Fractions

Why This Matters

Numbers like 1, 2, 3 are for counting whole things. One roti. Two friends. Three buses.

But life is not always made of whole things.

You eat half a roti. You drink a quarter of a glass of juice. You share one chocolate bar among three friends. None of those are whole numbers.

So we need a new kind of number. A number for parts of a thing. These numbers are called fractions.

You already use fractions every day, even if you do not call them that. “Give me half.” “I have a quarter of my homework left.” “We finished three-fourths of the match.” All of these are fractions.

In this chapter you will learn what a fraction really means. You will see fractions as pictures, as shares of food, and as points on a line. You will learn to compare them, to add them, and to subtract them. By the end, fractions will feel as easy as counting.

The Big Idea

A fraction tells you how much you get when one whole thing is split into equal parts. If you cut one roti into 4 equal pieces and take 3 of them, you have 3/4 (three-fourths) of the roti. The bottom number (4) says how many equal parts the whole was cut into. The top number (3) says how many of those parts you took. That is the whole idea — equal parts of a whole.

Let’s Break It Down

What is a fraction?

Imagine one roti. You and a friend want to share it equally. So you cut it into 2 equal parts. You take 1 part.

How much roti did you get? You got one-half. We write it as 1/2.

Now imagine sharing the roti among 4 friends. Cut it into 4 equal parts. Each friend gets 1 part. Each one gets one-fourth, written 1/4.

The most important word here is equal. The parts must all be the same size. If you cut a roti into 2 pieces but one piece is big and one is tiny, those are not halves. Halves must be equal.

A fraction has two numbers, one on top and one on the bottom, with a line between them.

• The bottom number is the denominator. It tells you how many equal parts the whole is cut into.
• The top number is the numerator. It tells you how many of those parts you take.

Let’s look at a real example. Figure 7.1 below shows one chocolate bar cut into 4 equal parts, with 3 parts shaded.

Tap to zoom

So in 3/4: the denominator 4 means “cut into 4 equal parts”, and the numerator 3 means “take 3 of them”.

A helpful way to read 3/4 is “3 times 1/4”. This reminds you that you have 3 little pieces, and each piece is 1/4. The piece 1/4 is called a fractional unit — it is one single equal part.

Here is one more idea that surprises many students. When you share more, each share is smaller. Share 1 roti between 2 people, each gets 1/2 (a big share). Share the same roti among 4 people, each gets 1/4 (a smaller share). So 1/2 is bigger than 1/4, even though 4 is a bigger number than 2. The bigger the denominator, the smaller each piece.

Fractions on the number line

You already know how to put whole numbers on a number line: 0, 1, 2, 3, and so on, each one step apart.

Fractions live on the number line too. They sit between the whole numbers.

Here is how. Take the gap between 0 and 1. This gap is 1 whole unit long. Now cut this gap into equal steps, just like cutting a roti.

Figure 7.2 below splits the gap from 0 to 1 into 4 equal steps.

Tap to zoom

So to place a fraction like 3/4 on the line: the denominator 4 tells you to make 4 equal steps between 0 and 1. The numerator 3 tells you to count 3 steps from 0. You land on 3/4.

Notice that 4/4 lands exactly on 1. That makes sense: 4 quarters make one whole.

A fun fact: between 0 and 1 there are endless fractions. You can always cut the gap into more and more steps. So there is no “next” fraction after 1/2 the way 3 comes after 2.

Mixed numbers and improper fractions

So far our fractions were smaller than 1. But fractions can be bigger than 1 too.

Think about it. If 1 roti is 2/2, then 3/2 must be more than 1 roti. It is 3 half-rotis: one whole roti (2 halves) plus 1 more half. So 3/2 = 1 whole and 1/2 left over.

There are two ways to write a number like this.

• Improper fraction: the numerator is bigger than (or equal to) the denominator. Example: 3/2, 7/4, 9/5. (“Improper” just means top-heavy. It is not wrong — it is a normal, correct fraction.)
• Mixed number: a whole number written next to a fraction. Example: 1 1/2 (read “one and a half”). It has a whole part (1) and a fraction part (1/2).

Both write the same amount. 3/2 and 1 1/2 are equal. One is just dressed differently.

How do you tell if a fraction is bigger than 1? Easy. If the numerator is bigger than the denominator, the fraction is more than 1 whole. Like 7/4: you have 7 quarters, and 4 quarters already make 1 whole, so there is more left over.

Let’s practise turning an improper fraction into a mixed number.

Now the other way: turning a mixed number back into an improper fraction.

Equivalent fractions (and why)

Here is something amazing. Different-looking fractions can be the same amount.

Look: 1/2 of a roti, 2/4 of a roti, and 4/8 of a roti are all exactly the same amount of roti. They just use different-sized pieces to say it.

Fractions that mean the same amount are called equivalent fractions. “Equivalent” means “equal in value”.

But why are they equal? Do not just believe it — let’s see it. Figure 7.3 below shows three bars of the same length, cut into different numbers of parts.

Tap to zoom

See it? When you cut each piece into 2 smaller pieces, you get twice as many pieces (the denominator doubles), but you also take twice as many (the numerator doubles). So the amount of roti does not change.

