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compulsory · CSS study notes

2. Percentage

CSS GSA — Paper 7, Part-II, Section VI, Sub-topic 2 (Percentage)

Key Formulas

Percentage=PartWhole×100\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100 Value=Percentage100×Whole\text{Value} = \frac{\text{Percentage}}{100} \times \text{Whole}
  • Fraction → Percentage: multiply the fraction by 100
  • Decimal → Percentage: multiply the decimal by 100 (shift decimal point two places right)
Percentage change=New−OldOld×100\text{Percentage change} = \frac{\text{New} - \text{Old}}{\text{Old}} \times 100 Successive % change=x−y−xy100\text{Successive \% change} = x - y - \frac{xy}{100}

(for an x% increase followed by a y% decrease, or vice versa)

Keeping Expenditure Constant — Price vs Consumption

  • Classic exam scenario: if the price of an item rises or falls by a given %, find the % change needed in consumption so total expenditure (Price × Consumption) stays the same.
  • If price increases by r%, consumption must be decreased by:
% decrease in consumption=r100+r×100\text{\% decrease in consumption} = \frac{r}{100+r} \times 100
  • If price decreases by r%, consumption can be increased by:
% increase in consumption=r100−r×100\text{\% increase in consumption} = \frac{r}{100-r} \times 100
  • Why this works: Expenditure = Price × Consumption. If Price becomes P′ = P(1+r/100), then for P′×C′ = P×C to hold, C′ must shrink by exactly the fraction that offsets the price rise — the formula above is just this relationship solved for the % change in C.

Worked Example 1 (Price Increase → Reduce Consumption)

  • The price of sugar rises by 25%. By what percentage must a household reduce its sugar consumption to keep its monthly sugar expenditure unchanged?
    • Step 1: Identify r = 25 (price increase %).
    • Step 2: Apply the formula: % decrease in consumption = r/(100+r) × 100 = 25/(100+25) × 100 = 25/125 × 100.
    • Step 3: Calculate: 25/125 = 0.20, so 0.20 × 100 = 20% decrease in consumption.
    • Check: Old expenditure = P×C. New price = 1.25P. New consumption = 0.80C. New expenditure = 1.25P × 0.80C = 1.00PC — unchanged ✓.

Worked Example 2 (Price Decrease → Increase Consumption)

  • The price of petrol falls by 20%. By what percentage can a family increase its petrol consumption while keeping the same monthly petrol budget?
    • Step 1: Identify r = 20 (price decrease %).
    • Step 2: Apply the formula: % increase in consumption = r/(100−r) × 100 = 20/(100−20) × 100 = 20/80 × 100.
    • Step 3: Calculate: 20/80 = 0.25, so 0.25 × 100 = 25% increase in consumption.
    • Check: New price = 0.80P. New consumption = 1.25C. New expenditure = 0.80P × 1.25C = 1.00PC — unchanged ✓.

Worked Example 3 (Reverse — Given Consumption Change, Find Price Change)

  • A family had to cut its rice consumption by 10% to keep its rice budget the same. By what percentage did the price of rice rise?
    • Step 1: This is the price-increase formula rearranged: 10 = r/(100+r) × 100.
    • Step 2: Solve for r: 10(100+r) = 100r → 1000 + 10r = 100r → 1000 = 90r.
    • Step 3: r = 1000/90 ≈ 11.11% price increase.

MCQ Traps

  • The price-increase and price-decrease formulas are not symmetric: a 25% price rise requires only a 20% consumption cut to balance it, but a 20% price fall allows a 25% consumption increase — don't assume the same percentage works both ways
  • Always keep the denominator as (100 + r) for a price increase and (100 − r) for a price decrease — mixing these up is the most common error in this topic

Quiz (with answers)

  1. 25% of 150 = 37.5 (B)
  2. 0.175 as a percentage = 17.5% (A)
  3. 1/5 as a percentage = 20% (C)
  4. 75% of 200 = 150 (B)
  5. 100 is what % of 300 = 33.33% (A)

Worked Questions (with answers)

  1. Batsman scored 110 runs with 3 boundaries (4×3=12) and 8 sixes (6×8=48) → 60 runs from boundaries; runs by running between wickets = 110–60 = 50 → 50/110 × 100 ≈ 45.45%
  2. Votes 1000, 5000, 10000 (total 16,000); winning candidate's % = 10000/16000×100 = 62.5%
  3. Ratio 5:4 expressed as a percentage (5/4 × 100) = 125%
  4. 25% of 80 = 20
  5. 50 is what % of 500 = 10%
  6. If 16% of 40% of a number is 8: 0.16 × 0.40 × x = 8 → x = 125
  7. 80% of 40 (=32) minus 4/5 of 25 (=20) = 12
  8. 10% of 80 = 8

Practice Questions (with answers)

  1. Oven bought at Rs. 36,800, sold at 8.5% gain → SP = 36,800 × 1.085 = Rs. 39,928
  2. Osman gave away 110+75=185 cards, 315 left → total = 500 → % given away = 185/500×100 = 37%
  3. Total votes 7,500; invalid 20% (=1,500) → valid = 6,000; winner got 55% of valid (=3,300) → other candidate got 6,000–3,300 = 2,700 votes
  4. Higher scorer's marks = 56% of the sum (S); difference = 9 → 0.56S – 0.44S = 0.12S = 9 → S = 75 → marks are 42 and 33
  5. 5%A + 4%B = ⅔(6%A + 8%B) → solving gives A : B = 4 : 3

