Fractions in Disguise

Every rupee problem in this chapter is really a fraction problem wearing a different coat. The words per cent come from the Latin per centum, which means out of a hundred. So 37 per cent is just the fraction 37/100, and the symbol % is shorthand for a denominator of 100. Once you accept that, half the work is already done: a percentage is a fraction whose denominator has been fixed at 100 so that different quantities can be compared fairly.

Why fix the denominator at all? Suppose Aarti scored 42 out of 60 in one test and 51 out of 75 in another. Which was the better performance? The fractions 42/60 and 51/75 cannot be compared at a glance. Turn both into percentages and the answer jumps out: 42/60 = 0.7 = 70%, and 51/75 = 0.68 = 68%. The first test was the stronger one. Percentages are a common scale, in the same way that you would convert two distances to kilometres before deciding which is longer.

Fraction to per cent. Multiply by 100 and write the % sign. So 3/4 becomes (3/4 × 100)% = 75%, 7/20 becomes 35%, and 5/8 becomes 62.5%. Per cent to fraction. Divide by 100 and reduce. So 40% = 40/100 = 2/5, 12.5% = 12.5/100 = 125/1000 = 1/8, and 150% = 150/100 = 3/2. Yes, a percentage can be more than 100 — it simply means more than the whole thing you started with.

A few pairs are worth keeping in your head, because they turn ugly multiplications into easy ones: 1/2 = 50%, 1/4 = 25%, 3/4 = 75%, 1/5 = 20%, 1/8 = 12.5%, 1/10 = 10%, 1/20 = 5%, 1/3 = 33 1/3 % and 2/3 = 66 2/3 %. When a question mentions a 12.5% discount, an alert student immediately thinks one-eighth off and divides instead of multiplying.

Finding a percentage of an amount. The word of means multiply. To find 15% of ₹2,400, write 15/100 × 2400 = ₹360. If mental arithmetic suits you better, use the ten-per-cent trick: 10% of ₹2,400 is ₹240 (move the decimal point one place to the left), 5% is half of that, ₹120, so 15% is 240 + 120 = ₹360. Same answer, no calculator needed.

Here is a school-sized example. A school has 1,250 students and 44% of them are girls. Number of girls = 44/100 × 1250 = 550. So the number of boys is 1250 − 550 = 700. Check it: 700 out of 1250 is 700/1250 × 100 = 56%, and 44% + 56% = 100%, so nobody has gone missing. Getting into the habit of a closing check like this will catch most slips before they cost you anything.

Writing one quantity as a percentage of another. Put the part on top, the whole underneath, then multiply by 100. If a batsman scores 91 runs out of a team total of 260, his share is 91/260 × 100 = 35%. If 18 out of 40 students in a class own a bicycle, that is 18/40 × 100 = 45%. The order matters enormously: 91 out of 260 is 35%, while 260 out of 91 would be a completely different and much larger number. Read the sentence carefully and ask yourself which quantity is the whole.

Two cautions before we move on. First, a percentage means nothing on its own until you know what it is a percentage of: 20% of ₹500 is ₹100, while 20% of ₹5,000 is ₹1,000. The same 20% has produced answers ten times apart. Second, everything that follows in this chapter — GST, discount, profit, loss and interest — is this one skill applied to money, with only the choice of base changing. If taking a percentage of a rupee amount still feels shaky, practise it until it does not.

Prices go up, populations grow, phone batteries lose capacity, and shopkeepers cut rates before a festival. All of these are changes, and the fair way to describe a change is as a percentage of what you started with. One rule settles every question in this section: the change is always compared with the original value, never with the value you ended up at.

Suppose a bus fare rises from ₹25 to ₹29. The increase is 29 − 25 = ₹4, and the original fare was ₹25, so increase % = (4/25) × 100 = 16%. Now suppose onions fall from ₹60 a kilogram to ₹48. The decrease is ₹12 on an original of ₹60, so decrease % = (12/60) × 100 = 20%.

