Adding and subtracting fractions

Worked solution

1. 1

   Choose sixths as the shared unit.

   $\frac{2}{3}=\frac{2\times 2}{3\times 2}=\frac{4}{6}$

   Each third contains two sixths. Multiplying both numbers by $2$ keeps the fraction the same size.

2. 2

   Add the numerators and keep the denominator.

   $\frac{4}{6}+\frac{1}{6}=\frac{5}{6}$

   Four sixths plus one sixth makes five sixths. The pieces are still sixths.

3. 3

   Check the answer.

   $\frac{2}{3}<\frac{5}{6}<1$

   The sum is larger than the starting fraction and smaller than one whole. The answer is already in simplest form.

Worked solution

1. 1

   Read the size of the removed piece.

   $\frac{1}{3}=\frac{2}{6}$

   The complete bar has six equal parts. Two of those parts make one third of the whole.

2. 2

   Subtract the sixths that were removed.

   $\frac{5}{6}-\frac{2}{6}=\frac{3}{6}$

   There were five shaded sixths. Crossing out two leaves three shaded sixths.

3. 3

   Simplify the remaining fraction.

   $\frac{3}{6}=\frac{1}{2}$

   Divide both numbers by $3$. Three of six equal parts cover half the whole, which is less than the starting amount.

Worked solution

1. 1

   Rename the full length in quarters.

   $1⁤\frac{1}{2}=\frac{3}{2}=\frac{6}{4}$

   One whole contains four quarters and one half contains two more quarters.

2. 2

   Subtract the length that was cut off.

   $\frac{6}{4}-\frac{3}{4}=\frac{3}{4}$

   Both lengths now count quarters of a metre, so subtract their numerators.

3. 3

   State the unit and check the length.

   $\frac{3}{4}\mathrm{m}<1⁤\frac{1}{2}\mathrm{m}$

   The remaining length is smaller than the original ribbon. The fraction is already in simplest form.