Sugar Price Reduction Problem Explained
This question asks for the reduced price per kg of sugar when a 10% price decrease allows purchasing more quantity for the same total cost.
Calculating the Reduced Price Per Kilogram
Let the original price per kg be \(P_{orig}\) and the reduced price per kg be \(P_{red}\).
Given:
- Total amount spent = \(₹2790\)
- Price reduction = 10%
- Extra quantity bought = 6.2 kg
The reduced price is 10% less than the original price:
\(P_{red} = P_{orig} \times (1 - 0.10) = 0.9 \times P_{orig}\)
The quantity that can be bought is inversely proportional to the price for a fixed total amount.
Original Quantity (\(Q_{orig}\)) = \(\frac{2790}{P_{orig}}\)
New Quantity (\(Q_{new}\)) = \(\frac{2790}{P_{red}} = \frac{2790}{0.9 \times P_{orig}}\)
The difference in quantity is given:
\(Q_{new} - Q_{orig} = 6.2\)
Substituting the quantity expressions:
\(\frac{2790}{0.9 \times P_{orig}} - \frac{2790}{P_{orig}} = 6.2\)
Factor out \(\frac{2790}{P_{orig}}\):
\(\frac{2790}{P_{orig}} \left( \frac{1}{0.9} - 1 \right) = 6.2\)
Simplify the term in parenthesis:
\(\frac{2790}{P_{orig}} \left( \frac{10}{9} - \frac{9}{9} \right) = 6.2\)
\(\frac{2790}{P_{orig}} \left( \frac{1}{9} \right) = 6.2\)
\(\frac{310}{P_{orig}} = 6.2\)
Solve for the original price:
\(P_{orig} = \frac{310}{6.2} = 50\)
The original price was \(₹50\) per kg.
Now, calculate the reduced price:
\(P_{red} = 0.9 \times P_{orig} = 0.9 \times 50 = 45\)
The reduced price per kilogram is \(₹45\).
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