Fractions, decimals, percentages and operations · 分数、小数、百分数及运算
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| fraction/ˈfrækʃn/ | 分数 | fēn shù |
| decimal/ˈdesɪml/ | 小数 | xiǎo shù |
| percentage/pəˈsentɪdʒ/ | 百分比 | bǎi fēn bǐ |
| reciprocal/rɪˈsɪprəkl/ | 倒数 | dào shǔ |
| common denominator/ˈkɒmən dɪˈnɒmɪneɪtə/ | 公分母 | gōng fēn mǔ |
| order of operations/ˈɔːdə ɒv ˌɒpəˈreɪʃnz/ | 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 · 数值形式练习
Classify equivalent number forms and operations. · 对等价的数值形式和运算进行分类。
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. · 将 3/8 写成小数。
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. · 将 45% 转换为小数。
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? · 2/3 × 4/5 等于多少？
Multiply tops and bottoms: (2×4)/(3×5) = 8/15. · 分子分母分别相乘：(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. · 1/3 + 1/4 = 4/12 + 3/12 = 7/12，而不是 2/7。相加前必须先通分。
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. · 利用运算顺序，计算 −6 × −3 + 7 × 2。
Multiply first: −6×−3 = 18 and 7×2 = 14; then 18 + 14 = 32. · 先乘法：−6×−3 = 18 且 7×2 = 14；然后 18 + 14 = 32。
Ordering mixed numbers
- To order a list mixing fractions, decimals, and percentages: convert everything to decimals first, then compare.
- $\dfrac{2}{3}=0.666\ldots$, $65\%=0.65$, and $0.6\dot6=0.666\ldots$. The two recurring forms are equal; 0.667 is only an approximation (recurring notation is Extended).
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. · 将这些数按从小到大排序：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. · 2/3 ≈ 0.667, 65% = 0.65, 0.7 = 0.7, 7/10 = 0.7。排序：0.65, 0.667, 0.7, 0.7。
Mixed numbers and negative comparisons
- Convert a mixed number before arithmetic: $2\frac13=(2\times3+1)/3=7/3$. Then $2\frac13-\frac56=14/6-5/6=9/6=1\frac12$.
- Convert unlike forms to compare: $-1\frac14=-1.25<-1.2$. On a number line the more negative number is further left; $\le$ includes the endpoint while $<$ excludes it.
Calculate 2⅓ − ⅚ as a decimal. · 将2⅓ − ⅚化为小数。
Convert to sixths: 14/6 − 5/6 = 9/6 = 1.5. · 化为六分之几：14/6 − 5/6 = 9/6 = 1.5。
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, ÷×, +−)