Question:
The price of tea increased by 40% and then decreased by 20%. If a person now reduces his consumption of tea by \(6\frac{1}{4}\%\), then what is the percentage change in his expenditure?
चाय की कीमत में 40% की वृद्धि हुई और फिर 20% की कमी आई। यदि एक व्यक्ति अब अपनी चाय की खपत को \(6\frac{1}{4}\%\) कम कर देता है, तो उसके व्यय में कितने प्रतिशत का परिवर्तन होता है?
Options:
- Decrease, 5 %
- Increase, 15 %
- Increase, 5 %
- Decrease, 15 %
- कमी, 5%
- वृद्धि, 15%
- वृद्धि, 5%
- कमी, 15%
Correct Answer: c)
Explanation:
Step 1: Calculate the net change in the price of tea:
Let the original price per unit be \(P_0 = 100\).
Price after a 40% increase:
\[ 100 \times 1.40 = 140 \]
Price after a subsequent 20% decrease:
\[ 140 \times (1 – 0.20) = 140 \times 0.80 = 112 \]
So, the new price is \(P_1 = 112\).
Step 2: Calculate the change in consumption:
Let original consumption be \(C_0 = 100\).
Reduction in consumption = \(6\frac{1}{4}\% = \frac{25}{4}\% = \frac{1}{16}\).
New consumption:
\[ C_1 = 100 \times \left(1 – \frac{1}{16}\right) = 100 \times \frac{15}{16} \]
Step 3: Calculate the original and new expenditures:
\[ \text{Original Expenditure } E_0 = P_0 \times C_0 = 100 \times 100 = 10,000 \]
\[ \text{New Expenditure } E_1 = P_1 \times C_1 = 112 \times \left(100 \times \frac{15}{16}\right) = (112 \div 16) \times 1500 = 7 \times 1500 = 10,500 \]
Step 4: Find the percentage change in expenditure:
\[ \text{Percentage Change} = \frac{E_1 – E_0}{E_0} \times 100 = \frac{10,500 – 10,000}{10,000} \times 100 = \frac{500}{10,000} \times 100 = +5\% \]
This represents an increase of 5% (Increase, 5 %).
Hence, the correct answer is option c).
चरण 1: चाय की कीमत में शुद्ध परिवर्तन ज्ञात करें:
माना मूल कीमत \(P_0 = 100\) रुपये है।
40% वृद्धि के बाद कीमत:
\[ 100 + 40 = 140 \]
फिर 20% कमी के बाद कीमत:
\[ 140 \times 0.80 = 112 \]
अतः नई कीमत \(P_1 = 112\) है।
चरण 2: खपत में परिवर्तन ज्ञात करें:
माना मूल खपत \(C_0 = 100\) इकाई है।
खपत में कमी = \(6\frac{1}{4}\% = \frac{25}{4}\% = \frac{1}{16}\)।
नई खपत:
\[ C_1 = 100 \times \left(1 – \frac{1}{16}\right) = 100 \times \frac{15}{16} \]
चरण 3: मूल और नया व्यय ज्ञात करें:
\[ \text{मूल व्यय } E_0 = 100 \times 100 = 10,000 \]
\[ \text{नया व्यय } E_1 = 112 \times \frac{1500}{16} = 7 \times 1500 = 10,500 \]
चरण 4: व्यय में प्रतिशत परिवर्तन ज्ञात करें:
\[ \text{प्रतिशत परिवर्तन} = \frac{10,500 – 10,000}{10,000} \times 100 = \frac{500}{10,000} \times 100 = 5\% \text{ की वृद्धि} \]
अतः सही उत्तर विकल्प c) है।