Class 7 Mathematics Chapter 8: Working with Fractions

Three quarters of eight metres

A rope is 8 metres long and a job needs three quarters of it. How much rope is that?

Split the 8 metres into four equal parts of 2 metres, then take three of them. That is 6 metres.

The word "of" is doing a job

Notice what happened. We divided by the bottom number and multiplied by the top one.

In arithmetic, "three quarters of 8" is written 3/4 x 8, and it comes to 6. The word "of" between a fraction and a quantity means multiply.

Multiplying a fraction by a whole number

The same two steps work every time. For 2/5 of 10, split 10 into five parts of 2 and take two of them, giving 4.

For 5/6 of 6, split 6 into six parts of 1 and take five, giving 5.

Why the top number does the counting

A fraction is a count of equal pieces. In 3/4 the bottom number says how big each piece is, and the top number says how many to take.

So 3/4 x 8 is three pieces, each of them a quarter of 8. Written in one line, it is 3 x 8 divided by 4.

Multiplying two fractions needs a picture

Now take 2/3 x 3/5. There is no whole number here to split, so the rope will not help us.

Draw a square and call it one whole. We will cut it twice.

Two cuts make a grid

First cut the square into 5 equal columns, because of the fifths. Then cut it into 3 equal rows, because of the thirds.

The square is now in 15 small rectangles, since 3 rows of 5 make 15. Each one is a fifteenth of the whole.

Reading the shaded part

Shade 3 of the 5 columns, which is the 3/5. Then, inside that shading, take 2 of the 3 rows, which is the 2/3 of it.

The doubly shaded block is 2 rows by 3 columns, so it holds 6 small rectangles. The answer is 6/15, which is 2/5.

Multiply across the top, and across the bottom

Look at where those numbers came from. The 6 on top is 2 x 3, the two top numbers. The 15 underneath is 3 x 5, the two bottom numbers.

That is the whole rule. Multiply the top numbers together for the new top, and the bottom numbers together for the new bottom.

Checking the rule elsewhere

Take 1/4 x 1/3. The tops give 1 and the bottoms give 12, so the answer is 1/12.

The grid agrees. Cutting a square into 4 columns and 3 rows gives 12 pieces, and one of them is picked out.

A whole number is a fraction too

The rule also covers the rope. Write the 8 as 8/1, and 3/4 x 8/1 gives 24/4, which is 6.

So there is only one rule to learn, not two.

The order makes no difference

Multiplying 2/3 by 3/5 gives 6/15, and multiplying 3/5 by 2/3 gives 6/15 as well. The tops are multiplied either way round, and so are the bottoms.

In the grid, swapping the order just turns the picture on its side.

Must the product be the bigger number?

With counting numbers it always is. Three times five is 15, which is more than both 3 and 5.

With fractions that stops being true, and which way it goes depends on whether the numbers are above or below 1.

Both numbers above 1

Take 7/4 x 3. That is 21/4, which is 5 and a quarter.

Twenty-one quarters is more than 7/4 and more than 3. When both numbers are above 1, the product beats both of them.

Both numbers below 1

Take 2/3 x 3/5 again, which came to 2/5. Compare them as fifteenths: 2/3 is 10/15, 3/5 is 9/15, and the answer is 6/15.

So 2/5 is smaller than both. When both numbers are below 1, the product is under both of them.

One of each

Take 3/4 x 8, which is 6. That is more than 3/4 and less than 8, so it lands between the two.

The pattern is really one idea. Multiplying by a number above 1 stretches, multiplying by a number below 1 shrinks, and multiplying by 1 leaves a number alone.

What carries forward

The word "of" means multiply. To multiply two fractions, multiply the top numbers for the new top and the bottom numbers for the new bottom, which the grid of a unit square explains. A whole number can be written over 1, so the same rule covers it.

The order never matters. The product beats both numbers when both are above 1, falls below both when both are below 1, and lands between them when there is one of each.

Scenario:

A shop sells ribbon by the metre. A customer orders three quarters of a roll holding 8 metres. The assistant writes 3 x 8 and then divides by 4, and asks whether that is the same as what the fraction says. A second customer wants 2/3 of the 3/5 of a roll that is left, and the assistant cannot see how to start without a whole number to split. A third customer is told that the second order must be larger than 3/5 of a roll, because multiplying makes things bigger.

Analysis:

One order needs the two steps, one needs the grid, and the third rests on a rule that only holds above 1.

Activity:

Price up five ribbon orders, each a fraction of a roll, and mark which ones came out smaller than the roll you started from.

Title:

Cut a square twice

Safety:

Nothing hazardous here beyond a sharp pencil.

What You Notice:

The grid holds 15 pieces and 6 of them carry both colours, giving 6/15, which is 2/5. Doing it the other way round gives 6/15 again, with the picture turned on its side, which shows the order makes no difference.

Multiplying two fractions needs no common bottom number at all, which is what makes it easier than adding them.

The grid behind 2/3 x 3/5 has exactly 15 pieces, and the 15 is the two bottom numbers multiplied.

Multiplying by 1 is the dividing line: anything above 1 stretches a number and anything below 1 shrinks it.

