Big picture
A ₹200 reduction is large on a ₹500 item but relatively small on a ₹10,000 item. Comparing quantities requires a reference, not just a difference. Ratios and percentages express how one quantity relates to another and make fair comparisons between different totals.
The same reasoning explains examination marks, shop discounts, profit and simple interest. Before calculating, identify the base quantity: original price, cost price, marked price or principal. Most errors in this chapter come from choosing the wrong base rather than from difficult arithmetic.
Learning goals
Form and simplify ratios using consistent units.
Convert fractions, decimals and percentages.
Calculate a percentage of a quantity and recover the whole.
Determine percentage increase and decrease.
Solve profit, loss, discount and simple-interest problems.
Core ideas
1. Ratios and equivalent units
A ratio compares two quantities by division. The ratio a:b means a/b with b nonzero. Compare like quantities in the same units before simplifying. For 1.5 m ribbon and 60 cm ribbon, convert 1.5 m to 150 cm, giving 150:60 = 5:2.
Order matters: boys:girls differs from girls:boys. If a group has 12 boys and 18 girls, boys:girls = 2:3, but boys:total = 12:30 = 2:5. The total is not the second group alone. Equivalent ratios arise by multiplying or dividing both terms by the same nonzero number, such as 2:3 = 8:12. Adding the same number to both terms does not generally preserve a ratio.
2. Sharing in a given ratio
To divide a total in ratio m:n, calculate total parts m + n, find one part and multiply by each share. For ₹840 divided in ratio 3:4, there are 7 parts, each worth 840/7 = ₹120. Shares are ₹360 and ₹480.
Check both the sum and ratio: 360 + 480 = 840 and 360:480 = 3:4. Do not divide the total by 3 and 4 separately. For three shares in ratio 2:3:5, use 10 parts. This method works because a ratio specifies relative shares, while the total sets their actual scale.
3. Percentages as hundredths
Percent means per hundred. Thus 27% = 27/100 = 0.27. To convert a fraction to a percentage, multiply by 100%. For 3/8, 3 ÷ 8 = 0.375, so 37.5%. To convert a decimal to a percentage, multiply by 100; 0.06 = 6%, not 0.06%.
Form | Fraction | Decimal | Percentage |
|---|---|---|---|
One-half | 1/2 | 0.5 | 50% |
One-quarter | 1/4 | 0.25 | 25% |
Three-fifths | 3/5 | 0.6 | 60% |
One-eighth | 1/8 | 0.125 | 12.5% |
A percentage can exceed 100% when a quantity exceeds its reference whole. For example, 150% of ₹200 is ₹300. This does not violate arithmetic; it means one and a half times the base.
4. Finding a part or the whole
To find p% of a quantity Q, calculate pQ/100. For 18% of 250 students, 18 × 250/100 = 45 students. If the result counts people, interpret whether the given data should yield a whole-number count.
To recover the whole when a part and percentage are known, divide by the percentage fraction. If 42 marks are 70% of the maximum, maximum = 42 ÷ 0.70 = 60 marks. Multiplying 42 by 70/100 would instead find 70% of the part and answer a different question. Identify which number represents 100% before choosing an operation.
5. Comparing unequal groups
Raw numbers can mislead. In Class A, 18 of 30 learners join a library; in Class B, 24 of 50 do. The percentages are 18/30 × 100 = 60% and 24/50 × 100 = 48%. Class B has more members in absolute number, but Class A has the higher participation proportion.
A percentage comparison must use appropriate totals. If a school survey includes absent learners in one class's denominator but excludes them from another's, the comparison may be unfair. Read case-study wording carefully and state assumptions. Percentages summarise proportions; they do not automatically explain why a difference exists.
6. Percentage increase and decrease
Percentage change = change/original quantity × 100. A price increasing from ₹800 to ₹920 changes by ₹120, giving 120/800 × 100 = 15% increase. A decrease from 920 to 800 is 120/920 × 100 ≈ 13.04% decrease, not 15%, because the starting base differs.
