8.6 Tricky Percentages

Percentages compare proportions, not absolute amounts, and forgetting that is where nearly everyone trips.

Would You Rather? Option A: put in ₹100, get back ₹300 (a 200% gain). Option B: put in ₹1000, get back ₹1500 (a 50% gain). A boasts the bigger percentage, but B drops more rupees (₹500 versus ₹200) into your hand. Which would you take, and why? The honest answer depends on whether you care about the rate or the rupees.

Beware: Successive Changes Don’t Add A common trap: if something rises by a percentage and then falls by the same percentage, you do not end up where you began. Picture a salary of ₹50,000 that gets a 10% raise and, a year later, a 10% cut: 50000 \times 1.10 \times 0.90 = ₹49{,}500, a 1% net loss, not break-even. The reason is that the cut acts on a larger base (₹55,000) than the raise acted on. The same warning underlies every “trap” in this section: percentage changes multiply, they do not add.

Exam Tip When a question chains two percentage changes, multiply the factors instead of adding the percentages. A 20% rise then a 10% rise is 1.20 \times 1.10 = 1.32, a 32% increase, not 30%. Convert each step to a multiplier (1+r for a rise, 1-r for a fall), multiply them, then read off the single overall change. This one habit prevents the most common percentage error in the exam.

Worked Example: why 40% then 20% is not 60%

Cake Corner offers “40% + 20% off”; Sweet Spot offers “55% off”. On a ₹300 cake, which is cheaper?

“40\% + 20\%” means the discounts compound, the second is taken on the already-cut price:

  • Cake Corner: after 40% off, ₹300 → ₹180; then 20% off ₹180 → 0.8 \times 180 = ₹144.
  • Sweet Spot: 55% off ₹300 → 0.45 \times 300 = ₹135.

So Sweet Spot (₹135) is cheaper. Combining the two discounts gives only 0.6 \times 0.8 = 0.48, a 52% reduction, not 60%.

Worked Example: a costly mishap

A seller marks goods up 50%, then offers a 50% discount, expecting to break even. Is she right?

Let cost = x. Marked price = 1.5x. A 50% discount drops the sale price to 0.5 \times 1.5x = 0.75x, only three-quarters of what she paid. That is a 25% loss, not break-even! On goods bought for ₹16,000 she would sell for ₹12,000, dropping ₹4,000. To truly break even she should have discounted by just \tfrac{1}{3} \approx 33.33\%, since 1.5x \times (1 - \tfrac13) = x.

Figure it Out: Tricky Percentages

Practice

  1. Pune’s 2025 population is about 300% of its 2000 population. If 2000 was 40 lakh, what is 2025?
  2. A ₹7,500 phone has 18% GST added. Which expression gives the final price?   (i) 7500 + 18   (iii) 7500 + \tfrac{18}{100}   (v) 7500 \times 1.18   (vi) 7500 + 7500 \times 0.18   (vii) 1.8 \times 7500.
  3. A shopkeeper sets a 40% profit margin, then gives a 35% discount on the selling price. Profit or loss?
  4. What is 8% of 25? And 25% of 8? Try 15% of 60 vs 60% of 15. What do you notice, can you justify x\% of y = y\% of x with algebra?
  5. For a field trip, 35% of students are Grade 8; of these, 55% are girls.   (i) What percentage of all students are Grade 8 girls?   (ii) If 180 students go, how many are Grade 8 girls?
  6. A shopkeeper sells 4 pens for the cost of 7 pens. Profit or loss? What percentage?
  7. Bus fares rose 5% one year and 6% the next. What is the overall increase?
  8. If a rectangle’s length grows 20% but the area stays the same, by what exact percentage does the breadth shrink?
  9. In a room of 100 people, 95% wear glasses. How many glasses-wearers must leave so that glasses-wearers become 90%?
  1. 300\% of 40 lakh = 3.0 \times 40 = \mathbf{120\ \text{lakh}} (1.2 crore).
  2. The final price is 7500 \times 1.18 = ₹8850, given by both (v) 7500\times 1.18 and (vi) 7500 + 7500\times 0.18.
  3. Let cost = 100. Marked at 40% profit = 140; 35% discount \to 0.65 \times 140 = 91. Since 91 < 100, it is a loss of 9%.
  4. 8\% of 25 = 2 and 25\% of 8 = 2, equal! Likewise 15\% of 60 = 60\% of 15 = 9. In general x\% of y = \tfrac{x}{100}\,y = \tfrac{y}{100}\,x = y\% of x, so the two are always equal.
    1. 35\% \times 55\% = 0.35 \times 0.55 = 0.1925 = \mathbf{19.25\%} are Grade 8 girls.   (ii) 19.25\% of 180 = \mathbf{34.65}, i.e. about 35 girls.
  6. Let cost per pen be 1 unit. Selling 4 for the cost of 7 means 4 pens fetch 7 units, so SP per pen = \tfrac{7}{4}. Profit per pen = \tfrac{7}{4} - 1 = \tfrac{3}{4}, a profit of \tfrac{3}{4}\times 100 = 75\%.
  7. 1.05 \times 1.06 = 1.113, an overall increase of 11.3% (not 11%, because the second rise applies to the already-raised fare).
  8. New length = 1.2\times old. For the same area, new breadth = \dfrac{1}{1.2} of old = 0.8333\ldots, a decrease of 1 - \tfrac{1}{1.2} = \tfrac{1}{6} \approx \mathbf{16.67\%}.
  9. With 95 glasses-wearers and 5 others, the 5 others must become 10% of the room, so the room must shrink to 50 people. That means 100 - 50 = \mathbf{50} glasses-wearers leave.