See what happens if you are careless and divide by the new value instead. For the onions you would get 12/48 × 100 = 25%, which is not the answer to the question asked. The shopkeeper cut 20% off his old price; the fact that the cut also happens to be one quarter of the new price is a different statement altogether, and mixing the two is the commonest error in the whole chapter.

The multiplier method. Instead of finding the change and then adding it on, you can jump straight to the new value. An increase of 8% turns 100% into 108%, so multiply by 108/100 = 1.08. A decrease of 15% leaves 85%, so multiply by 85/100 = 0.85. This single idea saves a great deal of time and removes a whole class of mistakes.

Example: a monthly salary of ₹18,000 is raised by 12%. New salary = 18000 × 1.12 = ₹20,160. Check the long way: 12% of 18,000 = ₹2,160, and 18,000 + 2,160 = ₹20,160. The two routes agree.

Example: a laptop bought for ₹42,000 loses 15% of its value in a year. Value after one year = 42000 × 0.85 = ₹35,700. The long way: 15% of 42,000 = ₹6,300, and 42,000 − 6,300 = ₹35,700.

Working backwards from the new value. This is where most mistakes happen. After a 20% rise, a gas cylinder costs ₹840. What did it cost before? The wrong move is to take 20% off ₹840 and answer ₹672. The right move is to notice that ₹840 is not 100% of the old price — it is 120% of it. So 120% is 840, which gives 1% = 840/120 = ₹7, and 100% = ₹700. Check: 700 + 20% of 700 = 700 + 140 = ₹840, exactly right. In multiplier language you simply divide instead of multiplying: old value = 840 ÷ 1.20 = ₹700.

Two changes one after the other. Percentages do not add up, because the second percentage is taken on a value that has already moved. If a town of 25,000 people grows by 10% and then shrinks by 10%, you might expect to land back on 25,000. Work it through: 25000 × 1.10 = 27,500, then 27500 × 0.90 = 24,750. The town has ended up 250 people short, a net fall of (250/25000) × 100 = 1%. The reason is easy to see once stated: the 10% rise was taken on 25,000 but the 10% fall was taken on the larger 27,500, so the fall was worth more people than the rise.

One more everyday example. A stationery shop sold 250 notebooks in June and 320 in July. The increase is 70 notebooks on an original of 250, so the rise is (70/250) × 100 = 28%. If August sales then drop to 272, the fall is 48 on an original of 320, which is (48/320) × 100 = 15%. Note carefully that the 28% and the 15% are measured from different starting months, so you cannot subtract them to describe the change from June to August. For that you must compare 272 with 250 directly: an increase of 22 on 250, which is (22/250) × 100 = 8.8%.

Walk into any shop before a festival and you will often see two numbers on the same item: a printed price with a line through it, and a smaller price you actually pay. The printed one is the marked price (MP), also called the list price or the label price. The reduction the shop offers is the discount, and what you finally hand over is the selling price (SP).

The rule that governs everything here is that a discount is always calculated on the marked price — never on the selling price, and never on the cost price. So Discount = discount % of MP, and SP = MP − Discount.

Example: a jacket is marked ₹1,800 and the shop offers 25% off. Discount = 25/100 × 1800 = ₹450, so SP = 1800 − 450 = ₹1,350. With the multiplier method you skip a step, because paying after 25% off means paying 75% of the marked price: SP = 1800 × 0.75 = ₹1,350. Both routes must agree, and if they ever do not, one of your two calculations is wrong.

Finding the discount per cent. A shirt marked ₹2,500 is being sold for ₹2,125. The discount in rupees is 2500 − 2125 = ₹375, and as a percentage of the marked price that is (375/2500) × 100 = 15%. Note again that the base underneath is 2500, the marked price, not the 2125 the customer paid.

