The World of Numbers | Exercise 3.3

Question 7

Simplify the following using the distributive property:

$\frac{7}{9}\left(\frac{6}{7} - \frac{3}{4}\right)$

Solution

We will use the distributive property to simplify this expression.

Step 1 — Distribute the fraction

Let's multiply the outside fraction with each term inside. We apply the distributive property.

$\frac{7}{9}\left(\frac{6}{7} - \frac{3}{4}\right) = \left(\frac{7}{9} \times \frac{6}{7}\right) - \left(\frac{7}{9} \times \frac{3}{4}\right)$

$= \frac{42}{63} - \frac{21}{36}$

$= \frac{2}{3} - \frac{7}{12}$

$\boxed{\frac{2}{3} - \frac{7}{12}}$

Step 2 — Subtract the fractions

Now, we subtract these two fractions. First, find the Least Common Multiple (LCM) of the denominators. The denominators are 3 and 12. The LCM of 3 and 12 is 12. We convert the first fraction to have a denominator of 12.

$\frac{2}{3} = \frac{2 \times 4}{3 \times 4}$

$= \frac{8}{12}$

Now, we can subtract the fractions.

$\frac{8}{12} - \frac{7}{12}$

$= \frac{8 - 7}{12}$

$\boxed{\frac{1}{12}}$

Answer

(i) The simplified value is $\frac{1}{12}$ .

More questions in Exercise 3.3

Prove that the following rational numbers are equal:

(i) $\frac{2}{3}$ and $\frac{4}{6}$ (ii) $\frac{5}{4}$ and $\frac{10}{8}$ (iii) $-\frac{3}{5}$ and $-\frac{6}{10}$ (iv) $\frac{9}{3}$ and $3$

Find the sum:

(i) $\frac{2}{5} + \frac{3}{10}$ (ii) $\frac{7}{12} + \frac{5}{8}$ (iii) $-\frac{4}{7} + \frac{3}{14}$

Find the difference:

(i) $\frac{5}{6} - \frac{1}{4}$ (ii) $\frac{11}{8} - \frac{3}{4}$ (iii) $-\frac{7}{9} - \left(-\frac{2}{3}\right)$

Find the product:

(i) $\frac{2}{3} \times \frac{3}{10}$ (ii) $\frac{7}{11} \times \frac{5}{8}$ (iii) $-\frac{4}{7} \times \frac{5}{14}$

Find the quotient:

(i) $\frac{2}{3} \div \frac{3}{10}$ (ii) $\frac{7}{11} \div \frac{5}{8}$ (iii) $-\frac{4}{7} \div \frac{5}{14}$

Show that:

$\left(\frac{1}{2} + \frac{3}{4}\right) \times \frac{8}{3} = \frac{1}{2} \times \frac{8}{3} + \frac{3}{4} \times \frac{8}{3}$

Simplify the following using the distributive property:

$\frac{7}{9}\left(\frac{6}{7} - \frac{3}{4}\right)$

Find the rational number $x$ such that:

$\frac{5}{6}\left(x + \frac{3}{5}\right) = \frac{5}{6}x + \frac{1}{2}$