Fractions, decimals, percentages and operations
| English | Chinese | Pinyin |
|---|---|---|
| fraction | 分数 | fēn shù |
| decimal | 小数 | xiǎo shù |
| percentage | 百分比 | bǎi fēn bǐ |
| reciprocal | 倒数 | dào shǔ |
| common denominator | 公分母 | gōng fēn mǔ |
| order of operations | 运算顺序 | yùn suàn shùn xù |
Half a pizza, 50% of a pizza, 0.5 of a pizza
- They are the same thing — just three ways of writing the same number.
- Fractions 分数, decimals 小数, and percentages 百分比 are interchangeable. The skill is picking the right form and converting quickly.
Number form lab
Classify equivalent number forms and operations.
Fractions to decimals and back
- Fraction → decimal: divide the top by the bottom. $\dfrac{3}{8} = 3 \div 8 = 0.375$.
- Decimal → fraction: write the decimal over the appropriate power of 10, then simplify.
- $0.6 = \dfrac{6}{10} = \dfrac{3}{5}$.
Memorise the common ones. $\dfrac{1}{4} = 0.25 = 25\%$; $\dfrac{1}{3} = 0.\dot{3} = 33.\dot{3}\%$; $\dfrac{1}{5} = 0.2 = 20\%$; $\dfrac{3}{4} = 0.75 = 75\%$.
Shops use percentages for discounts, sales tax and profit margins
Write 3/8 as a decimal.
3 ÷ 8 = 0.375.
Decimals to percentages
- Decimal → percentage: multiply by 100. $0.375 \times 100 = 37.5\%$.
- Percentage → decimal: divide by 100. $45\% = 0.45$.
One value three ways: a fraction, a decimal and a percentage
Convert 45% to a decimal.
45% = 45 ÷ 100 = 0.45.
Multiplying and dividing fractions
- Multiply: tops × tops, bottoms × bottoms. $\dfrac{2}{3} \times \dfrac{4}{5} = \dfrac{8}{15}$.
- Divide: multiply by the reciprocal 倒数 of the second fraction. $\dfrac{2}{3} \div \dfrac{4}{5} = \dfrac{2}{3} \times \dfrac{5}{4} = \dfrac{10}{12} = \dfrac{5}{6}$.
- Add/subtract: find a common denominator 公分母 first.
Never add tops to tops. $\dfrac{1}{3} + \dfrac{1}{4} \neq \dfrac{2}{7}$. Find a common denominator: $\dfrac{4}{12} + \dfrac{3}{12} = \dfrac{7}{12}$.
What is 2/3 × 4/5?
Multiply tops and bottoms: (2×4)/(3×5) = 8/15.
1/3 + 1/4 = 2/7.
1/3 + 1/4 = 4/12 + 3/12 = 7/12, not 2/7. You must find a common denominator before adding.
Order of operations 运算顺序 (BIDMAS)
- Brackets → Indices → Division/Multiplication (left to right) → Addition/Subtraction (left to right).
- $-6 \times -3 + 7 \times 2 = 18 + 14 = 32$ (multiply before adding).
Using the order of operations, work out −6 × −3 + 7 × 2.
Multiply first: −6×−3 = 18 and 7×2 = 14; then 18 + 14 = 32.
Ordering mixed numbers
- To order a list mixing fractions, decimals, and percentages: convert everything to decimals first, then compare.
- $\dfrac{2}{3} = 0.667$, $\;65\% = 0.65$, $\;0.6\dot{6} = 0.667$.
Number lines make comparisons obvious — plot each value as a decimal and read left to right.
Put these in order from smallest to largest: 2/3, 65%, 0.7, 7/10.
2/3 ≈ 0.667, 65% = 0.65, 0.7 = 0.7, 7/10 = 0.7. Order: 0.65, 0.667, 0.7, 0.7.
You've got it
- fraction → decimal: divide top by bottom ($\dfrac{3}{8} = 0.375$)
- decimal → percentage: ×100; percentage → decimal: ÷100
- multiply fractions: tops × tops, bottoms × bottoms; divide: × the reciprocal
- order of operations: BIDMAS (brackets, indices, ÷×, +−)