Class 8 Mathematics Chapter 10: Proportional Reasoning – 2

Two cooks, one batter

Idli batter mixes rice and urad dal, and one common proportion is 2 cups of rice to 1 of dal, written 2 to 1. Viswanath uses 6 cups of rice to 3 of dal, while Puneet uses 4 to 2. Will the idlis taste the same?

The test

Their mixtures are 6 to 3 and 4 to 2. To check whether two ratios match, cross multiply: 6 times 2 is 12, and 3 times 4 is also 12. The two products agree, so the two ratios are proportional and the idlis should taste alike.

Writing the test down

In general, the ratios a to b and c to d are proportional exactly when a times d equals b times c. The same thing can be written as a over c equals b over d. Either form does the same work.

A ratio hiding in the corner of a map

Most maps carry a ratio in one corner, usually 1 against a very large number, such as 1 to 60,00,000. That ratio is called the representative fraction, and it compares a distance on the paper with the matching distance on the ground.

Making sense of the large number

A scale of 1 to 60,00,000 says that 1 cm on the map stands for 60,00,000 cm on the ground. Sorting out the units makes it friendly: sixty lakh centimetres is 60 kilometres. So 1 cm of paper is 60 km of country.

Using it

Measure the gap between two cities on the map with a ruler. If they are 6 cm apart on a map at 1 to 60,00,000, the real gap is 6 times 60, which is 360 km. A finer map at 1 to 50,000 puts each centimetre at only 500 metres.

One warning about map distances

What a map gives is the geographical distance, the gap a bird would fly. The road between the same two towns bends around hills and rivers, so the distance by road is almost always greater.

Four ingredients at once

Viswanath also makes a spice powder. Into it go 8 spoons of ground coriander, 4 dried red chillies, 2 spoons of split toor dal and one of fenugreek. That proportion is written 8 to 4 to 2 to 1, and it has four terms rather than two.

Nothing new is needed

A ratio can carry as many terms as there are quantities, so long as all of them change by one factor. That single condition is what keeps the mixture tasting the same, however many ingredients it has.

Scaling it down

Puneet has only 2 red chillies, which is half of what Viswanath used. So every other quantity must halve as well. That leaves 4 spoons of ground coriander, 2 chillies, one spoon of split toor dal, and fenugreek cut to half a spoon. That is the ratio 4 to 2 to 1 to 0.5.

Terms need not be whole

That half spoon is worth pausing on. A term of a ratio is allowed to be a fraction or a decimal, and nothing about the method changes. Insisting on whole numbers would simply make the mixture wrong.

Saying that two of them match

The two spice ratios are proportional, written 8 to 4 to 2 to 1 as against 4 to 2 to 1 to 0.5. In general, take a to b to c to d against p to q to r to s. They match when a over p, b over q, c over r and d over s are all the same.

Finding the rest from one term

Yasmin wants purple paint in the proportion red to blue to white as 2 to 3 to 5, and she has 10 litres of white. The white matches 5 parts, so one part is 10 divided by 5, which is 2 litres.

Filling in

Now every term follows. Red is 2 parts, which is 4 litres, and blue is 3 parts, which is 6 litres. The paint comes to 4 plus 6 plus 10, which is 20 litres in all.

Concrete for a pillar

Concrete mixes cement, sand and gravel, and the proportion depends on the strength wanted. For pillars, beams and roofs it is 1 to 1.5 to 3. Starting from 3 bags of cement, the factor is 3, so the mix is 3 to 4.5 to 9.

How much that makes

Adding those up gives 3 plus 4.5 plus 9, which is 16.5 bags of concrete from 3 bags of cement. The lesson in both examples is the same: find the factor from the term you know, then use it on every other term.

Scenario:

A site supervisor has three figures to settle before work begins. The plan shows the site 7 centimetres from the depot on a map at 1 to 60,00,000, and the driver wants to know the distance. Concrete for the pillars must be mixed at 1 to 1.5 to 3, and there are 4 bags of cement to use up. A painter needs purple at red to blue to white as 2 to 3 to 5, and has 15 litres of white. The supervisor’s assistant suggests dividing the 15 litres by 3, since the ratio has three terms.

Analysis:

One figure needs a unit conversion, one needs a factor applied to every term, and the last confuses the number of terms with their sum.

Title:

Mix a four-term powder

Safety:

Do not taste anything, and put every ingredient back where it came from.

What You Notice:

Halving gives 4 to 2 to 1 to 0.5 and scaling by one and a half gives 12 to 6 to 3 to 1.5. The fractional terms are not errors: they are what keeps the mixture the same.

A map scale of 1 to 60,00,000 looks forbidding until the units are sorted out. Sixty lakh centimetres is just 60 kilometres, so 1 cm of paper stands for 60 km of ground.

The distance a map gives you is the distance a bird would fly. The road between the same two towns is almost always longer.

One factor, every term — Find the factor from the term you know, then apply that same factor to every other term, leaving none out.

