Find the quotient: (i) (ii) (iii)

Understand the Question To divide one fraction by another, multiply the first fraction (dividend) by the reciprocal of the second fraction (divisor): Simplify the resulting fraction to its simplest form whenever common factors exist between the numerator and denominator. Part (i) Step 1 · Multiply by the Reciprocal Multiply by the reciprocal of , which is : Answer > (i) Part (ii) Step 1 · Multiply by the Reciprocal Multiply by the reciprocal of , which is : Answer > (ii) Part (iii) Step 1 · Multiply by the Reciprocal and Simplify Multiply by the reciprocal of , which is : Divide both numerator and denominator by their common factor : Answer > (iii) Common Mistakes Inverting the Wrong Fraction: Inverting the first fraction (dividend) instead of the second fraction (divisor). Forgetting to Invert: Multiplying the numerators and denominators directly without taking the reciprocal of the divisor first. Sign Errors: Dropping the negative sign when multiplying or reducing fractions in part (iii). Not Reducing to Lowest Terms: Leaving the answer as instead of simplifying to .

The World of Numbers | Exercise 3.3

Question 5

Find the quotient:

(i) $\dfrac{2}{3} \div \dfrac{3}{10}$

(ii) $\dfrac{7}{11} \div \dfrac{5}{8}$

(iii) $-\dfrac{4}{7} \div \dfrac{5}{14}$

Solution

Understand the Question

To divide one fraction by another, multiply the first fraction (dividend) by the reciprocal of the second fraction (divisor): $\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c} = \dfrac{a \times d}{b \times c}$
Simplify the resulting fraction to its simplest form whenever common factors exist between the numerator and denominator.

(i) $\dfrac{2}{3} \div \dfrac{3}{10}$

Step 1 · Multiply by the Reciprocal

Multiply $\dfrac{2}{3}$ by the reciprocal of $\dfrac{3}{10}$ , which is $\dfrac{10}{3}$ :

\begin{aligned} \dfrac{2}{3} \div \dfrac{3}{10} &= \dfrac{2}{3} \times \dfrac{10}{3} \\[0.6em] &= \dfrac{2 \times 10}{3 \times 3} \\[0.6em] &= \dfrac{20}{9} \end{aligned}

Answer

(i) $\dfrac{20}{9}$

(ii) $\dfrac{7}{11} \div \dfrac{5}{8}$

Step 1 · Multiply by the Reciprocal

Multiply $\dfrac{7}{11}$ by the reciprocal of $\dfrac{5}{8}$ , which is $\dfrac{8}{5}$ :

\begin{aligned} \dfrac{7}{11} \div \dfrac{5}{8} &= \dfrac{7}{11} \times \dfrac{8}{5} \\[0.6em] &= \dfrac{7 \times 8}{11 \times 5} \\[0.6em] &= \dfrac{56}{55} \end{aligned}

Answer

(ii) $\dfrac{56}{55}$

(iii) $-\dfrac{4}{7} \div \dfrac{5}{14}$

Step 1 · Multiply by the Reciprocal and Simplify

Multiply $-\dfrac{4}{7}$ by the reciprocal of $\dfrac{5}{14}$ , which is $\dfrac{14}{5}$ :

\begin{aligned} -\dfrac{4}{7} \div \dfrac{5}{14} &= -\dfrac{4}{7} \times \dfrac{14}{5} \\[0.6em] &= \dfrac{-4 \times 14}{7 \times 5} \\[0.6em] &= \dfrac{-56}{35} \end{aligned}

Divide both numerator and denominator by their common factor $7$ :

\begin{aligned} &= \dfrac{-56 \div 7}{35 \div 7} \\[0.6em] &= -\dfrac{8}{5} \end{aligned}

Answer

(iii) $-\dfrac{8}{5}$

Common Mistakes

Inverting the Wrong Fraction: Inverting the first fraction (dividend) instead of the second fraction (divisor).
Forgetting to Invert: Multiplying the numerators and denominators directly without taking the reciprocal of the divisor first.
Sign Errors: Dropping the negative sign when multiplying or reducing fractions in part (iii).
Not Reducing to Lowest Terms: Leaving the answer as $-\dfrac{56}{35}$ instead of simplifying to $-\dfrac{8}{5}$ .

Question 5

More questions in Exercise 3.3