Class 8 Mathematics Chapter 7: Proportional Reasoning – 1

Five pictures of a tiger

Here are five copies of one photograph, printed at different sizes. Three of them look right. The fourth looks stretched tall, and the fifth looks squashed and fat. All five show the same tiger, so what has gone wrong?

Measuring them

Take a ruler to each. The first is 60 millimetres wide and 40 high. The stretched one is 40 by 20. The third is 30 by 20, the fourth 90 by 60, and the squashed one 60 by 60.

The first guess

The squashed one is a square, so perhaps that is the trouble. But the stretched one is a rectangle like the three that look right. Being a rectangle is not the answer, so the numbers must be hiding something.

Comparing by division

Set the 60 by 40 picture against the 30 by 20 one. The width has halved and so has the height. Both measurements have been multiplied by the same number, and the picture survives.

Comparing by subtraction

Now set it against the 40 by 20 one. The width has dropped by 20 millimetres and so has the height, so the DIFFERENCE is the same for both. Yet this is the stretched one. Equal differences are not what the eye notices.

Checking the factor instead

Look at factors. The height has halved, from 40 to 20. The width has gone from 60 to 40, which is not halving. The two measurements have been multiplied by different numbers, and that is exactly why it looks wrong.

The rule, then

Two quantities change alike when both are multiplied by one and the same factor. Changing both by the same amount through subtraction will not do it. When the changes do match in that way, we call them proportional.

Writing it down

Mathematics has a notation for this. The width set against the height of the first picture is written 60 to 40, with a colon between. The two numbers in a ratio are called its terms, so the terms here are 60 and 40.

What a ratio says

Reading a ratio a to b aloud, it claims that wherever a of the first quantity are counted, b of the second are counted alongside. So the first picture carries 40 millimetres of height against 60 of width, however large or small the printed copy is.

Multiplying a ratio

Multiply both terms of 60 to 40 by a half and you get 30 to 20, which is the third picture. Multiply both by one and a half and you get 90 to 60, the fourth. Those three pictures are the ones that look alike.

A tidier way to compare

Hunting for the factor each time is clumsy. There is a neater test. Reduce every ratio to its simplest form and see whether the simplest forms agree.

Reducing to simplest form

Divide both terms by their highest common factor. For 60 and 40 that factor is 20, which leaves 3 to 2. For 90 and 60 it is 30, which leaves 3 to 2 again. The two ratios have the same simplest form.

Settling the five pictures

Run them all through. The 60 by 40, the 30 by 20 and the 90 by 60 all come to 3 to 2. The stretched 40 by 20 comes to 2 to 1, and the square 60 by 60 comes to 1 to 1. The three that match are the three that look alike.

In proportion

When two ratios have the same simplest form, we say they are in proportion, and the notation is a double colon. So 60 to 40 is in proportion with 30 to 20, and also with 90 to 60.

A warning about adding

One thing deserves its own warning. Adding or subtracting the same number from both terms of a ratio changes the ratio, and the new one is usually not in proportion with the old.

Ages make the point

When Neelima was 3, her mother was 30, so the ratio of their ages was 3 to 30, or 1 to 10. Nine years later Neelima is 12 and her mother is 39. That ratio is 12 to 39, which reduces to 4 to 13.

Why that happens

Four to 13 is nowhere near 1 to 10. Both ages went up by the same 9 years, and the ratio still moved. Nine years is a large share of a 3-year-old life and a small share of a 30-year-old one. Equal additions are not equal factors.

Scenario:

A school is ordering a banner from a photograph that measures 1200 pixels across and 800 down. The printer offers three sizes. The first is 600 by 400. The second is 600 by 200, which the printer says he reached by taking 600 off each measurement. The third is 1800 by 1200. A teacher asks which of the three will show the photograph correctly. The printer replies that the second must be fine, since he changed both measurements by exactly the same amount.

Analysis:

All three can be settled by reducing the ratios, and the printer has used the test that does not work.

Title:

Hunt for ratios at home

Safety:

Ask an adult before standing on anything to reach a window.

What You Notice:

Most things end up in groups of one, but any two objects sharing a simplest form will look like scaled copies of each other even when their sizes are nothing alike.

Two images can have exactly the same difference between width and height and still look nothing alike, because what the eye notices is the factor and not the gap.

A ratio of ages never stays put. A child of 3 with a mother of 30 is at 1 to 10, and nine years later the same pair is at 4 to 13.

Multiply together, or it bends — Both terms must be multiplied by the same number for the shape to survive. Subtracting the same amount from each is what stretches a picture.

Three numbers and a gap

Kesang makes 6 glasses of lemonade and stirs in 10 spoons of sugar. Her father asks for 18 glasses, at the same sweetness. How many spoons does she need? Three numbers are known and one is missing.

Setting it up

Sweetness stays the same when the ratio of glasses to spoons stays in proportion. So the proportion to solve is 6 to 10, in proportion with 18 to something. Writing it that way is most of the work.

