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.

Convert every value to a decimal: $\dfrac{3}{5} = 0.6$, $0.58$, $62\% = 0.62$ and $\dfrac{7}{12} \approx 0.583$. The largest is $0.62$, which is $62\%$. (You must convert to a common form before comparing.)

$17\div 5 = 3$ 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 $\dfrac{8}{12}+\dfrac{3}{12}=\dfrac{11}{12}$.

Multiplication comes before addition: $\dfrac{1}{4} \times \dfrac{2}{3} = \dfrac{2}{12} = \dfrac{1}{6}$. Then $\dfrac{1}{2} + \dfrac{1}{6} = \dfrac{3}{6} + \dfrac{1}{6} = \dfrac{4}{6} = \dfrac{2}{3}$. (Adding first would break the order of operations.)

The number halfway between two values is their average: $\left(\dfrac{1}{4} + \dfrac{1}{2}\right) \div 2 = \dfrac{3}{4} \div 2 = \dfrac{3}{8}$. (Check: $\dfrac{3}{8} = 0.375$, exactly between $0.25$ and $0.5$.)

Number of pieces $= \dfrac{3}{4} \div \dfrac{1}{8} = \dfrac{3}{4} \times 8 = \dfrac{24}{4} = 6$. (Dividing by a fraction smaller than 1 gives more pieces, not fewer.)

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.

Round each number to something easy: $4.87 \approx 5$ and $19.6 \approx 20$. Then $5 \times 20 = 100$, so the product is about 100. (The exact value, 95.4, is close to this estimate.)

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 $\dfrac{35}{100}=\dfrac{7}{20}$.

If 35% of the number is 14, then 1% is $14 \div 35 = 0.4$, so 100% is $0.4 \times 100 = 40$. (Or divide directly: $14 \div 0.35 = 40$.)

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

First apply the 15% discount: $15\%$ of $60 is $9, so $60 - 9 = 51$. Then take off the extra $5: $51 - 5 = 46$. So the final price is $46.

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.