Class 8 Mathematics Chapter 8: Fractions in Disguise

A symbol everyone has seen

Shop windows announce a sale of up to 50% off. A report card says a pupil scored 83%. The symbol turns up everywhere, so it is worth asking exactly what it says.

What the word means

It is read as per cent, from the Latin per centum, meaning by the hundred. So 25% means 25 out of every hundred: 25 people in every hundred people, or 25 rupees in every hundred rupees, or 25 marks in every hundred marks.

A fraction in disguise

Take that definition seriously and the symbol becomes ordinary. A count of 25 out of every 100 is the fraction 25 over 100. So a percentage is simply a fraction whose denominator has been fixed at 100, which is where this chapter gets its name.

Simplifying one

Being a fraction, a percentage can be reduced. Twenty per cent is 20 over 100, which is 2 over 10, which is one fifth. And 24% is 24 over 100, or 12 over 50, or 6 over 25. Any equivalent fraction will do.

Going the other way

If a percentage is a fraction, a fraction should be expressible as a percentage. Surya mixes paint for a sunset and red makes up three quarters of the mixture. What percentage of the colour is red?

Climbing to a hundred

Three quarters means 3 in each group of 4. Double the group and it becomes 6 in each 8. Take five such groups and it is 30 in each 40. Double once more and 75 sit in each 100, so the answer is 75%, with yellow making up the other 25%.

The same step, done once

Rather than climb, look for the equivalent fraction with 100 underneath in one move. Three quarters is 3 times 25 over 4 times 25, which is 75 over 100. Multiplying top and bottom by the same number has not changed the fraction at all.

The rule behind both methods

A fraction measures a part of one unit, while a percentage measures a part of a hundred. So to express a fraction as a percentage, multiply it by 100. Three quarters times 100 is 75, and two fifths times 100 is 40.

Checking it on a harder one

Try 15 of the 80 pupils in a school who walk to school. Multiplying 15 over 80 by 100 gives 18.75, so 18.75% walk. The answer need not be a whole number, and often is not.

Why bother at all

If a percentage is just a fraction, why not stay with fractions? Consider a biscuit factory trying two recipes. Sugar makes up 9 over 34 of the first and 13 over 45 of the second. Which is sweeter?

The reason in one line

Nothing about those two fractions answers the question at a glance. Say instead that sugar is 26.47% of the first and 28.89% of the second, and the answer is immediate. Fixing the denominator is the whole point.

Why a hundred, though

In principle any denominator would serve. Something could be quoted per ten, or per thousand, and statistics really do say per lakh. But 100 has advantages the others lack.

The case for it

Our number system counts in tens, so 10, 100 and 1,000 sit comfortably with decimals: 31% is 31 over 100, which is 0.31 straight away. A hundred is also round and easy to hold in the head. Ten would be too coarse for most purposes.

Three ways of writing one number

That decimal step deserves its own name. A half can be written as the fraction one over two, the decimal 0.5, or 50%. These are three spellings of the same number, not three different numbers.

Moving between the three

Going from a percentage to a decimal, divide by 100: so 75% is 0.75 and 1% is 0.01. Going from a fraction to a decimal, divide out: five eighths is 0.625, which is 62.5%. Each conversion is one short step.

Reading one straight off a grid

A ten by ten grid of squares holds exactly a hundred of them, which makes it the natural picture for this chapter. Shade three of every four squares and 75 end up shaded, so the count of shaded squares IS the percentage, with no conversion left to do.

A quick comparison without a calculator

The three forms let you judge sizes quickly. Is 3 over 11 bigger than 61%? Three elevenths is less than a third, and a third is about 33%, so the answer is no, with no arithmetic needed.

Scenario:

Two friends are comparing packets in a shop and keep running into the same difficulty. A juice carton says fruit makes up 17 over 50 of the contents, while a second carton says 34 per cent. One friend insists these must be different because one is a fraction. A biscuit packet lists sugar as 9 over 16 of the weight, and they want that as a percentage. A third packet says 12 per cent nuts, and they want that as a fraction in its simplest form. Finally, one of them claims that since 15 of 80 grams is not a whole number of hundredths, no percentage can be written for it at all.

Analysis:

Three of the four difficulties are about moving between the forms, and the fourth is about whether percentages have to be whole.

Title:

Build a hundred grid

Safety:

Nothing here is risky, but keep the pencils away from the younger ones in the house.

What You Notice:

The grid has 100 squares, so the count of shaded squares IS the percentage. That is the whole idea behind fixing the denominator at a hundred.

The idea of counting by the hundred is older than the decimal fraction. Kautilya’s Arthashastra, from the fourth century BCE, already quotes interest rates in panas per month per hundred.

Fifteenth-century Italian manuscripts write what we would spell as 20 per cent in the form xx p cento, which is where the symbol eventually came from.

Per cent, per hundred — Read the symbol as out of a hundred. Going from a fraction, multiply by 100. Coming back, divide by 100.

Two biscuits, two percentages

Madhu ate biscuits with 25% sugar and Madhav ate a different variety with 35% sugar. Who ate more sugar? The obvious answer is Madhav, and with no further information the obvious answer is not safe.

