Fractions, Decimals and Percentages

The bar shows 2 of 3 equal parts shaded, which is $\dfrac{2}{3}$. Multiply the top and bottom by 2: $\dfrac{2}{3}=\dfrac{4}{6}$, so $\dfrac{4}{6}$ is equivalent.

Use a common denominator of 15: $\dfrac{2}{3} = \dfrac{10}{15}$ and $\dfrac{3}{5} = \dfrac{9}{15}$. Since $\dfrac{10}{15} > \dfrac{9}{15}$, $\dfrac{2}{3}$ is larger. (A bigger denominator does not make a bigger fraction.)

$17\div 5 = 3$ remainder 2, so the mixed number is $3\dfrac{2}{5}$. The remainder becomes the numerator and the divisor stays as the denominator.

After the decimal point the places are tenths, hundredths, thousandths, ten-thousandths. The 7 is the second digit, so it is in the hundredths place.

You cannot add the numerators and the denominators straight across. Using the common denominator 6: $\dfrac{1}{2} + \dfrac{1}{3} = \dfrac{3}{6} + \dfrac{2}{6} = \dfrac{5}{6}$.

Use a common denominator of 12: $\dfrac{2}{3}=\dfrac{8}{12}$ and $\dfrac{1}{4}=\dfrac{3}{12}$. So $\dfrac{8}{12}+\dfrac{3}{12}=\dfrac{11}{12}$.

Use a common denominator of 12: $\dfrac{5}{6}=\dfrac{10}{12}$ and $\dfrac{1}{4}=\dfrac{3}{12}$. So $\dfrac{10}{12}-\dfrac{3}{12}=\dfrac{7}{12}$.

Multiply across: $\dfrac{2}{3}\times\dfrac{3}{5}=\dfrac{6}{15}$. Divide top and bottom by 3 to simplify: $\dfrac{2}{5}$.

Keep, change, flip: $\dfrac{3}{4}\div\dfrac{2}{5}=\dfrac{3}{4}\times\dfrac{5}{2}=\dfrac{15}{8}$. Flip the second fraction, not the first.

Pad each to the same number of places: 0.300, 0.310, 0.301, 0.400. In order that is 0.300, 0.301, 0.310, 0.400. More digits does not mean a larger number.

Line up the decimal points: $5.20-1.45=3.75$. Borrow from the tenths column when subtracting the hundredths.

Multiply $3\times 4 = 12$, then count decimal places: one in 0.3 and one in 0.4, so the answer has two: 0.12.

Divide it out: $3\div 8 = 0.375$.

$0.24=\dfrac{24}{100}$. The HCF of 24 and 100 is 4, so $\dfrac{24}{100}=\dfrac{6}{25}$.

Look at the third decimal place, 8. Since $8\ge 5$, round the second place up: $4.67\to 4.68$.

$\dfrac{1}{4}=0.25$, $\dfrac{3}{8}=0.375$ and $\dfrac{5}{16}=0.3125$ all terminate. $\dfrac{1}{3}=0.333\ldots$ recurs, because 3 does not divide a power of 10.

25% means 25 per hundred, which is one quarter. So 25% of $80 = $80 ÷ 4 = $20.

$\dfrac{3}{4}=0.75$, and $0.75\times 100 = 75$, so $\dfrac{3}{4}=75\%$.

To turn a decimal into a percentage, multiply by 100 (move the point two places to the right): $0.45\to 45\%$.

$35\%=\dfrac{35}{100}$. The HCF of 35 and 100 is 5, so $\dfrac{35}{100}=\dfrac{7}{20}$.

20% of $50 = $10. New price = $50 + $10 = $60. (Or use the multiplier: $50 × 1.20 = $60.)

Walkers: $\dfrac{2}{5}\times 30 = 12$. Bus: $\dfrac{1}{3}\times 30 = 10$. Together that is 22, so the lift group is $30-22=8$.

Total cost = 3 × $14.95 = $44.85. Change = $50 − $44.85 = $5.15.

If 25% is taken off, the sale price is 75% of the original.

• original × 0.75 = $90
• original = $90 ÷ 0.75 = $120
• Check: 25% of $120 is $30, and $120 − $30 = $90.

Three-quarters of two-fifths means $\dfrac{3}{4}\times\dfrac{2}{5}=\dfrac{6}{20}=\dfrac{3}{10}$. Marcus ate $\dfrac{3}{10}$ of the whole pizza.