The World of Numbers | Exercise 3.3

Question 4

Find the product:

(i) 23×310\dfrac{2}{3} \times \dfrac{3}{10}

(ii) 711×58\dfrac{7}{11} \times \dfrac{5}{8}

(iii) −47×514-\dfrac{4}{7} \times \dfrac{5}{14}

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Solution
Understand the Question
  • To multiply fractions or rational numbers, multiply the numerators together and the denominators together: ab×cd=a×cb×d\dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{a \times c}{b \times d}
  • Simplify the resulting fraction to its simplest form by dividing both the numerator and denominator by their greatest common divisor.
  • Remember the sign rule for multiplication: a negative number multiplied by a positive number gives a negative result (−×+=−-\times + = -).

(i) 23×310\dfrac{2}{3} \times \dfrac{3}{10}

Step 1 · Multiply and Simplify

23×310=2×33×10=630=15\begin{aligned} \dfrac{2}{3} \times \dfrac{3}{10} &= \dfrac{2 \times 3}{3 \times 10} \\[0.6em] &= \dfrac{6}{30} \\[0.6em] &= \dfrac{1}{5} \end{aligned}
Answer

(i) 15\dfrac{1}{5}

(ii) 711×58\dfrac{7}{11} \times \dfrac{5}{8}

Step 1 · Multiply Numerators and Denominators

711×58=7×511×8=3588\begin{aligned} \dfrac{7}{11} \times \dfrac{5}{8} &= \dfrac{7 \times 5}{11 \times 8} \\[0.6em] &= \dfrac{35}{88} \end{aligned}
Answer

(ii) 3588\dfrac{35}{88}

(iii) −47×514-\dfrac{4}{7} \times \dfrac{5}{14}

Step 1 · Multiply and Simplify

−47×514=−4×57×14=−2098=−1049\begin{aligned} -\dfrac{4}{7} \times \dfrac{5}{14} &= \dfrac{-4 \times 5}{7 \times 14} \\[0.6em] &= \dfrac{-20}{98} \\[0.6em] &= -\dfrac{10}{49} \end{aligned}
Answer

(iii) −1049-\dfrac{10}{49}

Common Mistakes
  • Cross-multiplying instead of multiplying straight across: Multiplying fractions requires multiplying numerator by numerator and denominator by denominator, not cross-multiplying (which is used for solving equations or dividing fractions).
  • Sign Errors: Forgetting that multiplying a negative fraction by a positive fraction yields a negative result.
  • Leaving Fractions Unsimplified: Forgetting to divide numerator and denominator by common factors to express the final answer in simplest form (e.g., leaving 630\frac{6}{30} instead of reducing to 15\frac{1}{5}).

More questions in Exercise 3.3

Q1

Prove that the following rational numbers are equal:

(i) 23\dfrac{2}{3} and 46\dfrac{4}{6}

(ii) 54\dfrac{5}{4} and 108\dfrac{10}{8}

(iii) −35-\dfrac{3}{5} and −610-\dfrac{6}{10}

(iv) 93\dfrac{9}{3} and 33

Q2

Find the sum:

(i) 25+310\dfrac{2}{5} + \dfrac{3}{10}

(ii) 712+58\dfrac{7}{12} + \dfrac{5}{8}

(iii) −47+314-\dfrac{4}{7} + \dfrac{3}{14}

Q3

Find the difference:

(i) 56−14\dfrac{5}{6} - \dfrac{1}{4}

(ii) 118−34\dfrac{11}{8} - \dfrac{3}{4}

(iii) −79−(−23)-\dfrac{7}{9} - \left(-\dfrac{2}{3}\right)

Q4

Find the product:

(i) 23×310\dfrac{2}{3} \times \dfrac{3}{10}

(ii) 711×58\dfrac{7}{11} \times \dfrac{5}{8}

(iii) −47×514-\dfrac{4}{7} \times \dfrac{5}{14}

Q5

Find the quotient:

(i) 23÷310\dfrac{2}{3} \div \dfrac{3}{10}

(ii) 711÷58\dfrac{7}{11} \div \dfrac{5}{8}

(iii) −47÷514-\dfrac{4}{7} \div \dfrac{5}{14}

Q6

Show that:

(12+34)×83=12×83+34×83\left(\dfrac{1}{2} + \dfrac{3}{4}\right) \times \dfrac{8}{3} = \dfrac{1}{2} \times \dfrac{8}{3} + \dfrac{3}{4} \times \dfrac{8}{3}

Q7

Simplify the following using the distributive property:

79(67−34)\dfrac{7}{9}\left(\dfrac{6}{7} - \dfrac{3}{4}\right)

Q8

Find the rational number xx such that:

56(x+35)=56x+12\dfrac{5}{6}\left(x + \dfrac{3}{5}\right) = \dfrac{5}{6}x + \dfrac{1}{2}

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