Fractions - Four Operations

Find what 5 was multiplied by to make 20, then multiply 3 by the same number.

The denominators match, so count the total number of sevenths.

Subtract the numbers of ninths, keeping each piece the same size.

Write both fractions as tenths before adding.

Thirds can be rewritten as sixths.

Each whole contains five fifths.

1. Two wholes contain 2\times5=10 fifths.
2. Add the extra three fifths: 10+3=13. So 2\frac{3}{5}=\frac{13}{5} and the missing numerator is 13.

Multiply top by top and bottom by bottom, then simplify.

1. \frac{2}{3}\times\frac{3}{7}=\frac{2\times3}{3\times7}=\frac{6}{21}.
2. Divide top and bottom by 3: \frac{6}{21}=\frac{2}{7}.

Keep the first fraction and multiply by the reciprocal of the second.

1. The reciprocal of \frac{2}{3} is \frac{3}{2}.
2. \frac{4}{5}\div\frac{2}{3}=\frac{4}{5}\times\frac{3}{2}=\frac{12}{10}.
3. Simplify: \frac{12}{10}=\frac{6}{5}, which is 1\frac{1}{5}.

Divide the numerator and denominator by their highest common factor.

1. The highest common factor of 12 and 18 is 6.
2. \frac{12}{18}=\frac{12\div6}{18\div6}=\frac{2}{3}.
3. The fractions \frac{6}{9} and \frac{4}{6} have the same value but can still be simplified.

You are subtracting a larger amount from a smaller amount, so the answer will be negative.

1. The denominators already match. Subtract the numerators: 1-3=-2.
2. \frac{1}{4}-\frac{3}{4}=-\frac{2}{4}=-\frac{1}{2}.

Change both lengths into quarters, then subtract.

1. 3\frac{1}{2}=\frac{7}{2}=\frac{14}{4} metres.
2. 1\frac{3}{4}=\frac{7}{4} metres.
3. Length left: \frac{14}{4}-\frac{7}{4}=\frac{7}{4} metres, or 1\frac{3}{4} metres.

Convert the mixed number to an improper fraction before multiplying.

1. 1\frac{2}{5}=\frac{1\times5+2}{5}=\frac{7}{5}.
2. \frac{7}{5}\times\frac{5}{6}=\frac{35}{30}.
3. Divide top and bottom by 5: \frac{35}{30}=\frac{7}{6}, or 1\frac{1}{6}.

Divide the total mass of jam by the mass in one jar.

1. 2\frac{1}{4}=\frac{9}{4} kg.
2. Number of jars: \frac{9}{4}\div\frac{3}{8}=\frac{9}{4}\times\frac{8}{3}=\frac{72}{12}=6.
3. Check: 6\times\frac{3}{8}=\frac{18}{8}=\frac{9}{4} kg, so all the jam fills exactly 6 jars.

Add the two painted fractions, then subtract their total from one whole wall.

1. \frac{2}{7}=\frac{4}{14}. Total painted: \frac{4}{14}+\frac{3}{14}=\frac{7}{14}=\frac{1}{2}.
2. The whole wall is 1. Unpainted fraction: 1-\frac{1}{2}=\frac{1}{2}.

Check how many twelfths are left after taking one whole from fourteen twelfths.

1. The conversion 1\frac{3}{4}=\frac{7}{4} is correct, and multiplying gives \frac{14}{12}.
2. One whole is \frac{12}{12}. The remainder is 14-12=2 twelfths, not 1 twelfth.
3. So \frac{14}{12}=1\frac{2}{12}=1\frac{1}{6}. The mistake is in the final conversion to a mixed number.

Divide the amount of water by the amount in one cup. Then work out how much the full cups use.

Part (a)

1. 2\frac{1}{2}=\frac{5}{2}.
2. \frac{5}{2}\div\frac{1}{3}=\frac{5}{2}\times3=\frac{15}{2}=7\frac{1}{2}.
3. There is enough for 7 full cups and half of another cup. The greatest number of completely full cups is 7.

Part (b)

1. The 7 full cups use 7\times\frac{1}{3}=\frac{7}{3} litres.
2. Water left over: \frac{5}{2}-\frac{7}{3}=\frac{15}{6}-\frac{14}{6}=\frac{1}{6} litre.

Find the flour used for all the bread and for all the biscuits before subtracting from 3 kg.

1. Bread uses 2\times\frac{3}{4}=\frac{6}{4}=\frac{3}{2} kg.
2. Biscuits use 3\times\frac{2}{5}=\frac{6}{5} kg.
3. Total used: \frac{3}{2}+\frac{6}{5}=\frac{15}{10}+\frac{12}{10}=\frac{27}{10} kg.
4. Flour left: 3-\frac{27}{10}=\frac{30}{10}-\frac{27}{10}=\frac{3}{10} kg.

Undo the multiplication first by dividing by three quarters. Then undo the addition.

1. The amount inside the brackets is \frac{1}{2}\div\frac{3}{4}=\frac{1}{2}\times\frac{4}{3}=\frac{2}{3}.
2. Subtract \frac{1}{6}: \square=\frac{2}{3}-\frac{1}{6}=\frac{4}{6}-\frac{1}{6}=\frac{3}{6}=\frac{1}{2}.
3. Check: \left(\frac{1}{2}+\frac{1}{6}\right)\times\frac{3}{4}=\frac{2}{3}\times\frac{3}{4}=\frac{1}{2}.

Find the fraction of the total width and total height occupied by the unshaded rectangle.

1. The unshaded rectangle has 1-\frac{1}{4}=\frac{3}{4} of the total width.
2. Its height is 1-\frac{2}{5}=\frac{3}{5} of the total height.
3. The area is therefore \frac{3}{4}\times\frac{3}{5}=\frac{9}{20} of the whole rectangle.
4. The two shaded strips overlap, so adding their fractions would count the bottom-left corner twice.

The one third refers to the distance left after the morning walk, not to the whole route.

1. After the morning walk, the fraction left is 1-\frac{2}{5}=\frac{3}{5}.
2. The second walk is \frac{1}{3} of this remainder: \frac{1}{3}\times\frac{3}{5}=\frac{1}{5} of the whole route.
3. Total completed: \frac{2}{5}+\frac{1}{5}=\frac{3}{5}.
4. The fraction still left is 1-\frac{3}{5}=\frac{2}{5}.