The two-term case, recalled

Splitting 12 in the ratio 2 to 1 was done by adding the terms to get 3 parts, dividing 12 by 3 to get 4, and multiplying each term by that: 8 and 4. Nothing about that method needed there to be only two terms.

Concrete again, with a total

Concrete mixes cement, sand and gravel at 1 to 1.5 to 3, and a job needs 110 units of it. How much of each goes in? The terms add to 1 plus 1.5 plus 3, which is 5.5 parts.

Finding the factor

The whole is 110 units and one round of the recipe is 5.5 units, so the recipe has to be repeated 110 divided by 5.5, which is 20 times. Every term is then multiplied by 20.

Reading off the three amounts

That gives 1 times 20, which is 20 units of cement; 1.5 times 20, which is 30 units of sand; and 3 times 20, which is 60 units of gravel. Checking, 20 plus 30 plus 60 comes to 110.

The rule in general

Write it for any number of terms. Take a quantity x to be divided in the ratio a to b to c and so on. The first share is x times a over the sum of all the terms. The second is x times b over that same sum, and so on for every term there is.

Using the rule directly

For 50 ml of purple paint at red to blue to white as 2 to 3 to 5, the sum of the terms is 10. So red is 50 times 2 over 10, which is 10 ml. Blue is 50 times 3 over 10, which is 15 ml. And white is 50 times 5 over 10, which is 25 ml.

A whole that was fixed for us

Sometimes the whole is not given but known. The angles of any triangle always total 180 degrees, so asking for angles in the ratio 1 to 3 to 5 is asking for 180 divided in that ratio.

Working the triangle out

The terms add to 9, so one part is 20 degrees. The three angles are 1 part, which is 20 degrees; 3 parts, which is 60; and 5 parts, which is 100. Checking, 20 plus 60 plus 100 is 180.

What a pie chart shows

A pie chart shows the proportions of a whole as slices of a circle, which makes a table of data readable at a glance. Suppose 40 pupils scored grades A to E in the counts 12, 10, 8, 6 and 4.

The whole being divided

To draw it, each slice must take an angle proportional to its count. The angles at the centre of a circle total 360 degrees, so the job is to divide 360 in the ratio 12 to 10 to 8 to 6 to 4.

Lightening the arithmetic

That ratio can be reduced first. Dividing every term by their highest common factor, which is 2, turns it into 6 to 5 to 4 to 3 to 2. The proportion is unchanged and the numbers are smaller.

The angles

Those terms add to 20 parts, so one part is 360 divided by 20, which is 18 degrees. The five angles are therefore 6 times 18, which is 108; then 90; then 72; then 54; then 36. They total 360, as they must.

Drawing it, step by step

Draw a circle and one radius out to a point B. Measure 108 degrees round from that radius and draw the next one. From there measure 90 degrees and draw another, then 72, then 54. The last slice closes the circle by itself.

Finishing the chart

Colour and label every slice so that a reader can tell which grade is which. The chart now carries the same information as the table, but the comparison between grades can be seen without reading any numbers.

Starting from percentages instead

Data sometimes arrives as percentages, which changes nothing. Shares of 50, 25, 15 and 10 per cent still divide 360 in that ratio, giving angles of 180, 90, 54 and 36 degrees.

Counts that arrive as a total

Sometimes a table gives a total and leaves one count out. A survey of 360 people found 90 liking summer and 120 liking the rains, with the rest liking winter. Subtracting gives 150 for winter, and only then can the ratio 90 to 120 to 150 be turned into angles.

A ratio a triangle cannot have

One warning about ratios of SIDES rather than angles. No triangle can have sides in the ratio 1 to 3 to 5, because 1 part and 3 parts together come to 4 parts, which is less than 5 and so cannot reach across.

Scenario:

A teacher is writing up the sports day and has four figures to settle. The 150 minute programme is split for warm-up, races, field events and prizes in the ratio 3 to 4 to 3 to 5. A pie chart is wanted for the house points, which came to 12, 10, 8, 6 and 4. Her assistant proposes dividing 180 degrees among the five houses. She also wants a banner triangle with angles in the ratio 4 to 5 to 9, and a second with SIDES in the ratio 1 to 3 to 5.

Analysis:

One figure needs the sum of the terms, one needs the right whole, and the last is impossible.

Title:

Build a triangle from a ratio

Safety:

Nothing here is risky, though the compass point is sharp if you use one.

What You Notice:

The angles work out at 30, 45 and 105 degrees. The sides do not work at all, since 2 cm and 3 cm together make 5 cm and cannot reach across 7 cm. Anything under 5 cm closes it.

A pie chart is nothing but this chapter’s rule applied to a circle. The whole being divided is the 360 degrees at the centre, and the ratio is whatever the data happens to be.

A hundred coins in the proportion 4 to 3 to 2 to 1 of ten, five, two and one rupee pieces come to exactly 600 rupees.