Finding the marked price from the selling price. After a discount of 20%, a pair of shoes sells for ₹1,560. What was printed on the tag? Since the customer pays 80% of the marked price, 80% is ₹1,560, so 1% is 1560/80 = ₹19.50 and 100% is ₹1,950. Check it forwards: 20% of 1,950 = ₹390, and 1950 − 390 = ₹1,560. Correct.

Two discounts one after the other. A board says 20% off, and at the counter you are given a further 10% off. That is not 30% off. Take a marked price of ₹4,000. After the first discount: 4000 × 0.80 = ₹3,200. The second discount is calculated on ₹3,200, not on ₹4,000, so the price becomes 3200 × 0.90 = ₹2,880. The total reduction is 4000 − 2880 = ₹1,120, which as a percentage of the marked price is (1120/4000) × 100 = 28%. Successive discounts of 20% and 10% therefore work out to a single discount of 28%, and the shop has quietly kept the other 2%.

GST is charged after the discount. On most bills, tax is applied to the discounted price rather than the label price. Take the jacket above, whose selling price after discount was ₹1,350. If GST is 12%, the tax is 12/100 × 1350 = ₹162, and the bill total is 1350 + 162 = ₹1,512. Notice that this is exactly the percentage-increase multiplier again: 1350 × 1.12 = ₹1,512. A discount is a percentage decrease and a tax is a percentage increase, so the same two tools handle a whole shop bill.

Two final points about the language of shop signs. When a shutter announces up to 50% off, all it promises is that no discount in the shop is more than 50%; many items will be discounted by less, and some may not be discounted at all. And a discount by itself tells you nothing about whether the shopkeeper is still making money. That depends on what the goods cost him, which is exactly what the next section is about.

A shopkeeper buys goods at one price and sells them at another. The price he pays is the cost price (CP); the price the customer pays him is the selling price (SP). If the selling price is larger there is a profit; if the cost price is larger there is a loss. In symbols, Profit = SP − CP when SP is greater than CP, and Loss = CP − SP when CP is greater than SP. At most one of the two can be positive; if the selling price and the cost price are equal there is neither a profit nor a loss, and the deal is said to break even.

Rupees alone do not tell the whole story. A profit of ₹500 on goods costing ₹1,000 is excellent; the same ₹500 on goods costing ₹50,000 is barely worth the trouble. That is why traders talk in percentages, and here is the rule to burn into memory: profit per cent and loss per cent are always calculated on the cost price. The cost price is the money the shopkeeper actually put at risk, so it is the fair base to compare against.

Example: a cycle is bought for ₹4,500 and sold for ₹5,400. Profit = 5400 − 4500 = ₹900, so profit % = (900/4500) × 100 = 20%. Had you divided by the selling price you would have got (900/5400) × 100 = 16.67%, which is a real number but not the answer to this question.

Example of a loss: a phone bought for ₹12,000 is sold second-hand for ₹10,200. Loss = 12000 − 10200 = ₹1,800, and loss % = (1800/12000) × 100 = 15%.

Going forwards, from CP to SP. Selling at a profit of r% means selling at (100 + r)% of the cost price. A pen set costing ₹850 sold at 16% profit fetches SP = 850 × 116/100 = ₹986. Check: 16% of 850 = ₹136, and 850 + 136 = ₹986. Selling at a loss of r% means selling at (100 − r)% of cost, so goods costing ₹2,400 sold at a 12% loss fetch SP = 2400 × 88/100 = ₹2,112.

Going backwards, from SP to CP. This is the reverse question, and it is very easy to get inside out. A table is sold for ₹8,280 at a profit of 15%. What did it cost? That ₹8,280 is 115% of the cost price, so 115% is 8,280, giving 1% = 8280/115 = ₹72 and 100% = ₹7,200. Check forwards: 15% of 7,200 = ₹1,080, and 7,200 + 1,080 = ₹8,280. Correct. Taking 15% off ₹8,280 would have produced ₹7,038, which is wrong, because the 15% was never a percentage of ₹8,280 in the first place.