What the percentages are of

A percentage is a proportion, so it only means something alongside the quantity it is a proportion of. If both ate 100 grams of biscuits, then Madhav really did eat more: 35 grams of sugar against 25 grams.

Change the amounts

Now suppose Madhu ate 120 grams and Madhav ate 95 grams. The percentages have not changed, but the question is open again, because the two proportions are proportions of different wholes.

Working out Madhu’s sugar

Twenty-five per cent means 25 grams of sugar in every 100 grams of biscuit. That is 5 grams in every 20 grams, so 120 grams carries 30 grams. Written as a proportion, 25 over 100 equals s over 120, giving s as 30.

Working out Madhav’s

His numbers are less friendly. A hundred grams holds 35 grams of sugar, so one gram holds 35 over 100 of a gram, and 95 grams holds 35 over 100 times 95, which is 33.25 grams. So Madhav ate more after all, but only just.

The rule in general

That working gives the rule. To find y per cent of a quantity, multiply the quantity by y over 100. So 45 per cent of some number z is 45 over 100 times z. The word OF is a multiplication sign in disguise.

Doing it in your head

Some percentages are easy and others can be built from them. Twenty-five per cent is a quarter, because 25 over 100 reduces to one over four, so 25 per cent of 40 is simply 10.

Doubling and adding

Twenty per cent of a number is always double 10 per cent of it, since 20 parts in a hundred is twice 10 parts. Percentages also add: 20 per cent of a number plus 5 per cent of it gives 25 per cent of it, because those fractions add.

Using that on a hard one

Fifteen per cent looks awkward but is 10 per cent plus 5 per cent. For 240, a tenth is 24 and half of that is 12, so 15 per cent is 36. Ninety per cent and 70 per cent come just as easily by subtracting from the whole.

The decimal route

Fifty per cent of a number can be found by halving it, and halving is multiplying by 0.5. Since 50 over 100 is 0.5, the two are the same operation. So 80 per cent of 75 is 0.8 times 75, which is 60.

Starting from a ratio

Millet kanji mixes millet and water in the ratio 2 to 7. What percentage of the mixture is millet? The ratio of millet to the whole mixture is 2 to 9, so millet is two ninths of it.

Finishing the kanji

Two ninths times 100 is 22.22, so millet is 22.22% of the mixture and water the other 77.78%. For 500 ml of mixture, the millet needed is five times 22.22, which is 111.1 ml.

Estimate before you calculate

Estimating first catches mistakes. Half of 9 is 4.5, so two ninths is clearly under 50 per cent. Half of 4.5 is 2.25, so it is under 25 per cent too. And 20 per cent of 9 is 1.8, so the answer sits between 20 and 25 per cent.

Working backwards

A cyclist has covered 40 per cent of the road to Agra, which is 92 km. How much further must he ride? This runs the other way: a part is known and the whole is wanted.

Two ways to finish it

The short way scales down. Forty per cent is 92, so 20 per cent is 46, and 60 per cent is 92 plus 46, which is 138 km. The long way finds the whole first: 92 divided by 0.4 is 230, and 230 less 92 is 138 again.

Going past a hundred

Kishanlal aims at daily sales of 5,000 rupees. On one day he takes 2,000, which is 2,000 over 5,000 of the target, or 40 per cent. On another he takes 6,000. A thousand is 20 per cent of 5,000, so he has reached 120 per cent.

What above a hundred means

There is nothing strange in it. A percentage above 100 simply means more than the whole, and the arithmetic is unchanged: 6,000 over 5,000 times 100 is 120. Taking 9,550 against the same target gives 191 per cent.

Reading it as a multiplier

A farmer harvested 260 kg last year and 650 kg this year. Dividing and multiplying by 100 gives 250 per cent of last year, and 250 per cent means two and a half times. So a percentage above a hundred is just a multiplier bigger than one.

Scenario:

Kishanlal sets himself a daily sales target of 5,000 rupees. On Monday he took 2,000 rupees. On Thursday he took 6,000. He also needs a 500 millilitre batch of kanji for the staff, mixing millet and water at 2 to 7. His assistant has written down that millet makes up two sevenths of the mixture. Separately, a delivery driver has covered 40 per cent of his round. That came to 92 kilometres, and he wants to know how much is left.

Analysis:

One figure goes past a hundred per cent, one confuses a part with the whole, and one runs the calculation backwards.

Title:

Mix a kanji by percentage

Safety:

Do not boil anything without an adult present, and keep away from the stove.

What You Notice:

The ratio has to be compared with the whole mixture, not with the other ingredient. That one step is where most of these problems are won or lost.

Working out 15 per cent in your head is usually easier than reaching for a calculator. Take a tenth, halve that tenth, and add the two.

A shopkeeper who makes 6,000 rupees against a target of 5,000 has reached 120 per cent of the target, which is another way of saying he was 20 per cent over it.

Of means times — Read y per cent OF a quantity as y over 100 TIMES that quantity. The word of is the multiplication sign in disguise.

