Curriv

Working with Fractions

Chapter 8 · Mathematics · Class 7 26 min read

Why This Matters

You use fractions every single day, even when you do not notice.

Half a glass of milk. A quarter of a pizza. Three-fourths of an hour to reach school. You already share, split and measure with fractions all the time.

But up to now you have mostly added and subtracted fractions. In this chapter we do two new things: we multiply fractions and we divide them.

Why care? Because real life keeps asking these exact questions. “If a tap fills 7/10 of a tank in one hour, how much fills in 1/3 of an hour?” “If I have 2 litres of oil and each bottle holds 1/4 litre, how many bottles can I fill?” Both of these need multiplying or dividing fractions.

Here is the best part. We will not just hand you the rules. We will show you, with pictures, why each rule works. By the end you will know why “of” means multiply, why multiplying by a small fraction makes a number smaller (it sounds backwards!), and why dividing by a fraction means you flip it over. Once you see the why, you never forget the how.

The Big Idea

To find a fraction of a quantity, you multiply. Multiplying fractions is just “a part of a part”, and a simple grid picture shows the answer every time. Dividing by a fraction asks “how many of these fit inside?” — and that turns out to be the same as multiplying by the fraction flipped upside down (its reciprocal). Every rule in this chapter comes from a picture, so once you see the picture, the rule makes sense.

Let’s Break It Down

Before we start, let us quickly refresh two fraction ideas from earlier classes. You will lean on them all chapter.

A quick reminder of the words too: in a fraction like 3/4, the top number (3) is the numerator and the bottom number (4) is the denominator. The denominator tells you how many equal parts the whole is cut into. The numerator tells you how many of those parts you take.

A fraction OF something (“of” means multiply)

Suppose a class has 12 sweets and you want 2/3 of them. What does “of” mean here?

Think about what 2/3 tells you to do. The denominator 3 says: split the 12 into 3 equal groups. The numerator 2 says: take 2 of those groups.

Split 12 into 3 groups: each group has 4 sweets (because 12 ÷ 3 = 4). Take 2 groups: that is 4 + 4 = 8 sweets. Figure 8.1 shows it.

A row of 12 sweets split into 3 equal groups of 4. Two of the three groups are shaded green to show two thirds of 12 is 8.
Figure 8.1 — Two thirds of 12 sweets. The 12 sweets are split into 3 equal groups, with 4 sweets in each group (12 divided by 3 is 4). The denominator 3 told us to make 3 groups. The numerator 2 tells us to take 2 of those groups, shown shaded green: that is group 1 and group 2, which is 4 plus 4 equals 8 sweets. The third group, shown plain, is left out. So two thirds of 12 is 8. Finding a fraction OF a quantity is the same as multiplying: 2/3 of 12 is 2/3 times 12 equals 8.

Now look closely at what we actually did. We divided 12 by the denominator 3 (giving 4), then multiplied by the numerator 2 (giving 8). That is exactly the same as the multiplication 2/3 × 12 = (2 × 12)/3 = 24/3 = 8.

So here is the rule, and it is one of the most useful in all of maths:

The word “of” means multiply.

2/3 of 12 = 2/3 × 12 = 8

Why does “of” become multiply? Because “a fraction of a quantity” means cutting the quantity into equal parts and taking some — and that cutting-and-taking is exactly what multiplying by a fraction does. The word “of” is just everyday English for the same action.

Concept check

Why does taking '2/3 of 12' give the same answer as the multiplication 2/3 × 12?

Multiplying a fraction by a whole number

Often you need a fraction copied several times. Aaron’s pet tortoise walks 1/4 km in 1 hour. How far does it walk in 3 hours?

It walks 1/4 km, then another 1/4 km, then another 1/4 km. So the total is 1/4 + 1/4 + 1/4 = 3/4 km. That repeated adding is just multiplying:

3 × 1/4 = (3 × 1)/4 = 3/4

So the rule is short: multiply the whole number by the numerator, and keep the same denominator.

whole number × (numerator/denominator) = (whole number × numerator) / denominator

Here is a real money example to see it at work.