Overhead expenses belong in the cost price. Money spent on repairs, transport, packing, cleaning or labour before the sale is added to the purchase price to give the true cost price. A dealer buys an old scooter for ₹18,000 and spends ₹2,000 on repairs, so his cost price is ₹20,000, not ₹18,000. If he sells it for ₹23,000, his profit is ₹3,000 and his profit per cent is (3000/20000) × 100 = 15%. Leaving out the repair bill would have suggested a profit of ₹5,000 on ₹18,000, which is about 27.8% — a comfortable-looking figure that does not exist.

Per unit or per lot, but never a mixture. When a trader buys many items you may work with the whole lot or with a single item, provided you stay consistent. If 40 kg of rice costs ₹45 per kilogram, the lot costs 40 × 45 = ₹1,800; if transport adds ₹200, the cost price of the lot is ₹2,000, which works out to ₹50 per kilogram. Selling at ₹56 per kilogram brings in 40 × 56 = ₹2,240, a profit of ₹240, and (240/2000) × 100 = 12%. Working per kilogram instead gives a profit of ₹6 on a cost of ₹50, and 6/50 × 100 = 12% as well. Consistency is what makes the two agree.

When you keep money in a bank, the bank pays you for the use of it; when you borrow money, you pay the lender for the same reason. That payment is called interest. Three quantities decide how much it comes to: the principal (P), which is the sum lent, borrowed or deposited; the rate (R), a percentage for each year; and the time (T), measured in years. The phrase per annum, usually shortened to p.a., means for each year.

In simple interest, the interest for every year is worked out on the original principal alone. If you deposit ₹10,000 at 6% p.a., you earn 6% of 10,000 = ₹600 in the first year, ₹600 again in the second year, and ₹600 in every year that follows. The interest never grows, because it is never added to the principal before the next year is calculated. Multiply the one-year interest by the number of years and the formula falls out by itself: SI = (P × R × T)/100. The total finally paid or received is the amount, A = P + SI.

Example: find the interest and the amount on ₹15,000 at 8% p.a. for 3 years. SI = (15000 × 8 × 3)/100. Take it in order: 15000 × 8 = 1,20,000; then 1,20,000 × 3 = 3,60,000; then divide by 100 to get ₹3,600. The amount is 15,000 + 3,600 = ₹18,600. A quick sanity check: one year would give ₹1,200, and three times that is ₹3,600, which matches.

Time given in months. The rate is a rate per year, so the time must be converted into years before it goes into the formula. Nine months is 9/12 = 3/4 of a year. Interest on ₹24,000 at 10% p.a. for 9 months = (24000 × 10 × 3/4)/100. Here 24000 × 10 = 2,40,000; three quarters of that is 1,80,000; dividing by 100 gives ₹1,800. If you forget to convert and put T = 9 into the formula you get ₹21,600, which is twelve times too large and should look absurd the moment you write it.

Rearranging the formula. Any one of the four quantities can be the unknown. From SI = PRT/100 you can get R = (100 × SI)/(P × T), T = (100 × SI)/(P × R), and P = (100 × SI)/(R × T). There is no need to memorise all four versions. Write the basic formula, substitute everything you know, and solve the small equation that is left.

Example: ₹9,000 earns ₹2,700 as simple interest in 3 years; find the rate. R = (100 × 2700)/(9000 × 3) = 2,70,000/27,000 = 10% p.a. Example: at 12% p.a., how long does ₹5,000 take to earn ₹1,800? T = (100 × 1800)/(5000 × 12) = 1,80,000/60,000 = 3 years. Check that one forwards: one year gives ₹600, so three years give ₹1,800.

