The price of sugar is increased by 20%. Manoj can increase his expenditure on sugar by only 15% from his income. In order to balance his expenditure, by what percentage should he reduce the quantity of sugar consumed?
$4 \frac{1}{6}$ %
To solve this problem, let's define the initial metrics:
- Initial price of sugar = \(P\)
- Initial quantity of sugar purchased = \(Q\)
- Initial expenditure on sugar = \(P \times Q\)
- New price of sugar after a 20% increase = \(1.2P\)
- New expenditure as allowed by Manoj's income (increased by 15%) is \(1.15 \times (P \times Q)\)
We need to find the new quantity, let's call it \(Q'\), that Manoj can purchase after the price increase such that:
\(Q' \times 1.2P = 1.15 \times P \times Q\)
We can simplify this equation by dividing both sides by \(P\):
\(Q' \times 1.2 = 1.15 \times Q\)
Solving for \(Q'\), we get:
\(Q' = \frac{1.15 \times Q}{1.2}\)
Simplifying further:
\(Q' = \frac{1.15}{1.2} \times Q\)
\(Q' = \frac{115}{120} \times Q\)
\(Q' = \frac{23}{24} \times Q\)
Therefore, the percentage reduction in quantity is calculated as:
\(\left(1 - \frac{23}{24}\right) \times 100\%\)
\(= \frac{1}{24} \times 100\%\)
\(= 4.1667\%\)
This is approximately \(4 \frac{1}{6}\%\). Thus, Manoj should reduce the quantity of sugar consumed by 4 1/6 % to balance his expenditure. Therefore, the correct answer is 4 1/6 %.
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