Cost of internet time

One hour of internet time costs ₹8. How much will 5/4 hours of internet time cost?

A quick tip you will use a lot: you can cancel before you multiply. In 5/4 × 8, the 8 and the 4 share a factor of 4. So 8 ÷ 4 = 2, and you get 5 × 2 = 10 straight away. Same answer, smaller numbers.

Multiplying a fraction by a fraction (the grid picture)

Now the new and powerful idea: multiplying two fractions, like 1/2 × 1/3.

What does this even mean? It means half of one third. “Of” means multiply, remember. So we want to take one-third of a whole, then take half of that.

Let us draw it. Take a square as one whole. Shade 1/3 of it (one column out of three). Then cut the whole square in half across the middle, and take the top half of that shaded part. Figure 8.2 shows both steps.

A unit square. In panel a, one third is shaded blue as one column. In panel b, the square is cut in half by a horizontal line, and the overlap (one box out of six) is shaded green to show one sixth.
Figure 8.2 — Half of one third equals one sixth, shown on a unit square. Panel (a): the whole square is cut into 3 equal columns and one column is shaded blue, which is 1/3. Panel (b): now we also cut the whole square in half with a horizontal dashed line, and take the top half of the shaded column — that small dark-green box is the overlap. Because the square is now cut into 3 columns and 2 rows, the whole has 6 equal boxes, and the overlap is exactly 1 of them. So half of one third is one sixth: 1/2 × 1/3 = 1/6.

Look at the final picture in panel (b). The whole square is now cut into 6 equal boxes (3 columns across, 2 rows down). The green overlap is just 1 box out of 6. So:

1/2 × 1/3 = 1/6

Now look at the numbers. The numerators were 1 and 1, and 1 × 1 = 1 (the top). The denominators were 2 and 3, and 2 × 3 = 6 (the bottom). The grid had to have 6 boxes, because cutting into 2 rows and 3 columns makes 2 × 3 = 6 boxes. That is the whole reason the rule works.

This gives us the rule for multiplying any two fractions:

To multiply two fractions, multiply the numerators together for the new top, and multiply the denominators together for the new bottom.

(a/b) × (c/d) = (a × c) / (b × d)

Let us prove it again with a harder pair, using the grid.

Multiplying with the grid picture

A faster tortoise covers 2/5 km in 1 hour. How far does it walk in 3/4 of an hour? (That is, find 3/4 × 2/5.)

Figure 8.3 below is exactly that grid — see how the orange overlap is 6 boxes out of 20.

A unit square split into 4 columns and 5 rows, making 20 equal boxes. An orange rectangle covering 3 columns and 2 rows highlights 6 boxes, showing three fourths times two fifths equals six twentieths.
Figure 8.3 — The area picture for 3/4 times 2/5. The whole square is cut into 4 equal columns and 5 equal rows, giving 4 times 5 equals 20 equal boxes. We pick 3 of the 4 columns (that is 3/4 across) and 2 of the 5 rows (that is 2/5 down). Where they overlap is the orange rectangle, and it covers exactly 3 times 2 equals 6 boxes. So the shaded part is 6 boxes out of 20, which is 6/20, and that simplifies to 3/10. This is why we multiply the tops together (3 times 2 is 6) and the bottoms together (4 times 5 is 20).

There is a neat bonus hidden here. The orange shape is a rectangle with sides 3/4 and 2/5, and its area is 6/20. So multiplying two fractions is the same as finding the area of a rectangle with those fractions as its sides. Pictures and numbers agree.

A time-saver again: you can cancel common factors before multiplying. For 3/4 × 2/5, the 2 (top) and the 4 (bottom) share a factor of 2. Cancel: 2 becomes 1, 4 becomes 2. Now multiply: (3 × 1)/(2 × 5) = 3/10. Same answer, no big numbers to simplify at the end.

