Question

If the price of gold increases by 30%, find by how much the quantity of ornaments must be reduced so that the expenditure may remain the same as before.

• A. 30%
• B. $23\frac{1}{3}$
• C. $27\frac{2}{13}$
• D. 19%

Answer 1

Answer: (B)

$23\frac{1}{3}$

Answer 2

Let the initial quantity of ornaments be $Q$ and the initial price be $P$.
$\therefore$ The total expenditure is $Q \times P$.

When the price increases by 30%, the new price becomes $1.3P$.
To keep the expenditure the same, the new quantity $Q'$ must satisfy:
$Q' \times 1.3P = Q \times P$
\Rightarrow$$Q' = \frac{Q \times P}{1.3P} = \frac{Q}{1.3}

The reduction in quantity is:
$Q - Q' = Q - \frac{Q}{1.3} = Q \left( 1 - \frac{1}{1.3} \right) = Q \left( \frac{0.3}{1.3} \right) = \frac{3}{13} Q$

Hence, the percentage reduction is
$\frac{\frac{3}{13} Q}{Q} \times 100 = \frac{3}{13} \times 100 \approx 23.08\%$

Therefore, the quantity must be reduced by approximately $23\frac{1}{13}\%$