Of means times, so write the sign.

Below one shrinks, above one stretches.

How many halves in six metres?

A tailor has 6 metres of cloth and each mask needs half a metre. How many masks can she cut?

Each metre gives 2 masks, and there are 6 metres, so she gets 12 masks.

Division is asking how many fit

That question was a division. We wrote 6 divided by 1/2, and the answer was 12.

Division always asks how many of one amount fit inside another. Nothing about that changes when the amounts are fractions.

The answer came out larger

Notice that 12 is larger than the 6 we started with. That can feel wrong, because dividing is supposed to make things smaller.

But halves are small, so a great many of them fit. Dividing by a small amount gives a large count.

Restating it as a multiplication

Look again at how we worked it out. Each metre holds 2 halves, so we multiplied the 6 by 2.

So 6 divided by 1/2 gave the same answer as 6 multiplied by 2. The division turned into a multiplication.

The number that was turned over

Where did that 2 come from? It is the 1/2 with its top and bottom swapped.

A fraction with its top and bottom swapped is called its reciprocal. The reciprocal of 1/2 is 2/1, which is 2.

Why turning it over is the right move

One whole contains exactly 2 halves, 4 quarters, or 8 eighths. In general, one whole holds as many pieces as the bottom number says.

So asking how many of a fraction fit into a number means counting pieces, and that count is what the reciprocal supplies.

The rule

To divide by a fraction, multiply by its reciprocal. Turn the second fraction over and multiply.

Only the second one is turned over. The first is left exactly as it is.

Checking the rule

Take 3/4 divided by 1/8. The reciprocal of 1/8 is 8, so this is 3/4 multiplied by 8, which is 6.

Count it directly to be sure. Three quarters is the same as 6/8, so there are 6 eighths inside it. The rule agrees.

Two fractions at once

Take 2/5 divided by 3/4. The reciprocal of 3/4 is 4/3, so we want 2/5 multiplied by 4/3.

Multiplying tops and bottoms gives 8/15. So 8/15 of a 3/4 fits inside 2/5.

A whole number divided by a fraction

Take 5 divided by 2/3. Write the 5 as 5/1 and turn the 2/3 over to get 3/2.

Then 5/1 multiplied by 3/2 gives 15/2, which is 7 and a half.

A fraction divided by a whole number

Now the other way round. Take 7/8 divided by 7, and write the 7 as 7/1.

Its reciprocal is 1/7, so we want 7/8 multiplied by 1/7, which is 7/56. That is 1/8, and it makes sense: sharing 7 eighths among 7 leaves one eighth each.

When the answer grows, and when it shrinks

Divide by something below 1 and the answer beats what you started with, as in all three of the cases above.

Divide by something above 1 and the answer is smaller, as 7/8 divided by 7 shows. Dividing by exactly 1 changes nothing.

The three names

The number being divided is the dividend. The number you divide by is the divisor. The answer is the quotient.

In 2/5 divided by 3/4, the dividend is 2/5, the divisor is 3/4 and the quotient is 8/15.

A check that always works

Multiply your quotient by the divisor and you should get the dividend back. Division and multiplication undo each other.

For the last case, 8/15 multiplied by 3/4 gives 24/60, which is 2/5. The answer is confirmed.

Why that check is worth doing

The commonest slip in this work is turning over the wrong fraction. That mistake gives a wrong answer which still looks tidy.

The check catches it at once, because the wrong answer will not multiply back to the dividend.

What carries forward

Division asks how many of one amount fit inside another, and a fraction changes none of that. To divide by a fraction, turn that second fraction over and multiply, because one whole holds as many pieces as its bottom number says.

Dividing by a number below 1 makes the answer larger and dividing by one above 1 makes it smaller. Multiplying the quotient by the divisor must give the dividend back, which catches the slip of turning over the wrong one.

Scenario:

A tailor has 6 metres of cloth and cuts masks needing half a metre each. She then has 3/4 of a metre of ribbon left and wants to know how many eighth-metre lengths she can cut from it. A trainee works out 2/5 divided by 3/4 for a third job and writes 15/8. He had turned over the first fraction instead of the second. The tailor then tells him that every division must make a number smaller, and he is unsure.

Analysis:

Two jobs are straight divisions, one answer was flipped the wrong way, and one rule stated as always is wrong.

Activity:

Work five divisions of fractions, check every one by multiplying back, and note whether each answer grew or shrank and what its divisor was.

Title:

Count the pieces, then use the rule

Safety:

Take the usual care with the scissors. No other hazards.

What You Notice:

The folding gives 12 halves in six wholes and 6 eighths in three quarters. The rule gives the same two answers, 6 times 2 and 3/4 times 8, and multiplying each answer back by its divisor returns the number that was divided.

A division can give an answer larger than the number divided, as 6 divided by 1/2 giving 12 shows.

Writing a whole number over 1 means the division rule needs no separate case for whole numbers.

The commonest slip in this work gives a tidy-looking answer, which is why multiplying back is worth the few seconds it costs.

Turn over the second, then multiply.