Multiplying by a fraction less than 1 makes things SMALLER

Here is something that surprises many students. We are taught that “multiplying makes things bigger”. 3 × 5 = 15, and 15 is bigger than both 3 and 5. So multiplying grows numbers, right?

Not always! When you multiply by a fraction that is less than 1, the answer comes out smaller.

Look: 8 × 1/4 = 2. The answer 2 is smaller than 8. Figure 8.4 shows why this is not strange at all.

A long bar showing 8, and below it a short bar showing 8 times one fourth which is 2. Multiplying by a fraction less than 1 gives a smaller result.
Figure 8.4 — Why multiplying by a fraction below 1 makes a number smaller. The top blue bar is split into 8 equal parts and stands for 8. Below it, we take only 1/4 of that bar — that is one quarter of the length, shown in green, which comes to 2 parts. The dashed empty part is the rest that we did not take. Since 1/4 means 'a small piece of', the result (2) is smaller than the 8 we started with. Multiplying by a fraction less than 1 always shrinks a number, because you are taking only a part of it.

Once you see it, it is obvious. Multiplying by 1/4 means “take one-fourth of it”. Taking a part of something gives you less than the whole thing. That is just what “a fraction of” means.

So here is the full picture you can trust:

  • Multiply by a number bigger than 1 → the answer grows (8 × 3 = 24).
  • Multiply by exactly 1 → the answer stays the same (8 × 1 = 8).
  • Multiply by a number between 0 and 1 (a proper fraction) → the answer shrinks (8 × 1/4 = 2).
Concept check

Without calculating exactly, is 20 × 2/3 bigger than 20 or smaller than 20? Why?

Dividing by a whole number

Dividing a fraction by a whole number is really just sharing it out equally.

Leena made 5 cups of tea using 1/4 litre of milk in total. How much milk is in each cup? We must share 1/4 litre equally among 5 cups. That is 1/4 ÷ 5.

Sharing 1/4 among 5 means cutting that 1/4 into 5 equal pieces. Each piece is much smaller. In fact:

1/4 ÷ 5 = 1/(4 × 5) = 1/20

So each cup has 1/20 litre of milk. The denominator got bigger (from 4 to 20) because we cut the part into more, smaller pieces.

Here is the short rule: to divide a fraction by a whole number, multiply the denominator by that whole number (keep the numerator the same). This is the same as multiplying by 1 over the whole number: 1/4 ÷ 5 = 1/4 × 1/5 = 1/20. That little fact — dividing by 5 is the same as multiplying by 1/5 — is the doorway to the next idea.

Dividing by a fraction — multiply by the reciprocal (and WHY)

Now the trickiest-sounding rule, made simple with a picture.

Question: 2 ÷ 1/4 = ? In words: “how many quarter-pieces fit inside 2 wholes?”

Do not reach for a rule yet. Just count. Take 2 whole bars. Cut each into 4 equal quarter-pieces. Now count all the pieces. Figure 8.5 shows it.

Two whole bars, each cut into 4 equal quarter-pieces, giving 8 quarter-pieces in all, showing 2 divided by one fourth equals 8.
Figure 8.5 — How many quarter-pieces fit in 2? Each of the two whole bars is cut into 4 equal pieces, and every piece is one quarter (1/4). Counting all the pieces from both bars gives 1, 2, 3, 4, 5, 6, 7, 8 — that is 8 quarter-pieces in 2 wholes. So 2 divided by 1/4 is 8. Notice this is the same as 2 times 4: dividing by 1/4 is exactly the same as multiplying by 4, which is 1/4 flipped upside down.

We counted 8 quarter-pieces. So 2 ÷ 1/4 = 8.

Now here is the magic. Notice that 8 is also just 2 × 4. And 4 is the fraction 1/4 flipped upside down. So:

2 ÷ 1/4 = 2 × 4 = 8

Why does flipping work? Think about it. Each whole has 4 quarters in it. So in 2 wholes there are 2 × 4 = 8 quarters. Dividing by 1/4 asks “how many quarters?”, and each whole contains 4 of them — which is why we multiply by 4. The smaller the piece you divide by, the more of them fit, so the answer gets bigger. That matches: dividing by a tiny fraction multiplies by a big flipped number.

