Level 4 · Ages 13–15
Read it again any time, including while you work through the exercises.
You can read a fraction and take a fraction of a quantity. Time to calculate with them: adding, subtracting, multiplying and dividing fractions. Each operation has its own rule, and the biggest danger is mixing the rules up.
To add or subtract, you need parts of the same size, so the same denominator. 2/7 + 3/7 is 2 parts plus 3 parts of the same size: 5/7. Careful, you never add the denominators! If the denominators differ, like 1/2 + 1/3, first find a common denominator: 1/2 = 3/6 and 1/3 = 2/6, so the sum is 5/6.
Multiplying is more direct: multiply the numerators together and the denominators together. 3/4 × 2/3 = 6/12, which simplifies to 1/2. Taking 3/4 of 2/3 of something really does give you half of it.
Dividing uses an elegant idea: dividing by a fraction means multiplying by its reciprocal. The reciprocal of 2/5 is 5/2. So 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8. You only flip the fraction you divide by, never the first one.
In this lesson you will practise all four operations, with carefully chosen traps: they are exactly the mistakes that well understood rules help you avoid.
Compute 1/4 + 2/3.
4 × 3 = 12
The denominators 4 and 3 are different: I look for a common denominator. 12 is a multiple of both 4 and 3.
1/4 = 3/12
I convert the first fraction: I multiply top and bottom by 3.
2/3 = 8/12
I convert the second: I multiply top and bottom by 4.
3/12 + 8/12 = 11/12
Now that the parts are the same size, I add the numerators. The denominator does not change.
Result: 1/4 + 2/3 = 11/12
Compute 3/4 ÷ 2/5.
3/4 ÷ 2/5 = 3/4 × 5/2
Dividing by 2/5 means multiplying by its reciprocal 5/2. I only flip the fraction that divides.
3 × 5 = 15 ; 4 × 2 = 8
I multiply the numerators together and the denominators together.
15/8
15 and 8 share no common divisor: the fraction is already in simplest form.
Result: 3/4 ÷ 2/5 = 15/8