Adding and taking away fractions

Half a litre of milk and then a quar­ter litre more. Three quar­ters of an hour on home­work and two thirds of an hour on cricket. Every time some­thing is shared and put together again, you are adding or tak­ing away frac­tions. This les­son starts with frac­tions whose pieces are the same size, then shows what to do when they are not, and ends with whole num­bers and longer sums.

When the denom­i­na­tors are the same#

The denom­i­na­tor tells you the size of each piece. When two frac­tions have the same denom­i­na­tor, the pieces are the same size, so you just count them: add or take away the numer­a­tors and keep the denom­i­na­tor.

$$\displaystyle \frac{2}{3} + \frac{5}{3} = \frac{2 + 5}{3} = \frac{7}{3}$$

The denom­i­na­tor does not change, because the cake was never cut again. You sim­ply have more thirds.

Remem­ber. $\displaystyle \frac{a}{b} + \frac{c}{b} = \frac{a + c}{b}$ and $\displaystyle \frac{a}{b} - \frac{c}{b} = \frac{a - c}{b}$.

Exam­ple 1#

Work out $\displaystyle \frac{3}{7} + \frac{2}{7} - \frac{1}{7}$ and $\displaystyle \frac{5}{6} - \frac{1}{6} + \frac{2}{6}$.

$$\displaystyle \frac{3}{7} + \frac{2}{7} - \frac{1}{7} = \frac{3 + 2 - 1}{7} = \frac{4}{7}$$

$$\displaystyle \frac{5}{6} - \frac{1}{6} + \frac{2}{6} = \frac{5 - 1 + 2}{6} = \frac{6}{6} = 1$$

Exam­ple 2#

Work out $\displaystyle \frac{30}{40} - \frac{15}{40} + \frac{10}{40}$, and give the answer in sim­plest form.

$$\displaystyle \frac{30 - 15 + 10}{40} = \frac{25}{40} = \frac{5}{8}$$

The H.C.F of 25 and 40 is 5, so $\displaystyle \frac{25}{40}$ can­cels to $\displaystyle \frac{5}{8}$. Always fin­ish in sim­plest form.

Exam­ple 3#

Work out $\displaystyle \frac{54}{70} - \frac{18}{70} + \frac{12}{70} - \frac{8}{70}$.

$$\displaystyle \frac{54 - 18 + 12 - 8}{70} = \frac{40}{70} = \frac{4}{7}$$

When the answer is neg­a­tive#

The numer­a­tors are inte­gers, so use the rules for adding inte­gers. Group the pos­i­tive num­bers and the neg­a­tive num­bers.

Exam­ple 4#

Work out $\displaystyle \frac{5}{3} - \frac{7}{3} + \frac{4}{3} - \frac{11}{3}$.

$$\displaystyle \begin{aligned}\frac{5 - 7 + 4 - 11}{3} &= \frac{9 - 18}{3} \\ &= \frac{-9}{3} \\ &= -3\end{aligned}$$

The pos­i­tives make $5 + 4 = 9$ and the neg­a­tives make $7 + 11 = 18$. The big­ger group is neg­a­tive, so the answer is neg­a­tive. Remem­ber that $\displaystyle \frac{a}{1} = a$, so $\displaystyle \frac{-3}{1} = -3$.

Exam­ple 5#

Work out $\displaystyle -\frac{12}{5} + \frac{33}{5} - \frac{27}{5} + \frac{9}{5}$.

$$\displaystyle \frac{-12 + 33 - 27 + 9}{5} = \frac{42 - 39}{5} = \frac{3}{5}$$

When the denom­i­na­tors are dif­fer­ent#

Now the pieces are dif­fer­ent sizes. First cut them again so that every piece is the same size. The new denom­i­na­tor is the L.C.M of the denom­i­na­tors.

Exam­ple 6#

Work out $\displaystyle \frac{5}{4} + \frac{7}{6}$.

