Reciprocals

Worked example

1. 1

   $8=\frac{8}{1}$ — Write the whole number as a fraction. Any number divided by 1 is itself, so 8 is 8 over 1.

2. 2

   $\frac{1}{8}$ — Flip it: the top and the bottom swap places. This is the reciprocal.

3. 3

   $8\times \frac{1}{8}=\frac{8}{8}=1✓$ — Check: a number times its reciprocal must give 1. It does, so the answer is right.

Worked example

1. 1

   $\frac{2}{7}$ — The question. The top is 2 and the bottom is 7.

2. 2

   $\frac{7}{2}$ — Flip it: the top goes down and the bottom goes up.

3. 3

   $\frac{2}{7}\times \frac{7}{2}=\frac{14}{14}=1✓$ — Check: the product is 1, so the flip is right.

4. 4

   $\frac{7}{2}=3\frac{1}{2}$ — You may leave it as 7 over 2, or change it to a mixed number: 7 ÷ 2 = 3 remainder 1. Both are accepted.

Worked example

1. 1

   $2\frac{1}{2}=\frac{2\times 2+1}{2}$ — Change the mixed number to one fraction. Multiply the whole number by the bottom (2 × 2), then add the top (1).

2. 2

   $=\frac{5}{2}$ — 4 + 1 = 5, so two and a half is five halves.

3. 3

   $\frac{2}{5}$ — Now it is one fraction, so flip it: the top and the bottom swap.

4. 4

   $\frac{5}{2}\times \frac{2}{5}=\frac{10}{10}=1✓$ — Check: the product is 1, so the reciprocal is $\frac{2}{5}$.

Worked example

1. 1

   $0.2=\frac{2}{10}$ — Change the decimal to a fraction. One decimal place means tenths, so 0.2 is 2 tenths.

2. 2

   $\frac{2}{10}=\frac{1}{5}$ — Simplify: divide the top and the bottom by 2. Smaller numbers are easier to flip.

3. 3

   $\frac{5}{1}=5$ — Flip it: the 1 goes down and the 5 goes up. 5 over 1 is just 5.

4. 4

   $0.2\times 5=1✓$ — Check: the product is 1, so the reciprocal of 0.2 is 5.

Worked example

1. 1

   $\frac{1}{0.625}$ — A reciprocal is 1 divided by the number. Write this line down: in a 2-mark question it earns a mark on its own.

2. 2

   $0.625=\frac{625}{1000}$ — Change the decimal to a fraction. Three decimal places means thousandths, so put it over 1000.

3. 3

   $\frac{625}{1000}=\frac{5}{8}$ — Simplify: divide the top and the bottom by 125, because 625 = 5 × 125 and 1000 = 8 × 125.

4. 4

   $\frac{8}{5}$ — Flip it: the 5 goes down and the 8 goes up.

5. 5

   $\frac{8}{5}=1.6$ — Divide 8 by 5 to get a decimal: 8 ÷ 5 = 1 remainder 3, and $\frac{3}{5}=0.6$.

6. 6

   $0.625\times 1.6=1✓$ — Check: the product is 1, so the reciprocal of 0.625 is 1.6.

Worked example

1. 1

   $\frac{6}{7}\text{flips to}\frac{7}{6}$ — Flip first. Write $\frac{7}{6}$ down: it earns a mark even if a later line goes wrong.

2. 2

   $7÷6=1.16666\dots$ — Divide the top by the bottom to change it to a decimal. Write down more digits than you need before rounding.

3. 3

   $1.1666\dots$ — To round to 3 decimal places, look at the 4th decimal place. It is 6, which is 5 or more, so round up.

4. 4

   $1.167$ — The 3rd decimal place goes up from 6 to 7. This is the answer to 3 decimal places.