Add, divide, multiply — Add the terms for the parts. Divide the whole by that for one part. Multiply each term by it for each share.

The familiar kind first

The rule of three handles proportions where two ratios match: if a to b matches c to d, then d is b times c over a. Those are called direct proportions, and they let a fourth quantity be found from three known ones.

A direct one, worked

If 5 workers move 4,500 bricks in a day, how many are needed for 18,000 bricks? More bricks means more workers, so the quantities move together. Writing 4,500 to 18,000 as 5 to x gives x as 18,000 times 5 over 4,500, which is 20 workers.

A question that behaves differently

Now a journey. A ride from Lucknow to Kanpur takes 3 hours on a motorcycle at 30 km an hour. How long in a car at 60? Writing 30 to 60 as 3 to x would give 6 hours, which is plainly absurd.

Why it is absurd

Going faster cannot take longer. As the speed rises, the time for the same journey falls. The two quantities are related, but they move in opposite directions, and the rule of three as written assumes they move together.

Reading a table instead

Walking at 5 km an hour takes 18 hours. A bicycle at 15 takes 6. The motorcycle at 30 takes 3. The car at 60 takes 1.5. Every increase in speed brings a decrease in time.

Checking the factor

A bicycle is 3 times faster than walking, since 15 over 5 is 3. And the time falls from 18 hours to 6, which is also a factor of 3. The speed went up by exactly the factor the time came down by.

Naming it

So the two quantities do change by the same factor, which is what makes them proportional, but they change in opposite directions. Such a relationship is called an inverse proportion.

What stays fixed

Look at the products in that table: 5 times 18 is 90, and 15 times 6 is 90, and 30 times 3 is 90, and 60 times 1.5 is 90. The product never changes, and 90 km is the distance from Lucknow to Kanpur.

The definition

A pair of quantities x and y stands in inverse proportion when x times y equals k for some constant k. In a direct proportion it is the quotient that stays fixed; in an inverse one it is the product.

The working rule

From that definition, if x1 and x2 are two values of x with matching y values y1 and y2, then x1 times y1 equals x2 times y2, since both equal k. Rearranging gives x1 over x2 equal to y2 over y1, with the y terms swapped.

Checking the rule

Take walking and the car. Here x1 is 5 and x2 is 60, so x1 over x2 is 5 over 60, which is 0.0833. And y1 is 18 with y2 is 1.5, so y2 over y1 is 1.5 over 18, which is 0.0833 as well.

Workers and days

Twenty workers take 4 days to lay a road. How long will 10 workers take? Fewer workers means more days, so the quantities are inversely proportional. From 20 times 4 equal to 10 times y, the answer is 8 days.

Reading that answer

The workers halved and the days doubled, which is the pattern in one line. If one of them moves by a factor n, its partner moves by one over n. That is as good a test as any for whether a proportion is inverse.

Pumps and provisions

Two pumps fill a tank in 18 hours; four of them take 2 times 18 over 4, which is 9 hours. And food for 80 students lasting 15 days will feed 100 students for 80 times 15 over 100, which is 12 days.

The assumption nobody states

Every one of these rests on an assumption. Each extra worker must work at the same rate as the first, each extra pump must pump as fast, and each extra student must eat the same amount. Without that, the product would not stay constant at all.

Two workers together

Ram cuts a quantity of vegetables in 1 hour and Shyam takes 1.5 hours. Together? Times cannot simply be added or averaged. What can be added are the RATES: in an hour Ram does 1 unit of the job and Shyam does 1 over 1.5, which is two thirds.

Finishing it

Between them they do 1 plus two thirds, which is five thirds of the job an hour. The job and the time are directly proportional here, so if five thirds takes 1 hour, then 1 whole job takes three fifths of an hour, which is 36 minutes.

Scenario:

A contractor has four decisions to make. Twenty painters would finish the hall in 4 days, but only 10 are available, and he wants the new figure. Two pumps fill the water tank in 18 hours, and he is thinking of hiring two more. The kitchen has provisions for 80 guests for 15 days, and 20 more guests have been invited. Finally a small pump takes 3 hours to fill a drum and a large one takes 2, and his foreman says that together they will manage it in two and a half hours.

Analysis:

Three of the four are inverse proportions, and the foreman has added two times that cannot be added.

Title:

Fill in an inverse table

Safety:

Nothing here is risky, but write the constant down before filling anything in.

What You Notice:

Every pair in a genuinely inverse table multiplies to the same number. A table where the products differ is not an inverse proportion at all, however the numbers rise and fall.

Every inverse proportion problem hides a constant. For speed and time it is the distance, for workers and days it is the amount of work, and for pumps and hours it is the size of the tank.

Rates of work add but times do not. One worker taking 3 hours and another taking 2 cannot finish in 2.5 hours together, but in only 6 fifths of an hour.

Direct divides, inverse multiplies — A direct proportion keeps the QUOTIENT fixed. An inverse one keeps the PRODUCT fixed. Ask which stays still.