Find the L.C.M of 4 and 6 with a lad­der:

$$\begin{array}{r|l} 2 & 4, \; 6 \\ \hline & 2, \; 3 \end{array}$$

L.C.M $= 2 \times 2 \times 3 = 12$. Since $4 \times 3 = 12$ and $6 \times 2 = 12$, mul­ti­ply the top and bot­tom of $\displaystyle \frac{5}{4}$ by 3, and of $\displaystyle \frac{7}{6}$ by 2.

$$\displaystyle \begin{aligned}\frac{5}{4} + \frac{7}{6} &= \frac{5 \times 3}{4 \times 3} + \frac{7 \times 2}{6 \times 2} \\ &= \frac{15}{12} + \frac{14}{12} \\ &= \frac{15 + 14}{12} \\ &= \frac{29}{12}\end{aligned}$$

Exam­ple 7#

Work out $\displaystyle \frac{5}{8} - \frac{7}{12}$.

$$\begin{array}{r|l} 2 & 8, \; 12 \\ \hline 2 & 4, \; 6 \\ \hline & 2, \; 3 \end{array}$$

L.C.M $= 2 \times 2 \times 2 \times 3 = 24$. Since $8 \times 3 = 24$ and $12 \times 2 = 24$:

$$\displaystyle \frac{5}{8} - \frac{7}{12} = \frac{15 - 14}{24} = \frac{1}{24}$$

Exam­ple 8#

Work out $\displaystyle \frac{5}{6} + \frac{7}{12}$ and $\displaystyle \frac{5}{3} + \frac{8}{7}$.

The L.C.M of 6 and 12 is 12, so $\displaystyle \frac{5}{6} + \frac{7}{12} = \frac{10 + 7}{12} = \frac{17}{12}$.

3 and 7 share no fac­tor, so the L.C.M is $3 \times 7 = 21$ and $\displaystyle \frac{5}{3} + \frac{8}{7} = \frac{35 + 24}{21} = \frac{59}{21}$.

Exam­ple 9#

Work out $\displaystyle \frac{3}{4} + \frac{7}{8} + \frac{9}{4}$.

Join the like frac­tions first: $\displaystyle \frac{3}{4} + \frac{9}{4} = \frac{12}{4} = 3$. Then $\displaystyle 3 + \frac{7}{8} = \frac{24}{8} + \frac{7}{8} = \frac{31}{8}$.

In the same way, $\displaystyle \frac{5}{6} + \frac{3}{4} + \frac{7}{6} + \frac{5}{4} = \frac{12}{6} + \frac{8}{4} = 2 + 2 = 4$.

The rules in let­ters#

Remem­ber.

$$\displaystyle \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \qquad \frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}$$

These always work, because $bd$ is always a com­mon denom­i­na­tor. The L.C.M often gives smaller num­bers, so use it when you can.

Exam­ple 10#

$$\displaystyle \begin{aligned}\frac{2}{3} - \frac{5}{7} &= \frac{14 - 15}{21} = -\frac{1}{21} \\ \frac{5}{3} + \frac{7}{4} &= \frac{20 + 21}{12} = \frac{41}{12}\end{aligned}$$

A whole num­ber and a frac­tion#

Write the whole num­ber over 1, then use the rules.

Exam­ple 11#

Work out $\displaystyle 3 - \frac{7}{8}$ and $\displaystyle \frac{5}{6} - 8$.

$$\displaystyle 3 - \frac{7}{8} = \frac{3}{1} - \frac{7}{8} = \frac{24 - 7}{8} = \frac{17}{8}$$

$$\displaystyle \frac{5}{6} - 8 = \frac{5}{6} - \frac{8}{1} = \frac{5 - 48}{6} = -\frac{43}{6}$$

In the sec­ond one, $5 - 48$ is neg­a­tive: the dif­fer­ence is 43, and 48 is the big­ger num­ber.

Be care­ful with the order: $7 - 10 = -3$, not 3.

Three frac­tions with dif­fer­ent denom­i­na­tors#

Exam­ple 12#

Work out $\displaystyle \frac{7}{8} + \frac{5}{24} - \frac{7}{6}$.