The factor of change

Both terms of a ratio must take the same factor. The first term has gone from 6 to 18, so divide 18 by 6 to find the factor, which is 3. Apply the same 3 to the second term: 10 becomes 30.

Reading the answer

So 6 to 10 is in proportion with 18 to 30, and Kesang needs 30 spoons of sugar. A common wrong answer is 22, from adding 12 to the sugar because 12 glasses were added. That is the trap from the first topic in a new coat.

When the factor is awkward

Sometimes the factor is not a whole number. Suppose 14 to 21 must be matched by something to 42. The second term has doubled, so the first term doubles too, giving 28 to 42.

And when it is a fraction

Now match 14 to 21 by 6 to something. What carries 14 to 6? Not a whole number: the factor is 6 divided by 14, which is three sevenths. Apply three sevenths to 21 and you get 9, so the answer is 6 to 9.

A general method is wanted

Hunting for a fraction each time is awkward. There is a method that never needs it. Two ratios are in proportion exactly when their cross products agree: the first term times the fourth equals the second times the third.

Cross multiplication

Take 6 to 10 in proportion with 18 to x. Cross multiplying gives 6 times x equal to 10 times 18, which is 180. So x is 180 divided by 6, which is 30. The same answer, with no factor to find.

An old name for it

This is not new. In ancient India, Aryabhata and others called these Rule of Three problems, because three numbers are given and the fourth is wanted. Aryabhata was writing around 499 CE.

Aryabhata’s four names

He named the parts. The first number was the pramana, or measure. The second was the phala, or fruit. The third was the ichchha, or requisition. The answer was the ichchhaphala, or yield.

His instruction

His rule was a single sentence: multiply the phala by the ichchha, and divide the product by the pramana. In our letters, the fourth number is b times c, divided by a. That is exactly cross multiplication.

A trap with units

Here is a car problem. It covers 90 kilometres in 150 minutes. How far does it go in 4 hours at the same speed? Writing 150 to 90, in proportion with 4 to something, looks reasonable and is wrong.

Why it is wrong

The 150 is in minutes and the 4 is in hours, so the two ratios are not comparing like with like. Four hours is 240 minutes. The proportion must read 150 to 90, in proportion with 240 to something.

Finishing it

Cross multiplying now gives 150 times x equal to 240 times 90, which is 21600. Dividing by 150 gives 144. So the car covers 144 kilometres in four hours. The arithmetic was never the hard part.

Comparing two prices

A farmer in Himachal sells 200 grams of tea for 200 rupees. An estate in Meghalaya sells a kilogram for 800. Do those two weight-to-price comparisons match, and which of the teas is dearer?

The units again

Work in grams throughout. The Himachal ratio is 200 to 200, which reduces to 1 to 1. The Meghalaya ratio is 1000 to 800, which reduces to 5 to 4. Those are different, so the two are not in proportion.

Which is dearer

The ratios say they differ but not which is worse value. For that, compare at the same weight. Two hundred grams is a fifth of a kilogram, so a kilogram of the Himachal tea costs five times 200, which is 1000 rupees. The Meghalaya kilogram costs 800.

The lesson of the tea

So the Himachal tea is the dearer of the two, at 1000 against 800 a kilogram. Notice that the cheaper-looking packet was the dearer tea. Prices can only be ranked once both are brought to one quantity.

Scenario:

A school office is making two decisions in one afternoon. The first is about rice. One shop sells 2 kilograms for 180 rupees and another sells 5 kilograms for 460. A clerk says the second shop is dearer, since 460 is more than 180. The second decision is about a coach. It covered 120 kilometres in 90 minutes on a trial run, and the office needs to know how far it will get in 3 hours at the same speed. A second clerk writes the proportion as 90 to 120, matched with 3 to something, and reaches an answer of 4.

Analysis:

One clerk has compared prices at different quantities, and the other has cross multiplied before making the units agree.

Title:

Solve one problem three ways

Safety:

No equipment is involved, so no safety step is needed.

What You Notice:

Both routes agree every time. When the factor is a fraction the cross product route is the easier one, which is why it is worth knowing both.

Aryabhata, writing around 499 CE, gave the rule in one sentence: multiply the phala by the ichchha and divide by the pramana. That is cross multiplication, fifteen centuries early.

A 200 gram packet of tea at 200 rupees works out dearer than a kilogram at 800 rupees, because the first comes to 1000 rupees a kilogram.

Units first, then cross — Make the units agree before anything else. Then the first times the fourth equals the second times the third, and the missing number falls out.

Twelve counters, shared unequally

Take 12 counters and share them between two people, not equally but in the ratio 3 to 1. Dealing by hand works. Give one person 3 and the other 1, then repeat, and repeat again. The counters run out after three rounds.

