Fractions

To find $\frac{1}{6}$ of 96, divide by 6.

$96 \div 6 = 16$

To find $\frac{1}{5}$ of 300, divide by 5.

$300 \div 5 = 60$

The shape is split into 10 equal triangles and 3 of them are shaded.

So the fraction shaded is $\frac{3}{10}$.

The grid has $3 \times 7 = 21$ squares and 6 of them are shaded.

Fraction shaded $= \frac{6}{21} = \frac{2}{7}$

One whole is $\frac{10}{10}$.

$\frac{10}{10} - \frac{3}{10} = \frac{7}{10}$

To find $\frac{1}{4}$ of an amount, divide by 4.

$28 \div 4 = 7$

The part goes on top and the whole on the bottom: $\frac{17}{30}$.

To find $\frac{1}{6}$ of an amount, divide by 6.

$66 \div 6 = 11$

One third means divide by 3.

$24 \div 3 = 8$

The highest common factor of 28 and 70 is 14.

$\frac{28}{70} = \frac{28 \div 14}{70 \div 14} = \frac{2}{5}$

To find $\frac{1}{3}$ of an amount, divide by 3.

$30 \div 3 = 10$

Simplify each: $\frac{4}{16} = \frac{1}{4}$, $\frac{2}{8} = \frac{1}{4}$, $\frac{15}{60} = \frac{1}{4}$, but $\frac{3}{9} = \frac{1}{3}$.

So the odd one out is $\frac{3}{9}$.

$\frac{1}{8}$ of 720 means $720 \div 8 = 90$

50 is $\frac{1}{16}$ of the total, so the total is 16 lots of 50.

$16 \times 50 = 800$

Large: $\frac{1}{4}$ of 12 = 3, so small = $12 - 3 = 9$.

Large weight: $3 \times 70 = 210$ g. Small weight: $9 \times 50 = 450$ g.

Total: $210 + 450 = 660$ grams.

Count the squares: 16 in total, 4 unshaded, so 12 shaded.

$\frac{12}{16} = \frac{3}{4}$

Harry has $20 - 7 = 13$ sweets left.

As a fraction of 20: $\frac{13}{20}$

(a) Cost: $13 \times 7.50 = £97.50$. Paid: $5 \times 20 = £100$.

Change $= 100 - 97.50 = £2.50$

(b) $\frac{1}{5}$ of 120 $= 24$, so the sale price is $120 - 24 = £96$.

A quarter of 52 is $52 \div 4 = 13$ litres already in the tank.

Fuel needed: $52 - 13 = 39$ litres (or $\frac{3}{4} \times 52 = 39$).

White bars $= \frac{2}{3} \times 24 = 16$

Milk bars $= 24 - 16 = 8$

Weight $= 8 \times 35 = 280$ g

Multiply tops and bottoms: $\frac{6 \times 5}{7 \times 12} = \frac{30}{84}$

Divide top and bottom by 6: $\frac{5}{14}$

(a) $\frac{1}{6} = \frac{2}{12}$, so $\frac{5}{12} + \frac{2}{12} = \frac{7}{12}$

(b) $\frac{3}{10} \times \frac{5}{8} = \frac{15}{80}$

Divide top and bottom by 5: $\frac{3}{16}$

(a) $132 \div 8 = 16.5$, and $16.5 \times 5 = 82.5$

(b) Use denominator 64: $\frac38 = \frac{24}{64}$, $\frac{9}{32} = \frac{18}{64}$, $\frac14 = \frac{16}{64}$, $\frac{21}{64}$.

Order: $\frac14, \frac{9}{32}, \frac{21}{64}, \frac38$

Large bars: $208 \div 4 = 52$, worth $52 \times £1 = £52$

Small bars: $208 - 52 = 156$, worth $156 \times £0.60 = £93.60$

Total: $£52 + £93.60 = £145.60$

Write it as a fraction: $\frac{60}{1000}$

Divide top and bottom by 20: $\frac{3}{50}$

Convert to decimals:

$\frac{5}{8} = 0.625$, $\frac{2}{3} = 0.667$, $\frac{4}{9} = 0.444$, $\frac{3}{5} = 0.6$

Order: $\frac{4}{9}, \frac{3}{5}, \frac{5}{8}, \frac{2}{3}$

The land is $1 - \frac{13}{15} = \frac{2}{15}$ of the cost.

$600 \div 15 = 40$

$40 \times 2 = 80$, so the land cost £80 million.

(a) $70 \div 5 = 14$

(b) Each whole contains 2 halves, so 48 wholes contain $48 \times 2 = 96$ halves. Dividing by $\frac{1}{2}$ is the same as multiplying by 2, so the answer should be 96, not 24.

$\frac{1}{10} = 0.1$ and $\frac{3}{5} = 0.6$

Halfway: $\frac{0.1 + 0.6}{2} = 0.35$ (or $\frac{7}{20}$)

Given away: $35 + 50 + 75 = 160$

Left: $400 - 160 = 240$

$\frac{240}{400} = \frac{3}{5}$

Both cats together eat $2 \times \frac{1}{4} = \frac{1}{2}$ tin a day.

For 14 days: $14 \times \frac{1}{2} = 7$ tins.

$7 < 8$, so yes, Sue has bought enough.

(a) $\frac{10}{16} = \frac{5}{8}$, not $\frac{3}{4}$. The others all simplify to $\frac{3}{4}$.

(b) $\frac{5}{6} = \frac{10}{12}$

$\frac{1}{12} + \frac{10}{12} = \frac{11}{12}$

Increase: $600 - 560 = 40$.

