Equivalent fractions

Lay the paper strips one under another to make a fraction wall. The strip for 1/2, the strip for 2/4 and the strip for 4/8 are exactly the same length, so 1/2 = 2/4 = 4/8. Fractions like these, which show the same length in different fractional units, are called equivalent fractions. The wall also shows 1/3 = 2/6, and that 3 pieces of 1/6 make 1/2 while 2 pieces of 1/6 make 1/3.

Sharing shows it too. 1 roti shared equally between 2 children, 2 rotis among 4 children and 3 rotis among 6 children: every child gets the same share, so 1/2 = 2/4 = 3/6. Anil's group shares 2 cakes among 5 children, so Anil gets 2/5 cake. Put two such groups together and there are 4 cakes and 10 children, and each child still gets the same: 2/5 = 4/10.

Look at the numbers: 2/5 becomes 4/10 when both the numerator and the denominator are multiplied by 2. Multiplying the numerator and the denominator by the same number always gives an equivalent fraction. Why is this safe? Multiplying by 2/2 is multiplying by 1, and multiplying by 1 changes nothing. So 5/7 = 10/14 = 15/21. 2 rotis among 3 children, 4 among 6 and 6 among 9 all give 2/3 each; 2/3 is the simplest form of 4/6 and of 6/9.

This helps with missing numbers. 5 glasses of juice shared equally among 4 friends is the same as 10 glasses among 8 friends: the friends are doubled, so the glasses are doubled, and 5/4 = 10/8. 4 kg of potatoes divided equally in 3 bags is the same as 12 kg in 9 bags: 12 is 3 times 4, so the bags are 3 times 3, and 4/3 = 12/9.