Multiply back and the dividend returns.

Which parcel is heavier?

One parcel is 4/5 of a 600 gram block. Another is 1/8 of a 5 kilogram sack. Which is heavier?

The fractions give no answer on their own. Four fifths is much more than an eighth, yet the sack is far bigger than the block.

Put them in the same unit first

Before anything else, make the units match. Five kilograms is 5000 grams.

Now both amounts can be worked out in grams and compared directly.

Working the two out

Four fifths of 600 is 480, since a fifth of 600 is 120 and four of those make 480.

One eighth of 5000 is 625, since 5000 divided by 8 is 625. So the second parcel is heavier, by 145 grams.

Why that is worth noticing

The larger fraction gave the smaller parcel. A fraction says nothing at all until you know what it is a fraction of.

So the first question in any such problem is always what the whole is.

A rectangle with fractional sides

A tile is 2 and a half feet by 3 and a fifth feet. What is its area?

Area is still length times width. Fractions do not change that.

Turn mixed numbers into fractions first

Two and a half is 5/2, since two wholes are 4 halves and one more makes 5. Three and a fifth is 16/5 in the same way.

Multiplying gives 80/10, which is 8. The tile is 8 square feet.

Count the gaps, not the posts

A fence has 5 posts in a line and the gap between neighbours is 2/3 of a metre. How long is the run from the first post to the last?

The tempting answer is 5 times 2/3. It is wrong.

Why five posts give four gaps

Walk along the line. There is a gap after the first post, after the second, after the third and after the fourth. The fifth post ends the run.

So 5 posts make 4 gaps, and the length is 4 times 2/3, which is 8/3 metres, or 2 and 2/3 metres.

Scaling a recipe down

A recipe uses 2/3 of a kilogram of flour and serves eight. You want to serve six, which is 3/4 of the number.

So you need 3/4 of 2/3 of a kilogram.

A fraction of a fraction

This is a fraction of a fraction, and the word "of" still means multiply. Three quarters times two thirds gives 6/12.

That is 1/2, so you need half a kilogram of flour.

Filling a tank

A tank is 2/5 full. You pour in enough to raise it by another 1/4 of its capacity. How full is it now?

Here the fractions are both of the SAME whole, the tank, and they are being put together.

Adding, not multiplying

So this one is an addition, not a multiplication. As twentieths, 2/5 is 8/20 and 1/4 is 5/20.

Together they make 13/20, so the tank is 13/20 full.

Which operation does the question ask for?

The two tank fractions were both of the tank, so they added. The two recipe fractions were not: the 3/4 was of the 2/3, so they multiplied.

That is the question to ask. Is the second fraction OF the first, or is it another part of the same whole?

A rough check before you start

A quick estimate catches a wrong operation. Three quarters of two thirds of a kilogram must come out under 2/3 of a kilogram.

Half a kilogram is under 2/3, so the answer survives the check. Had it come out above 2/3, the operation would have been wrong.

A rough check for the comparison

The same habit helps with the parcels. An eighth of 5 kilograms is in the region of 600 grams, while 4/5 of 600 grams is under 600.

So the sack piece is probably heavier, which is what the exact working confirmed.

An estimate is not an answer, and it never replaces the working. What it does is tell you roughly where the answer should land, so that a wrong operation stands out before you have written much down.

What carries forward

Make the units match before comparing, and remember that a fraction means nothing until you know its whole. Mixed numbers become fractions before multiplying, and a row of posts has one gap fewer than it has posts.

Fractions of the SAME whole are added, while a fraction OF another fraction is multiplied. A rough estimate first will usually catch a wrong operation.

Scenario:

A baker compares two bags of flour. One is 4/5 of a 600 gram block and the other is 1/8 of a 5 kilogram sack. She says the first must be heavier, because four fifths beats an eighth. She then scales a recipe down from eight people to six. It uses 2/3 of a kilogram, and she adds the 3/4 to the 2/3. Outside, a worker prices a fence of 5 posts with gaps of 2/3 of a metre, and multiplies 5 by 2/3. The baker also needs the area of a worktop 2 and a half feet by 3 and a fifth feet.

Analysis:

Three of the four pieces of reasoning are wrong, each in a different way, and the fourth needs mixed numbers converted.

Activity:

Collect five fraction problems from around the house, and before working any of them write down whether it needs adding or multiplying and why.

Title:

Weigh, pace and measure

Safety:

Take care lifting heavy weights, and keep the scales on a flat surface.

What You Notice:

The eighth of the sack comes out heavier at 625 grams against 480, although it is the smaller fraction. The row of 5 markers measures 8/3 of a metre, matching 4 gaps rather than 5, and the table area comes out as the product of the two sides written as fractions.

Four fifths of a 600 gram block weighs less than an eighth of a 5 kilogram sack, although four fifths is the far larger fraction.

A row of posts always has one gap fewer than it has posts, which is why the tempting multiplication is the wrong one.

A rough estimate costs seconds and catches a wrong operation, because multiplying by a fraction below 1 has to shrink an amount.

Ask what the whole is, before anything else.

Count the gaps, never the posts.