Describing a change

The price of a kilogram of tomatoes was 30 rupees three years ago and is 42 rupees now. The increase is 12 rupees. But 12 rupees means little on its own, so the rise is usually quoted as a percentage.

Which number is the base

Divide the increase by the ORIGINAL amount and multiply by 100. So 12 over 30 times 100 is 40, and the price has risen by 40 per cent. The original amount is the base because that is what the change is being measured against.

A fall works the same way

A cinema that used to draw 160 people now draws 100. The decrease is 60, so the percentage decrease is 60 over 160 times 100, which is 37.5 per cent. Dividing by the new figure instead would have given a different and wrong answer.

Two ways of saying one thing

Here are two statements. First, a population is 165 per cent of what it was. Second, it has risen by 65 per cent. These say the same thing: the first reads as 1.65 times the old figure, and the second as the old figure plus 0.65 of it, which is also 1.65 times it.

The three prices

Buying and selling brings three prices. The cost price is what the seller paid. The marked price is what the seller asks, which may be the MRP. The selling price is what the customer finally pays after any bargaining or discount.

Profit on a sweater

Kishanlal buys a sweater at 300 rupees, marks it at 480 and sells it at 430. The profit is the selling price less the cost price, which is 130 rupees. Had the selling price been below the cost price, the difference would have been a loss.

Profit as a percentage

Profit per cent is normally measured against the cost price, because the question it answers is how much was made on what was invested. So the profit here is 130 over 300 times 100, which is 43.33 per cent.

A loss, worked the same way

Raghu buys 10 kg of rice at 35 rupees a kilogram, so 350 rupees in all, and clears it at 300. The loss is 50 rupees, and the percentage loss is 50 over 350 times 100, which is 14.29 per cent.

Working from the percentage instead

Shyamala paid 2,650 rupees for a vase and must sell a damaged one at an 18 per cent loss. One route takes 18 per cent of 2,650, which is 477, and subtracts: 2,650 less 477 is 2,173 rupees.

The shorter route

The other route notices that losing 18 per cent leaves 82 per cent. So the selling price is 0.82 times 2,650, which is 2,173 rupees straight away. One multiplication has replaced a multiplication and a subtraction.

Discounts

A discount is a percentage taken off the marked price. A cooker marked at 1,800 rupees with 35 per cent off sells for 65 per cent of 1,800, which is 1,170 rupees. If the shop had paid 900 for it, the profit of 270 is 30 per cent on cost.

Interest without compounding

A bank offering 10 per cent a year on a deposit of 6,000 rupees pays 600 rupees a year. If that interest is handed over each year, the principal stays at 6,000 throughout, so three years pay 1,800 and the depositor ends with 7,800 rupees.

Interest with compounding

Now suppose the interest is left in. The first year ends at 6,600. The second year earns 10 per cent of 6,600, which is 660, ending at 7,260. The third earns 726 and ends at 7,986. Leaving the interest in has gained 186 rupees.

The two patterns

Without compounding, each year adds the same amount, so the total is the principal times one plus the rate times the number of years. With compounding, each year multiplies by one plus the rate, so the total is the principal times that factor raised to the number of years.

Value running downhill

The same arithmetic works downwards. A television bought at 21,000 rupees that loses 5 per cent of its value in a year is worth 95 per cent of 21,000, which is 19,950. That fall through age and use is called depreciation.

Why two changes never add

A bakery offers 30 per cent off and then 20 per cent more off. That sounds like 50 per cent, but the second discount is taken on the already reduced price. On a 200 rupee cake, 30 per cent off leaves 140, and 20 per cent off that leaves 112, not 100.

A costly mistake

Surbhi adds a 50 per cent margin to everything she buys, then offers a 50 per cent discount expecting to break even. Her selling price was 1.5 times cost, and half of that is 0.75 times cost. She is selling at three quarters of what she paid, a loss of 25 per cent.

Scenario:

A shopkeeper is clearing stock and has four figures to settle. He bought a jacket for 480 rupees and sold it for 552. He bought a vase for 2,650 and must sell a chipped one at an 18 per cent loss. He is advertising 30 per cent off and then 20 per cent more off, and tells customers this is half price. And he has 8,000 rupees spare, which the bank will take for two years at 10 per cent, either paying the interest out each year or leaving it in.

Analysis:

One figure is an ordinary profit, one is a loss worked backwards, one is a pair of changes that multiply, and one is a choice about compounding.

Title:

Race two deposits

Safety:

Nothing here is risky, but check each year against the calculator before moving on.

What You Notice:

The first line is straight and the second bends upwards. That is the difference between adding the same amount each year and multiplying by the same factor each year.

A shop offering 30 per cent off and then 20 per cent more is not offering half price. On a 200 rupee cake the two discounts leave 112 rupees, while a straight 50 per cent would leave 100.

Money at 10 per cent a year doubles in ten years without compounding but in only eight years with it, which is the difference between linear and exponential growth.

Change over original — Divide the change by what you STARTED with, never by what you ended with. Then multiply by 100.