MCQ Traps

  • "Percent of ratio a:b" typically means express a/b as a percentage (not a/(a+b)) — read the question wording carefully, as CSS papers sometimes intend either
  • A percentage increase then decrease of the same magnitude does not cancel out — there is always a net decrease (successive % change formula)
  • Converting a decimal to a percentage: multiply by 100, don't just append a "%" sign to the decimal digits

Self-Test Questions

  1. Convert 3/8 to a percentage.
  2. A price is increased by 20% then decreased by 20%. Is the final price equal to the original? Show the calculation.
  3. If 30% of a number is 45, find the number.
  4. A wins an election with 60% of 8,000 valid votes. How many votes did the losing candidate get?
  5. The price of cooking oil rises by 15%. By what percentage should a household cut its oil consumption to keep its monthly oil budget unchanged? Show the formula and full working.
  6. The price of a vegetable falls by 30%. By what percentage can a family increase its consumption while keeping the same budget? Show the formula and full working.

Past-Paper MCQs (FPSC)

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The FPSC MCQ papers you shared (2024 and 2025) had no direct percentage question (the 2024 circle-area MCQ is in Plane Geometry).

Past-Paper Questions with Answers (FPSC)

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Every question below appeared in the FPSC papers you shared; the year and question number are in each title. Answers are worked out by me.

  • CSS 2025 Q6(a): A 15% sale. Saima pays Rs. 57,120 for a fridge and a freezer. The freezer cost Rs. 40,000 before the sale. What was the fridge's price before the sale? (5)
    1. Price paid is 85% of the original total, so original total = 57,120 ÷ 0.85 = Rs. 67,200
    2. Fridge = 67,200 − 40,000 = Rs. 27,200
    • Check: 67,200 × 0.85 = 57,120
  • CSS 2025 Q7(a): Marked price of a sculpture sold for Rs. 79,950 after an 18% discount. If instead 10% and then a further 8% are given, is the price still Rs. 79,950? (5)
    1. 82% of marked price = 79,950, so marked price = 79,950 ÷ 0.82 = Rs. 97,500
    2. After 10%: 97,500 × 0.90 = 87,750
    3. After a further 8%: 87,750 × 0.92 = Rs. 80,730
    4. No: the price would be Rs. 80,730, which is Rs. 780 more (successive discounts of 10% and 8% total only 17.2%, less than 18%)
  • CSS 2025 Q7(b): Majid pays Rs. 51,520 for car insurance, made up of a basic charge plus 15% sales tax. Find the sales tax. (5)
    1. Total = 115% of basic charge, so basic = 51,520 ÷ 1.15 = Rs. 44,800
    2. Sales tax = 51,520 − 44,800 = Rs. 6,720
    • Check: 15% of 44,800 = 6,720
  • CSS 2022 Q6(b): A company's income fell 10% and then 15% in two successive years. What is the overall percentage decrease? (5)
    1. Remaining after year 1 = 0.90; after year 2 = 0.85
    2. Overall remaining = 0.90 × 0.85 = 0.765
    3. Overall decrease = 1 − 0.765 = 23.5% (never just add 10 + 15)
  • CSS 2022 Q8(c): A sweater's price was raised 20% instead of being lowered 20%, so the tag was Rs. 80 too high. What was the original price? (5)
    1. Wrong price = 1.2P; correct price = 0.8P
    2. Difference = 0.4P = 80
    3. P = Rs. 200
  • CSS 2021 Q6(c): Dinner bill: marked price Rs. 15,000; 10% discount, 10% service charge, 17% GST. Find the total. (5)
    1. After 10% discount: 15,000 × 0.90 = 13,500
    2. Add 10% service charge: 13,500 × 1.10 = 14,850
    3. Add 17% GST: 14,850 × 1.17 = Rs. 17,374.50
  • CSS 2018 Q6(A): 3 litres of paint cover 24 m². What percentage increase in paint is needed for 50.4 m²? (5)
    1. Paint per m² = 3 ÷ 24 = 0.125 litre
    2. Paint for 50.4 m² = 50.4 × 0.125 = 6.3 litres
    3. Increase = 6.3 − 3 = 3.3 litres
    4. Percentage increase = 3.3 ÷ 3 × 100 = 110%
  • CSS 2019 Q6(c): A car hire costing Rs. 264 in 2018 was 20% more than in 2013. What was the cost in 2013? (5)
    1. 2018 cost = 120% of 2013 cost
    2. 2013 cost = 264 ÷ 1.2 = Rs. 220
  • CSS 2017 Q7(A): A man buys 5 kg of meat at Rs. 500 per kg plus 6% consumption tax. Total? (5)
    1. Cost = 5 × 500 = 2,500
    2. Tax = 6% of 2,500 = 150
    3. Total = Rs. 2,650
  • CSS 2023 Q8(a): Ali buys an oven for Rs. 36,800 and sells it at an 8.5% gain. Selling price? (5)
    1. Selling price = 36,800 × 1.085 = 36,800 + 3,128 = Rs. 39,928

CSS Exam Perspective

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  • Most repeated areas: discount and tax word problems, successive percentage change, reverse percentage (find the original)
  • Tip: for 'after a discount', divide by the remaining fraction (e.g. ÷ 0.85); for 'plus tax', divide by 1 plus the rate