Pariksha SaathiLesson 4 of 8 · Mathematics, Classes VI-VIII (Paper II)
Ratio, Proportion, Percentage, Profit and Loss, Simple and Compound Interest
Drafted with AI assistance and fact-checked on 28 October 2026; not yet read line by line by a human editor. Spotted a mistake? Tell us. Editorial standards & corrections
Ratio and proportion
A ratio compares two quantities of the same kind in the same unit by division and has no unit itself; it is written in simplest form, so 40 minutes : 2 hours = 40 : 120 = 1 : 3. Two ratios are compared as fractions with a common denominator: 3 : 4 = 21/28 and 5 : 7 = 20/28, so 3 : 4 is greater. Four numbers are in proportion, a : b :: c : d, when the product of the extremes equals the product of the means (a × d = b × c): 15 : 45 :: 40 : 120 because 15 × 120 = 45 × 40 = 1,800. If 4, 6, 10 and x are in proportion, 4x = 60 and x = 15. To share in a ratio, add the parts: 36 pupils with boys and girls in the ratio 4 : 5 make 9 parts of 4 pupils each, so 16 boys and 20 girls.
Unitary method
Find the value of one unit first, then of the number required. If 6 pens cost ₹90, one costs ₹15 and 10 cost ₹150; if a car runs 150 km on 12 litres of petrol, it runs 12.5 km on one litre and 250 km on 20 litres. The same reasoning underlies percentages, profit and loss, and interest.
Percentage
Per cent means per hundred, so a fraction or decimal is converted by multiplying by 100: 3/8 = 37.5% and 0.045 = 4.5%. The whole is found from a part by dividing: if 25% of a number is 30, the number is 30 × 100/25 = 120. Increase or decrease per cent is always worked on the original value: a price rising from ₹40 to ₹50 rises by 10/40 × 100 = 25%, but falling from ₹50 to ₹40 falls by 10/50 × 100 = 20%.
Profit, loss and discount
Profit = selling price (SP) − cost price (CP), loss = CP − SP, and both per cents are calculated on the CP. Overhead expenses such as repairs or transport after buying are added to the CP: a scooter bought for ₹12,000, repaired for ₹3,000 and sold for ₹13,500 gives a loss of ₹1,500 on ₹15,000, which is 10%. With CP and gain per cent known, SP = CP × (100 + gain%)/100, so ₹600 at a 15% gain sells for ₹690; with SP and loss per cent known, CP = SP × 100/(100 − loss%), so an article sold for ₹1,140 at a 5% loss cost ₹1,200. Discount is a reduction on the marked price (MP) and its per cent is calculated on the MP: an MP of ₹2,000 with a 12% discount gives an SP of ₹1,760.
Simple and compound interest
Simple interest is charged on the principal alone, SI = P × R × T/100: ₹5,000 at 8% per annum for 3 years earns ₹1,200, so the amount is ₹6,200. Compound interest adds each period's interest to the principal, so the amount is A = P(1 + R/100)ⁿ and CI = A − P. ₹10,000 at 10% per annum compounded annually for 2 years amounts to 10,000 × 1.1 × 1.1 = ₹12,100, giving CI ₹2,100 against SI ₹2,000; the extra ₹100 is the interest on the first year's interest, and for 2 years CI − SI = P(R/100)². Compounded half-yearly, the rate is halved and the number of periods doubled: ₹8,000 at 10% per annum for 1 year becomes 8,000 × 1.05² = ₹8,820, a CI of ₹820. The same formula gives growth and depreciation: a population of 20,000 growing at 5% a year reaches 22,050 in 2 years, and a machine worth ₹50,000 losing 10% a year is worth 50,000 × 0.9² = ₹40,500 after 2 years.
Direct and inverse proportion
Two quantities are in direct proportion when they increase or decrease together in the same ratio, so x/y stays constant: if 6 m of cloth costs ₹210, 10 m costs ₹350. They are in inverse proportion when one increases as the other decreases in the same ratio, so x × y stays constant: if 15 workers build a wall in 48 hours, 30 workers take 24 hours, and a school day of 8 periods of 45 minutes split into 9 periods gives periods of 40 minutes, since 8 × 45 = 9 × 40 = 360. For a fixed distance, speed and time are inversely proportional (60 km/h for 2 hours or 80 km/h for 1.5 hours). Not every pair that grows together is proportional: a child's height increases with age, but not in a fixed ratio.
In the classroom
Pupils often add the same number to both terms of a ratio instead of multiplying, work a percentage change on the new value, or calculate profit per cent on the SP. Ratio tables, double number lines and shop role plays with price tags and bills make the multiplicative idea visible, and estimating first (25% off ₹798 is about ₹200 off) helps pupils judge whether an answer is sensible.
Key facts
- A ratio needs the same unit: 40 minutes : 2 hours = 40 : 120 = 1 : 3.
- Proportion test: 15 : 45 :: 40 : 120 because 15 × 120 = 45 × 40 = 1,800; the fourth proportional to 4, 6 and 10 is 15.
- Percentage change is on the original value: ₹40 to ₹50 is a 25% rise, but ₹50 to ₹40 is a 20% fall.
- SP = CP × (100 + gain%)/100, so CP ₹600 at a 15% gain gives SP ₹690; CP = SP × 100/(100 − loss%), so SP ₹1,140 at a 5% loss gives CP ₹1,200.
- Overheads join the cost price: ₹12,000 + ₹3,000 repairs = ₹15,000; sold for ₹13,500, the loss is ₹1,500, or 10%.
- Discount is on the marked price: MP ₹2,000 with a 12% discount gives SP ₹1,760.
- SI = PRT/100: ₹5,000 at 8% per annum for 3 years gives SI ₹1,200 and amount ₹6,200.
- CI: ₹10,000 at 10% per annum for 2 years gives A = ₹12,100 and CI = ₹2,100; for 2 years CI − SI = P(R/100)² = ₹100.
- Half-yearly compounding: ₹8,000 at 10% per annum for 1 year is 5% for 2 half-years, giving ₹8,820.
- Inverse proportion: if 15 workers take 48 hours, 30 workers take 24 hours; 8 periods of 45 minutes equal 9 periods of 40 minutes.
Quick revision
Ratio: same kind, same unit, no unit of its own; a : b :: c : d when ad = bc; unitary method finds one unit first; percentage change on the original value; profit and loss per cent on CP, discount per cent on MP, overheads added to CP; SI = PRT/100; A = P(1 + R/100)ⁿ, CI = A − P; half-yearly: rate halved, periods doubled; 2-year CI − SI = P(R/100)²; direct proportion: x/y constant; inverse: xy constant.