The L.C.M of 8, 24 and 6 is 24.

$$\displaystyle \begin{aligned}\frac{7}{8} + \frac{5}{24} - \frac{7}{6} &= \frac{21 + 5 - 28}{24} \\ &= \frac{26 - 28}{24} \\ &= \frac{-2}{24} \\ &= -\frac{1}{12}\end{aligned}$$

Exam­ple 13#

Work out $\displaystyle \frac{5}{12} - \frac{7}{8} + \frac{3}{16}$.

$$\begin{array}{r|l} 4 & 12, \; 8, \; 16 \\ \hline 2 & 3, \; 2, \; 4 \\ \hline & 3, \; 1, \; 2 \end{array}$$

L.C.M $= 4 \times 2 \times 3 \times 1 \times 2 = 48$.

$$\displaystyle \frac{5}{12} - \frac{7}{8} + \frac{3}{16} = \frac{20 - 42 + 9}{48} = \frac{29 - 42}{48} = -\frac{13}{48}$$

Exam­ple 14#

Work out $\displaystyle \frac{7}{18} + \frac{5}{26} - \frac{8}{9}$.

The L.C.M of 18, 26 and 9 is $2 \times 3 \times 3 \times 13 = 234$. Since $18 \times 13 = 234$, $26 \times 9 = 234$ and $9 \times 26 = 234$:

$$\displaystyle \begin{aligned}\frac{7}{18} + \frac{5}{26} - \frac{8}{9} &= \frac{91 + 45 - 208}{234} \\ &= \frac{136 - 208}{234} \\ &= -\frac{72}{234} \\ &= -\frac{4}{13}\end{aligned}$$

The H.C.F of 72 and 234 is 18, so $\displaystyle \frac{72}{234}$ can­cels all the way to $\displaystyle \frac{4}{13}$. Do not stop halfway at $\displaystyle \frac{36}{117}$.

Exam­ple 15#

Work out $\displaystyle -\frac{7}{8} - \left(-\frac{5}{24}\right) + \left(-\frac{17}{32}\right)$.

Tak­ing away a neg­a­tive is adding, and adding a neg­a­tive is tak­ing away:

$$\displaystyle -\frac{7}{8} + \frac{5}{24} - \frac{17}{32}$$

The L.C.M of 8, 24 and 32 is 96.

$$\displaystyle \frac{-84 + 20 - 51}{96} = \frac{20 - 135}{96} = -\frac{115}{96}$$

Com­mon mis­takes#

• Adding the denom­i­na­tors. $\displaystyle \frac{2}{3} + \frac{5}{3}$ is $\displaystyle \frac{7}{3}$, not $\displaystyle \frac{7}{6}$.
• Adding before mak­ing the denom­i­na­tors the same. $\displaystyle \frac{5}{4} + \frac{7}{6}$ is not $\displaystyle \frac{12}{10}$.
• Get­ting the order wrong when tak­ing away. $\displaystyle \frac{7 - 10}{5} = -\frac{3}{5}$.
• Not sim­pli­fy­ing. Fin­ish every answer in sim­plest form.

Key terms#

• Like frac­tions: frac­tions with the same denom­i­na­tor.
• L.C.M: the small­est num­ber that all the denom­i­na­tors divide into.
• Com­mon denom­i­na­tor: a denom­i­na­tor shared by all the frac­tions in a sum.

Prac­tice#

1. $\displaystyle \frac{60}{100} - \frac{15}{100} + \frac{25}{100}$
2. $\displaystyle \frac{44}{85} + \frac{16}{85} - \frac{30}{85}$
3. $\displaystyle \frac{5}{6} + \frac{7}{6} - \frac{1}{6}$
4. $\displaystyle \frac{5}{3} - \frac{1}{4}$
5. $\displaystyle \frac{4}{7} + \frac{3}{5}$
6. $\displaystyle \frac{5}{6} + \frac{3}{4} - \frac{7}{8}$
7. $\displaystyle \frac{5}{8} + \frac{3}{4} - \frac{7}{6}$
8. $\displaystyle \frac{3}{5} - 8$
9. $\displaystyle 5 - \frac{3}{8}$
10. $\displaystyle 3 - \frac{7}{12}$
11. $\displaystyle -\frac{13}{15} + \frac{3}{7} + \frac{7}{15} - \frac{9}{7}$
12. $\displaystyle 3\frac{1}{2} - 5\frac{3}{4}$

Answers#