For a direct update, new value after p% increase is original × (1 + p/100); after decrease it is original × (1 − p/100). A ₹600 item increased by 10% becomes ₹660. A subsequent 10% decrease gives 660 × 0.9 = ₹594. Equal successive percentage changes do not cancel because their bases are different.
7. Profit and loss
Cost price, CP, is the purchase or production cost in the stated problem. Selling price, SP, is the sale amount. Profit = SP − CP when SP exceeds CP; loss = CP − SP otherwise. Profit% = profit/CP × 100 and loss% = loss/CP × 100.
A shopkeeper buys a bag for ₹750 and sells it for ₹900. Profit = ₹150 and profit percentage = 150/750 × 100 = 20%. Selling at ₹675 gives loss ₹75 and loss percentage 10%. The denominator is cost price, not selling price. To sell an ₹800 item at 15% profit, SP = 800 × 1.15 = ₹920. Include additional costs only when the problem states they form part of cost.
8. Marked price and discount
Marked price, MP, is the listed price before discount. Discount = MP − sale price, and discount% = discount/MP × 100. A jacket marked ₹1,250 with 12% discount has reduction 1,250 × 0.12 = ₹150, giving sale price ₹1,100.
Discount percentage uses marked price, while profit percentage uses cost price. The same sale can therefore involve both discount and profit. If CP is ₹900, MP ₹1,250 and sale price ₹1,100, profit is ₹200, so profit% = 200/900 × 100 ≈ 22.22%. Do not assume a discount automatically means the seller makes a loss. Tax questions must state a rate and base; do not assume an unstated current tax rate.
9. Simple interest
Principal, P, is the original sum. Simple interest, SI, is calculated on the same principal for each period: SI = PRT/100, where R is annual rate percent and T time in years. Amount A = P + SI. These school examples illustrate a model, not a claim about every real financial product.
For P = ₹4,800, R = 7.5% per year and T = 2 years, SI = 4,800 × 7.5 × 2/100 = ₹720, and amount = ₹5,520. For nine months, T = 9/12 = 0.75 year. Interest on ₹4,000 at 8% annually for nine months is 4,000 × 8 × 0.75/100 = ₹240. Keeping months as 9 without conversion would overstate the result twelvefold.
10. Reverse and mixed problems
If a sale price after 20% discount is ₹960, then 80% of MP = 960. MP = 960/0.80 = ₹1,200. Adding 20% of ₹960 gives the wrong original base. Similarly, if amount after two years of 5% simple interest is ₹3,300, then A = P(1 + 5 × 2/100) = 1.10P, giving P = ₹3,000.
A useful written solution names quantities before substitution. For a shop case, list CP, MP, discount and SP separately; for interest, list P, R, T and A. Check the final direction: a discounted price should be below the marked price, and positive simple interest should make amount greater than principal. These checks catch many accidental reversals.
A reliable approach to questions
Identify the whole or base quantity before calculating a percentage.
Convert compared quantities to common units.
Name CP, SP, MP or P, R, T as relevant.
Write the relation, substitute values and keep exact arithmetic where possible.
Check totals, direction of change and the required unit or whole-number restriction.
Connections that matter
Fractions and decimals provide equivalent representations of percentages.
Algebra solves reverse-price and unknown-principal problems.
Data handling uses proportions to compare groups of different sizes.
Common mistakes
Comparing metres with centimetres directly gives an incorrect ratio.
Using selling price as the base for profit% changes its meaning.
Adding equal increase and decrease percentages ignores changing bases.
Using months directly with an annual interest rate ignores time conversion.
Revision checklist
I can simplify ratios and divide totals proportionally.
I can convert and apply percentages.
I can recover an original whole from a part.
I can calculate profit, loss and discount using correct bases.
I can solve simple-interest problems with time conversion.