 + 3 = 13\)

    solve the given equation

    \(8(13) + 7 = 104 + 7 = 111\)

    substitute x by 13 in the given expression and evaluate