The flipped-over fraction has a special name: the reciprocal. To get the reciprocal of a fraction, just swap its top and bottom. Figure 8.6 shows it.

The fraction three fourths flips upside down to four thirds, its reciprocal. A fraction times its reciprocal equals one.
Figure 8.6 — The reciprocal of a fraction is found by flipping it upside down — swap the numerator and denominator. So the reciprocal of 3/4 is 4/3. The orange arrows show the flip. The reason this matters: a fraction multiplied by its reciprocal always gives 1, because the same numbers end up on top and bottom. Here 3/4 times 4/3 is 12/12, which equals 1. This is why dividing by a fraction is the same as multiplying by its reciprocal.

So here is the rule for dividing by any fraction:

To divide by a fraction, multiply by its reciprocal (the fraction flipped upside down).

(a/b) ÷ (c/d) = (a/b) × (d/c)

Watch it work:

2/3 ÷ 3/5 = 2/3 × 5/3 = (2 × 5)/(3 × 3) = 10/9

The reciprocal of the divisor 3/5 is 5/3, and we multiply by that. That is all there is to it.

Dividing by a fraction

A water tank fills 7/10 of the way in 1 hour. How long, in hours, to fill the whole tank? (Find 1 ÷ 7/10.)

Concept check

When you divide 6 by 1/3, do you get a number bigger or smaller than 6? Explain using the 'how many fit?' idea.

Word problems

Many real problems mix these skills. The trick is to read slowly and decide: am I finding a part of something (multiply), or am I asking how many fit / sharing equally (divide)?

Here is one from the textbook, using square bricks.

Counting bricks

You must cover an area of 7 and 1/2 square units using small square bricks. Each brick has sides of 1/5 unit. How many bricks are needed?

Common Mistakes

These slips trip up most students. Read them once and you will sidestep them.

⚠️ Common mistake
What students think

Multiplying two numbers always gives a bigger answer.

Why it seems right

Every multiplication you learned with whole numbers grew the result — 3 × 5 = 15, 6 × 4 = 24 — so it feels like a law that multiplying must make things larger.

What actually happens

That is only true when you multiply by a number bigger than 1. When you multiply by a fraction below 1, like 8 × 1/4 = 2, you are taking only a part of the number, so the answer comes out smaller, not bigger.

⚠️ Common mistake
What students think

To find '2/3 of 12' you add: 2/3 + 12.

Why it seems right

The word 'of' sits between the two numbers like a join word, and adding is the first thing students learned to do with fractions, so 'add them together' feels like the natural move.

What actually happens

'Of' means multiply, not add. 2/3 of 12 = 2/3 × 12 = 8. Splitting 12 into 3 groups and taking 2 of them gives 8 — adding 2/3 to 12 would give a number close to 12, which makes no sense as 'a part of 12'.

⚠️ Common mistake
What students think

To divide by a fraction, you flip the first fraction (the dividend).

Why it seems right

The rule 'flip and multiply' is easy to half-remember, and since the first fraction is what you read first, students flip that one by habit.

What actually happens

You flip only the second fraction — the divisor (the one you are dividing by). For 2/3 ÷ 3/5, flip 3/5 to 5/3, then multiply: 2/3 × 5/3 = 10/9. The first fraction stays exactly as it is.

⚠️ Common mistake
What students think

When multiplying fractions you must first make the denominators the same, like in addition.

Why it seems right

The very first big rule for fractions — adding and subtracting — needs a common denominator, so it feels like a step you always do before any fraction operation.

What actually happens

Multiplying needs NO common denominator. You just multiply tops together and bottoms together: 1/2 × 1/3 = 1/6. Finding a common denominator is only for adding and subtracting.

Quick Check

Try these to lock in the ideas. Pick an answer, then read why.

What is 3/4 × 8?

Without working it out fully, which is true about 30 × 1/5?