When the amount is given instead of the interest. Subtract first. If ₹11,500 grows to ₹13,570 in 2 years, the interest is 13,570 − 11,500 = ₹2,070, and then R = (100 × 2070)/(11500 × 2) = 2,07,000/23,000 = 9% p.a. If it is the principal that is unknown, use the amount directly through A = P × (100 + RT)/100. At 9% for 2 years, RT = 18 and the factor is (100 + 18)/100 = 1.18, so P = 13570 ÷ 1.18 = ₹11,500, which agrees with the figures we started from.

One last idea worth carrying away. How long does money take to double under simple interest? Doubling means the interest earned equals the principal itself, so P = (P × R × T)/100. The P on both sides cancels, leaving R × T = 100. At 12.5% p.a. that needs 8 years; at 10% p.a. it needs 10 years; at 8% p.a. it needs 12.5 years. Notice that the answer does not depend at all on how much money you started with, which is a surprising and rather pleasing result.

Real questions rarely stay inside one heading. A shopkeeper buys at a cost price, marks the goods up, prints that figure as the marked price, allows a discount, and ends at a selling price — and only then can anyone talk about his profit. The safest way through such a chain is to name the base for each percentage before calculating anything, because almost every mistake in this chapter comes from taking a percentage of the wrong number.

Here is the checklist worth writing in the margin. Percentage increase or decrease: the base is the original value. Discount: the base is the marked price. Profit or loss per cent: the base is the cost price. Simple interest: the base is the principal. Four percentages, four different bases — and in a single question they are usually four different numbers.

The full chain. A shopkeeper buys a water bottle for ₹800, marks it 40% above cost, and then allows a 10% discount at the counter. Find his profit per cent.

• The mark-up is on the cost price: MP = 800 × 140/100 = ₹1,120.
• The discount is on the marked price: SP = 1120 × 90/100 = ₹1,008.
• The profit is on the cost price: profit = 1008 − 800 = ₹208, so profit % = (208/800) × 100 = 26%.

The three percentages 40%, 10% and 26% sit on ₹800, ₹1,120 and ₹800 in turn. Combining them directly, as in 40 − 10 = 30%, gives an answer that is simply not the profit per cent.

Designing the marked price. Traders often run the chain backwards. Suppose a trader pays ₹1,200 for an item and intends to make 25% profit even after advertising a 20% discount. What should he mark it at? First fix the selling price he needs: SP = 1200 × 125/100 = ₹1,500. That ₹1,500 has to be 80% of the marked price, so MP = 1500 × 100/80 = ₹1,875. Now check the whole chain forwards: 20% of 1,875 is ₹375, so SP = 1875 − 375 = ₹1,500, and the profit on a cost of ₹1,200 is ₹300, which is (300/1200) × 100 = 25%. Everything ties up.

Interest inside a trading problem. A vendor borrows ₹25,000 for one year at 12% p.a. simple interest and uses it to buy stock. The interest is (25000 × 12 × 1)/100 = ₹3,000, so the money he must recover before he has gained anything is 25,000 + 3,000 = ₹28,000. That figure, and not ₹25,000, behaves as his real cost price. If he sells the entire stock for ₹31,360, his profit is 31,360 − 28,000 = ₹3,360, and (3360/28000) × 100 = 12%. Had he compared his takings with ₹25,000 alone he would have imagined a profit of ₹6,360 and forgotten that ₹3,000 of it already belongs to the lender.

Three habits that prevent mistakes. First, write down what each symbol stands for before substituting anything — CP = ₹800, MP = ?, and so on — so that you can see at a glance which base each percentage sits on. Second, use the unitary method whenever a percentage of an unknown quantity is given: if ₹1,008 is 126% of the cost price, then 1% is 1008/126 = ₹8 and 100% is ₹800. Third, test every answer for sense. A selling price below the cost price must give a loss, never a profit. A discount must make the price smaller, never larger. At simple interest, the interest for 3 years must be exactly three times the interest for 1 year. If any of these checks fails, you have almost certainly taken a percentage of the wrong base, and going back to the checklist above will show you where.