8.4 Using Percentages

Comparing proportions

Worked Example

Faisal scored 45 out of 55 in History and 68 out of 80 in Geography. In which did he do better?

The maximums differ, so raw marks cannot be compared head to head. Convert both to percentages:

\text{History} = \frac{45}{55}\times 100 = 81.82\%, \qquad \text{Geography} = \frac{68}{80}\times 100 = 85\%.

Geography (85%) edges out History (81.82%), so he did better in Geography, even though he “lost more marks” there.

Percentage increase and decrease

The Rule \text{percentage change} = \frac{\text{amount of change}}{\text{original amount (base)}}\times 100. The denominator is always the original value, the base you started from.

  • Tomatoes climbed from ₹25 to ₹40, a ₹15 rise: \dfrac{15}{25}\times 100 = \mathbf{60\%} increase.
  • A cinema’s footfall slipped from 200 to 130, a drop of 70: \dfrac{70}{200}\times 100 = \mathbf{35\%} decrease.

Two statements that sound unalike can carry the same meaning. “The 1991 population is 165% of the 1961 population” and “the population rose 65% from 1961 to 1991” both amount to q = 1.65\,p.

Profit and loss

Three prices are in play: the cost price (CP) the seller paid, the marked price (MP) quoted on the tag, and the selling price (SP) the buyer actually hands over. Profit or loss is always reckoned against the cost price.

Profit and Loss % \text{Profit \%} = \frac{\text{SP} - \text{CP}}{\text{CP}}\times 100, \qquad \text{Loss \%} = \frac{\text{CP} - \text{SP}}{\text{CP}}\times 100.

Worked Example

A furniture seller buys a sofa for ₹2500 and sells it for ₹3200. Find the profit percentage.

Profit = 3200 - 2500 = ₹700, reckoned against CP = ₹2500:

\frac{700}{2500}\times 100 = 28\%.

A grocer bought 12 kg of rice at ₹40/kg (CP = ₹480) and cleared it for ₹410. Loss = ₹70, so loss \% = \dfrac{70}{480}\times 100 = \mathbf{14.58\%}.

A discount is a percentage shaved off the marked price. Suppose a decorative lamp cost a shop ₹3400; sold at an 18% loss, it fetches 0.82 \times 3400 = ₹2788.

Taxes

Tax rates such as GST (Goods and Services Tax) are percentages tacked onto a price; that extra slice goes to the government. A ₹7500 phone with 18% GST ends up costing 7500 \times 1.18 = ₹8850.

Figure it Out: Using Percentages

Practice

  1. A shop buys a pencil box for ₹90 and sells it for ₹130. Find the profit % on cost.
  2. A carpenter’s materials for a stool cost ₹480 and she wants a 50% profit margin. What is the selling price?
  3. A firm’s revenue was ₹3 crore with a 20% profit margin. What was its total cost?
  4. A jacket priced ₹350 carries a 30% discount. How much does the buyer pay?
  5. Petrol went from ₹50 (2015) to ₹90 (2025). Percentage increase?   (i) 40%   (ii) 50%   (iii) 60%   (iv) 80%   (v) 140%   (vi) 180%.
  6. A buyer paid ₹5,10,000 for a scooter after a 15% discount. What was the original price?
  7. 2000 people voted; the winner got 600 votes. What percent did the winner get? What is the least possible number of candidates?
  8. A litre of milk was ₹40 in 2024 and ₹44 in 2025. What is the inflation rate?
  9. A number increased by 25% becomes 100. What is the number?
  10. A trader sold two scooters for ₹90,000 each, one at 8% profit, one at 12% loss. Overall profit or loss?
  1. Profit = 130 - 90 = 40, so \dfrac{40}{90}\times 100 \approx \mathbf{44.44\%}.
  2. 480 \times 1.5 = \mathbf{₹720}.
  3. A 20% margin means revenue = 1.20 \times cost, so cost = \dfrac{3\text{ cr}}{1.20} = \mathbf{₹2.5\ \text{crore}}.
  4. 350 \times 0.70 = \mathbf{₹245}.
  5. Increase = 40 on a base of 50: \dfrac{40}{50}\times 100 = \mathbf{80\%}, option (iv).
  6. Paid 85% of the original: original = \dfrac{510000}{0.85} = \mathbf{₹6{,}00{,}000}.
  7. \dfrac{600}{2000}\times 100 = \mathbf{30\%}. The remaining 1400 votes must be split so that no rival reaches 600, which is possible only with at least 3 candidates in all (the other 1400 spread across at least two others).
  8. Increase = 4 on base 40: \dfrac{4}{40}\times 100 = \mathbf{10\%}.
  9. 1.25 \times n = 100 \Rightarrow n = \dfrac{100}{1.25} = \mathbf{80}.
  10. CP of the 8%-profit scooter = \dfrac{90000}{1.08} \approx ₹83{,}333; CP of the 12%-loss one = \dfrac{90000}{0.88} \approx ₹1{,}02{,}273. Total CP \approx ₹1{,}85{,}606, total SP = ₹1{,}80{,}000, so an overall loss of about ₹5{,}606.