What is 2/3 ÷ 4/5?

How many 1/2-litre bottles can you fill from 3 litres of juice?

Practice Problems

Work each one yourself first, then tap to check. Every solution is fully worked.

Easy

Easy

Find 1/2 of 18.

Easy

Multiply 2/5 × 3/7.

Easy

A team makes 1/8 km of canal in one day. How much do they make in 5 days?

Medium

Medium

Tenzin drinks 1/2 glass of milk every day. How many glasses does he drink in the month of January (31 days)?

Medium

Find the area of a rectangle whose sides are 3/4 ft and 5/6 ft.

Medium

Manju and two neighbours buy 5 litres of oil every week and share it equally among the 3 families. How much oil does each family get in 4 weeks?

Challenge

Challenge

A miser gives 1/2 of 2/3 of 3/4 of a rupee to a beggar. What fraction of a rupee did the beggar get?

Challenge

You must cover an area of 4 and 1/2 square units with square bricks of side 1/3 unit. How many bricks are needed?

Summary

  • To find a fraction of a quantity, multiply — the word “of” means multiply. 2/3 of 12 = 2/3 × 12 = 8.
  • To multiply a fraction by a whole number, multiply the whole number by the numerator and keep the denominator: 3 × 1/4 = 3/4.
  • To multiply a fraction by a fraction, multiply tops together and bottoms together: (a/b) × (c/d) = (a × c)/(b × d). The grid picture shows why the boxes total b × d.
  • Multiplying two fractions is the same as finding the area of a rectangle with those fractions as its sides.
  • Multiplying by a fraction less than 1 makes a number smaller, because you take only a part of it (8 × 1/4 = 2).
  • The reciprocal of a fraction is that fraction flipped upside down (3/4 → 4/3). A fraction times its reciprocal is 1.
  • To divide by a fraction, multiply by its reciprocal: 2 ÷ 1/4 = 2 × 4 = 8. This works because each whole holds that many small pieces.
  • Multiplying needs no common denominator — that step is only for adding and subtracting.

What’s Next

You have now mastered all four operations on fractions — adding, subtracting, multiplying and dividing. Fractions will keep showing up everywhere, from ratios to percentages to measuring, so this skill pays off for years.

Next up is A Tale of Three Intersecting Lines, where we leave numbers for a while and explore shapes, lines and the surprising twin-like relationships hiding inside triangles. Get ready to draw and discover.

Frequently Asked Questions

What does 'fraction of a quantity' mean and how do you calculate it?

Finding a fraction of a quantity means taking that fraction-sized portion of the whole. You multiply the fraction by the quantity. For example, 3/4 of 20 = 3/4 x 20 = 60/4 = 15. The word 'of' in maths always signals multiplication.

How do you multiply two fractions together?

Multiply the numerators (top numbers) together and multiply the denominators (bottom numbers) together. For example, 2/3 x 3/5 = (2 x 3)/(3 x 5) = 6/15, which simplifies to 2/5. You can also cross-cancel common factors before multiplying to keep the numbers smaller.

Why does multiplying by a fraction less than 1 make the answer smaller?

A fraction less than 1 (like 1/2 or 3/4) represents a part of one whole — something less than the full amount. So taking that fraction 'of' a number means taking only a portion of it, which must be smaller than the original. For example, 1/2 x 8 = 4, which is less than 8.

What is the reciprocal of a fraction and how is it used in division?

The reciprocal of a fraction is what you get when you flip it — swap the numerator and denominator. The reciprocal of 3/4 is 4/3. To divide by a fraction, you multiply by its reciprocal instead. For example, 6 / (3/4) = 6 x 4/3 = 24/3 = 8.

How do you divide a fraction by another fraction?

Keep the first fraction as it is, change division to multiplication, and flip the second fraction (use its reciprocal). Then multiply normally. For example, (2/3) / (4/5) = (2/3) x (5/4) = 10/12 = 5/6. The rule 'Keep, Change, Flip' helps you remember the steps.