What the dealing showed

Counting up, one person has 9 counters and the other has 3. So 12 shared in the ratio 3 to 1 gives 9 and 3. Check it: 9 to 3 reduces to 3 to 1, so the shares really are in the ratio asked for.

Why dealing is not enough

Now try sharing 42 counters in the ratio 4 to 3. Dealing them out would take six rounds and plenty of counting, and a larger total would be worse. A shortcut is wanted.

Counting the groups

Look again at what the dealing did. Each round handed out 4 counters to one person and 3 to the other, which is 7 counters a round. So the 42 counters divide into groups of equal size, and one person wants 4 groups while the other wants 3.

Finding the group size

The total number of groups is 4 plus 3, which is 7. So each group holds 42 divided by 7, which is 6 counters. That one division replaces all the dealing.

Filling the shares

Now multiply. The first person gets 4 groups of 6, which is 24. The second gets 3 groups of 6, which is 18. Check the total: 24 plus 18 is 42. Check the ratio: 24 to 18 reduces to 4 to 3.

The rule in general

To split a quantity in the ratio m to n, add the terms to get m plus n groups. Divide the quantity by that to find one group. Then the first share is m groups and the second is n groups.

Why the sum, and not something else

The sum of the terms is the number of groups because that is what one round of dealing hands out. Nothing else in the problem counts rounds, which is why adding the terms is the step people most often skip.

Sharing a profit

Two friends start a food cart. One puts in 75,000 rupees and the other 25,000, and they agree to split profits in the same ratio. The first month’s profit is 4,000 rupees. How much does each get?

Working it through

The investment ratio is 75,000 to 25,000, which reduces to 3 to 1. The groups number 3 plus 1, which is 4, so one group is 4,000 divided by 4, or 1,000 rupees. The first friend takes 3 groups, which is 3,000, and the second takes 1,000.

Changing a mixture

Here is a harder one. A 40 kilogram mix of sand and cement is in the ratio 3 to 1. How much cement must be added to make the ratio 5 to 2?

What is in it now

First find the present amounts. The groups number 3 plus 1, which is 4, so one group is 40 divided by 4, or 10 kilograms. That gives 30 kilograms of sand and 10 of cement.

What changes and what does not

This is the step that catches people. Only cement is being added, so the sand stays at 30 kilograms throughout. The new ratio of 5 to 2 must be reached by moving the cement alone.

Finishing it

If 30 is to stand to the cement as 5 stands to 2, then the cement must be 12 kilograms, since 30 is 6 times 5 and 12 is 6 times 2. There are already 10 kilograms, so 2 kilograms more are needed.

Units get in the way

Proportion problems constantly come with mismatched units, so a few conversions are worth having. A metre is 3.281 feet and a square metre is 10.764 square feet. An acre is 43,560 square feet, while a hectare is 10,000 square metres, or 2.471 acres. For liquids, a millilitre is the same as a cubic centimetre.

A rate and a conversion together

Good practice is ten tonnes of manure to the acre. A farmer has a plot 200 feet by 500 feet. That is 100,000 square feet, which is 100,000 divided by 43,560 acres, or about 2.296. At ten tonnes an acre he needs about 22.96 tonnes.

Temperature is the odd one

Most conversions are a single multiplication, but temperature is not. Zero degrees Celsius is 32 Fahrenheit, so the scales do not share a zero. Fahrenheit is nine fifths of the Celsius figure, plus 32.

Checking it both ways

So 25 degrees Celsius is nine fifths of 25, which is 45, plus 32, giving 77 Fahrenheit. Going back, Celsius is five ninths of the Fahrenheit figure less 32. For 98.6 Fahrenheit that gives five ninths of 66.6, which is exactly 37 degrees.

Scenario:

Two cousins buy a plot measuring 300 feet by 290.4 feet to grow vegetables. They put in 60,000 and 40,000 rupees and agree to share everything in that ratio. Three questions arise on the first day. Good practice is ten tonnes of manure to the acre, and one cousin multiplies the area in square feet by ten and reads off 870,000 tonnes. The first season’s profit of 25,000 rupees has to be divided. And a 60 kilogram mix of sand and cement at 4 to 1 needs changing to 3 to 1, so they want to know how much cement to add.

Analysis:

One cousin has applied a per-acre rate to an area in square feet, and the mixture question needs the untouched ingredient as its anchor.

Title:

Convert a real plot

Safety:

Keep the tape out of the way of anyone walking past, and do not measure near a road.

What You Notice:

A school playground usually comes out between a quarter and half an acre. Doing the two conversions in turn shows why a single wrong step makes the final figure meaningless.

An acre is 43,560 square feet, which is a number nobody would choose on purpose. It comes from an old measure of the land one ox team could plough in a day.

A plot measuring 200 feet by 500 feet is just under two and a third acres, so at ten tonnes to the acre it needs nearly 23 tonnes of manure.

Count the groups, then fill them — Add the two terms to get the number of groups. Divide the total by that to get one group. Then give each share its own number of groups.