This gives us a simple rule.

To make an equivalent fraction, multiply the top and bottom by the same number.

1/2 = (1 × 2)/(2 × 2) = 2/4

1/2 = (1 × 4)/(2 × 4) = 4/8

It works the other way too. You can divide the top and bottom by the same number, and the value stays the same.

Simplest form

A fraction can be written many ways: 4/8, 2/4, 1/2. They are all equal. But the simplest one to read is 1/2.

A fraction is in its simplest form (also called lowest terms) when the numerator and denominator have no common factor except 1. In other words, you cannot make the numbers any smaller.

Before we simplify, let’s quickly refresh what a “factor” is.

To reach simplest form, divide the top and bottom by a common factor. Keep going until the only common factor left is 1.

Let’s simplify a fraction step by step.

Comparing fractions

Which is bigger, 2/3 or 3/4? This is tricky. The top numbers and bottom numbers are both different. You cannot just look and tell.

If two fractions have the same bottom number, comparing is easy. Just look at the top numbers. 5/8 is bigger than 3/8, because 5 pieces is more than 3 pieces (and each piece is the same size, 1/8).

So the trick is to make the bottom numbers the same first.

We do this by making equivalent fractions until both fractions have the same denominator (called a common denominator). A safe common denominator is just the two denominators multiplied together. For 2/3 and 3/4, that is 3 × 4 = 12.

• 2/3 = (2 × 4)/(3 × 4) = 8/12
• 3/4 = (3 × 3)/(4 × 3) = 9/12

Now both have the same bottom (12). Just compare tops: 9 is more than 8. So 3/4 is bigger than 2/3.

Figure 7.4 below shows this with bars, so you can see it.

Tap to zoom

So the rule to compare: make the denominators the same, then compare the numerators.

Adding and subtracting fractions

Now the big skill: adding and subtracting fractions.

When the bottom numbers are the same (like denominators)

This is the easy case. If the pieces are all the same size, you just count them.

Say you eat 1/4 of a chocolate bar, then 2/4 more. How much did you eat in total? You ate 1 quarter-piece, then 2 quarter-pieces. That is 3 quarter-pieces in all, which is 3/4.

Figure 7.5 below shows this with bars.

Tap to zoom

So the rule for like denominators:

Add (or subtract) the top numbers. Keep the bottom number the same.

1/4 + 2/4 = (1 + 2)/4 = 3/4

5/7 − 2/7 = (5 − 2)/7 = 3/7

A common worry: why doesn’t the bottom number add up too? Because the bottom number tells you the size of each piece. The pieces do not change size when you put them together. A quarter plus a quarter is two quarters, not “two-eighths”. Only the number of pieces grows.

When the bottom numbers are different (unlike denominators)

Now the harder case. What is 1/2 + 1/3?

You cannot just add them as they are. Here is the reason. A half-piece and a third-piece are different sizes. It is like adding 1 big spoon and 1 small spoon — you cannot say “2 spoons” because the spoons are not the same size.

So first we must make all the pieces the same size. We do this by finding a common denominator, exactly like we did for comparing.

For 1/2 and 1/3, a common denominator is 2 × 3 = 6. Let’s change both fractions to sixths.

• 1/2 = (1 × 3)/(2 × 3) = 3/6
• 1/3 = (1 × 2)/(3 × 2) = 2/6

Now both use the same-sized piece (a sixth). So we can add: 3/6 + 2/6 = 5/6.

Figure 7.6 below shows the whole idea.

Tap to zoom

Let’s do one more, including making the answer simple.

Subtraction works the very same way: make the bottoms the same, then subtract the tops.

Common Mistakes

These are the slip-ups almost every student makes at first. Knowing them now will save you marks later.

Quick Check

Try these to test yourself. Click an option to see if you are right.

Practice Problems

Try each one yourself first. Then tap “Show Solution” to check.

Easy

Medium

Challenge

Summary

Here is everything you should now be able to explain:

• A fraction shows how much you get when a whole is split into equal parts. The bottom number (denominator) is how many equal parts; the top number (numerator) is how many you take.
• A bigger denominator means smaller pieces. So 1/8 is smaller than 1/2.
• Fractions sit on the number line between whole numbers. To place a/b, make b equal steps from 0 to 1 and count a of them.
• A fraction bigger than 1 can be written as an improper fraction (top-heavy, like 7/4) or a mixed number (like 1 3/4). They mean the same amount.
• Equivalent fractions are equal in value, like 1/2 = 2/4 = 4/8. You make them by multiplying (or dividing) the top and bottom by the same number.
• A fraction is in simplest form when the top and bottom have no common factor except 1.
• To compare fractions, make the bottoms the same, then compare the tops.
• To add or subtract fractions, the bottoms must be the same first. With like bottoms, add or subtract the tops and keep the bottom. With unlike bottoms, change them to a common denominator first — because you can only add pieces of the same size.

What’s Next

You now understand fractions — parts of a whole, on the page and on the number line. Next, in Chapter 8 — Playing with Constructions, you will put down the calculator and pick up a ruler and compass. You will draw clean lines, exact angles, and neat shapes by hand. It is maths you can see and build, and the careful, step-by-step thinking you used with fractions will help you there too.