Adding/subtracting fractions

Worked solution

1. 1

   Rename the quarter using eighths.

   $\frac{1}{4}=\frac{2}{8}$

   A quarter contains two eighths, so both addends now count equal-sized parts.

2. 2

   Add the numerators.

   $\frac{3}{8}+\frac{2}{8}=\frac{5}{8}$

   Three eighths and two eighths make five eighths; the part size remains an eighth.

3. 3

   Check the result.

   $\frac{3}{8}<\frac{5}{8}<1,\mathrm{HCF}\left(5,8\right)=1$

   The positive addition increased the first amount, and the resulting fraction is already simplified.

Worked solution

1. 1

   Convert the mixed number.

   $2⁤\frac{1}{4}=\frac{2\times 4+1}{4}=\frac{9}{4}$

   The two wholes contribute eight quarters, plus the extra quarter.

2. 2

   Rename both fractions as twelfths.

   $\frac{9}{4}=\frac{27}{12},\frac{5}{6}=\frac{10}{12}$

   Twelve is a common multiple of 4 and 6, making the parts equal-sized.

3. 3

   Subtract and write a mixed number.

   $\begin{array}{rl}\frac{27}{12}-\frac{10}{12}& =\frac{17}{12}\\ & =1⁤\frac{5}{12}\end{array}$

   Seventeen twelfths contain one complete group of twelve and five twelfths left over.

4. 4

   Check by adding back the subtracted amount.

   $\frac{17}{12}+\frac{10}{12}=\frac{27}{12}=\frac{9}{4}$

   The inverse operation recovers the starting mixed number.