As a fraction of last year's cost: $\frac{40}{560} = \frac{1}{14}$.

Men: $\frac{7}{20} \times 500 = 175$

Women: $40\% \times 500 = 200$

Children: $500 - 175 - 200 = 125$

Use a denominator of 24:

$\frac{1}{2} = \frac{12}{24}$, $\frac{3}{4} = \frac{18}{24}$, $\frac{5}{12} = \frac{10}{24}$

Order: $\frac{10}{24}, \frac{12}{24}, \frac{17}{24}, \frac{18}{24}$, i.e. $\frac{5}{12}, \frac{1}{2}, \frac{17}{24}, \frac{3}{4}$

(a) $\frac{2}{5} = \frac{4}{10}$ and $\frac{1}{2} = \frac{5}{10}$, so the sum is $\frac{9}{10}$. Lethna changed the denominators but left the numerators alone.

(b) $1\frac{1}{2} \times 5\frac{1}{3} = \frac{3}{2} \times \frac{16}{3} = \frac{48}{6} = 8$. Dave split the mixed numbers apart, which does not work for multiplying.

(a) $3\frac{1}{2} - 1\frac{1}{6} = 3\frac{3}{6} - 1\frac{1}{6} = 2\frac{2}{6} = 2\frac{1}{3}$

(b) $5\frac{1}{4} = \frac{21}{4}$ and $2\frac{1}{3} = \frac{7}{3}$

$\frac{21}{4} \div \frac{7}{3} = \frac{21}{4} \times \frac{3}{7} = \frac{63}{28} = \frac{9}{4} = 2\frac{1}{4}$

(a) $3\frac{4}{5} - 1\frac{2}{3} = \frac{57}{15} - \frac{25}{15} = \frac{32}{15} = 2\frac{2}{15}$

(b) $35 - 24 = 11$, so $\frac{35}{24} = 1\frac{11}{24}$. Kevin wrote 9 instead of 11 in the numerator.

Keep, change, flip: $\frac{3}{5} \div \frac{1}{6} = \frac{3}{5} \times \frac{6}{1} = \frac{18}{5}$

$18 \div 5 = 3$ remainder 3, so $\frac{18}{5} = 3\frac{3}{5}$

Make improper fractions: $7\frac{3}{8} = \frac{59}{8}$, $2\frac{1}{2} = \frac{20}{8}$

$\frac{59}{8} - \frac{20}{8} = \frac{39}{8} = 4\frac{7}{8}$

Make improper fractions: $7\frac{3}{8} = \frac{59}{8}$, $2\frac{1}{2} = \frac{20}{8}$

$\frac{59}{8} - \frac{20}{8} = \frac{39}{8} = 4\frac{7}{8}$

(a) $1 + 2 = 3$; $\frac{3}{5} + \frac{1}{4} = \frac{12}{20} + \frac{5}{20} = \frac{17}{20}$

Answer $3\frac{17}{20}$

(b) $2\frac{2}{3} = \frac{8}{3}$

$\frac{8}{3} \div 6 = \frac{8}{3} \times \frac{1}{6} = \frac{8}{18} = \frac{4}{9}$

$4\frac{1}{5} = \frac{21}{5} = \frac{63}{15}$ and $2\frac{2}{3} = \frac{8}{3} = \frac{40}{15}$

$\frac{63}{15} - \frac{40}{15} = \frac{23}{15} = 1\frac{8}{15}$

Rachel: $\frac{2}{5} \times 600 = £240$. Left: $600 - 240 = £360$.

Samina: $\frac{1}{4} \times 360 = £90$.

Tom: $360 - 90 = £270$.

Equal share: $600 \div 3 = £200$. Tom already gets more, so he is not correct.

Change to improper fractions: $2\frac{1}{3} = \frac{7}{3}$ and $3\frac{3}{4} = \frac{15}{4}$.

Multiply: $\frac{7 \times 15}{3 \times 4} = \frac{105}{12} = \frac{35}{4}$.

$35 \div 4 = 8$ remainder 3, so $\frac{35}{4} = 8\frac{3}{4}$.

(a) $\frac{2}{3} - \frac{1}{5} = \frac{10}{15} - \frac{3}{15} = \frac{7}{15}$

(b) $\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}$

$3\frac{1}{2} = \frac{7}{2}$ and $1\frac{3}{5} = \frac{8}{5}$

$\frac{7}{2} \times \frac{8}{5} = \frac{56}{10} = \frac{28}{5} = 5\frac{3}{5}$

(a) Whole numbers: $2 + 1 = 3$. Fractions: $\frac{1}{7} + \frac{1}{4} = \frac{4}{28} + \frac{7}{28} = \frac{11}{28}$

Answer $3\frac{11}{28}$

(b) $1\frac{1}{5} = \frac{6}{5}$, then $\frac{6}{5} \times \frac{4}{3} = \frac{24}{15} = \frac{8}{5} = 1\frac{3}{5}$

The top strip of B matches the unshaded part of A: $1 - \frac{5}{8} = \frac{3}{8}$

The bottom strip of B matches the unshaded part of C: $1 - \frac{9}{11} = \frac{2}{11}$

Shaded part of B: $1 - \frac{3}{8} - \frac{2}{11} = \frac{88 - 33 - 16}{88} = \frac{39}{88}$

$1\frac{3}{4} = \frac{7}{4}$ and $1\frac{1}{3} = \frac{4}{3}$

$\frac{7}{4} \times \frac{4}{3} = \frac{28}{12} = \frac{7}{3} = 2\frac{1}{3}$