What are equivalent fractions?

What does a fraction wall show?

Take paper strips of the same length. Fold one into halves, one into quarters and one into eighths, and lay them one below the other. The edge of 1/2 lines up exactly with the edge of 2/4 and with 4/8. They are the same length, so

1/2 = 2/4 = 4/8

Fractions like these, which stand for the same length or share but use different fractional units, are called equivalent fractions. Keep adding strips for thirds, fifths, sixths and so on, and you get a fraction wall.

From the wall: 1/3 = 2/6, 2/3 = 4/6, and 3 pieces of 1/6 make 1/2.

Equal shares give equivalent fractions

Share 1 roti between 2 children, 2 rotis among 4 children, or 3 rotis among 6 children. In each case every child gets the same amount, so 1/2 = 2/4 = 3/6.

In the same way, 2 cakes shared among 5 children and 4 cakes shared among 10 children give the same share: 2/5 = 4/10. And 2 rotis among 3 children, 4 among 6 and 6 among 9 give 2/3 = 4/6 = 6/9.

Sharing can also be written as facts. When 3 rotis are shared by 4 children:

• division fact: 3 ÷ 4 = 3/4;
• addition fact: 3 = 3/4 + 3/4 + 3/4 + 3/4;
• multiplication fact: 3 = 4 × 3/4.

How do you make an equivalent fraction?

Look at 2/3 = 4/6 = 6/9. The numerator and the denominator were both multiplied by the same number:

2/3 = (2 × 2)/(3 × 2) = 4/6, and 2/3 = (2 × 3)/(3 × 3) = 6/9.

Why does this work? Multiplying the denominator by 2 cuts every part into 2 smaller parts. There are now twice as many parts, each half the size, so we need twice as many of them: the numerator doubles too. The share itself does not change.

Giving two fractions the same fractional unit

To compare 3/4 and 7/10, we want both in the same fractional unit. List equivalent fractions:

• 3/4 = 6/8 = 9/12 = 12/16 = 15/20 = … = 30/40
• 7/10 = 14/20 = 21/30 = 28/40

Both lists reach the denominator 20, so 3/4 = 15/20 and 7/10 = 14/20. The product of the denominators (4 × 10 = 40) always works as a common denominator, but a smaller common multiple, here 20, keeps the numbers small. You will use this in comparing fractions.

What is a fraction in lowest terms?

A fraction is in lowest terms (or in its simplest form) when its numerator and denominator have no common factor except 1.

Is 16/20 in lowest terms? No: 4 is a common factor of 16 and 20. Divide both by 4: 16/20 = 4/5. Now 4 and 5 have no common factor except 1, so 4/5 is the simplest form of 16/20.

You can also go in steps. For 36/60:

1. Both are even: 36/60 = 18/30.
2. Both are even again: 18/30 = 9/15.
3. Both are multiples of 3: 9/15 = 3/5.

Or notice at once that 36 = 3 × 12 and 60 = 5 × 12, so 36/60 = 3/5. Either way gives the same answer, because dividing by the highest common factor (here 12) gets there in one step.