Percentages · 百分数
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| multiplier/ˌmʌltɪˈplaɪə/ | 乘数 | chéng shù |
| percentage change/pəˈsentɪdʒ tʃeɪndʒ/ | 百分比变化 | bǎi fēn bǐ biàn huà |
| reverse percentage/rɪˈvɜːs pəˈsentɪdʒ/ | 逆百分比 | nì bǎi fēn bǐ |
| compound interest/ˈkɒmpaʊnd ˈɪntrest/ | 复利 | fù lì |
| simple interest/ˈsɪmpl ˈɪntrest/ | 单利 | dān lì |
The 70%-off mystery
- A shop advertises "70% off everything". A jacket originally priced at 120 now costs… $120 \times 0.30 = 36$? Or $120 - 70 = 50$?
- The answer is $36$. Percentages are multipliers 乘数, not amounts to subtract. Getting this right saves you from shopping disasters.
"七折优惠"之谜
- 一家商店宣传"一切七折优惠"(70% off)。一件原价 120 的夹克现在花……$120 \times 0.30 = 36$?还是 $120 - 70 = 50$?
- 答案是 $36$。百分数是乘数,不是要减去的量。把这个弄对让你免于购物灾难。
Percentage change lab · 百分比变化练习
new value = old value x multiplier · 新值 = 旧值 × 乘数
Change the multiplier and see the final value change. · 改变乘数并观察最终值的变化。
Percentage of an amount
- Convert the percentage to a decimal and multiply:
- $15\%$ of $80 = 0.15 \times 80 = 12$.
- $40\%$ of $250 = 0.40 \times 250 = 100$.
一个量的百分数
- 把百分数转换成一个小数并相乘:
- $80$ 的 $15\%$ $= 0.15 \times 80 = 12$。
- $250$ 的 $40\%$ $= 0.40 \times 250 = 100$。
What is 15% of 80? · 15% 的 80 是多少？
0.15 × 80 = 12.
One amount as a percentage of another
- Divide the part by the whole, then multiply by 100:
- $18$ out of $25 = \dfrac{18}{25} \times 100\% = 72\%$.
Percentage change 百分比变化 $= \dfrac{\text{change}}{\text{original}} \times 100\%$. A price rising from 40 to 50: change $= 10$, so $\dfrac{10}{40} \times 100\% = 25\%$ increase.
一个量作为另一个的百分数
- 用部分除以整体,然后乘以 100:
- $25$ 中的 $18$ $= \dfrac{18}{25} \times 100\% = 72\%$。
百分数变化 $= \dfrac{\text{change}}{\text{original}} \times 100\%$。一个价格从 40 上升到 50:变化 $= 10$,所以 $\dfrac{10}{40} \times 100\% = 25\%$ 增加。
A price rises from 40 to 50. What is the percentage increase? · 价格从 40 上涨到 50。百分比涨幅是多少？
change/original × 100% = 10/40 × 100% = 25%. · 变化量/原值 × 100% = 10/40 × 100% = 25%。
Multipliers and reverse percentages 逆百分比
- A multiplier encodes a percentage change in one number:
- $+15\%$ → multiplier $1.15$; $\;-20\%$ → multiplier $0.80$.
- Reverse percentage (find the original from the changed value): divide by the multiplier.
Multipliers shift values: 1.15 moves up by 15%, 0.85 moves down by 15%.
Don't just subtract. A coat costs 60 after a 20% increase. The original was $60 \div 1.2 = 50$, not $60 - 20\% = 48$. You must undo the multiplier, not apply a new one.
乘数与逆百分数
- 一个乘数(multiplier)把一个百分数变化编码进一个数:
- $+15\%$ → 乘数 $1.15$;$\;-20\%$ → 乘数 $0.80$。
- 逆百分数(reverse percentage,从变化后的值求原值):除以乘数。
乘数移动值:1.15 上移 15%,0.85 下移 15%。
不要只是相减。 一件外套在 20% 增加后花 60。原值是 $60 \div 1.2 = 50$,不是 $60 - 20\% = 48$。你必须撤销乘数,而不是应用一个新的。
A coat costs 60 after a 20% increase. What was the original price? · 一件大衣的价格是 60 ，在上涨 20% 之后。原价是多少？
60 is 120% of the original, so 60 ÷ 1.2 = 50. · 60 是原价的 120%，因此 60 ÷ 1.2 = 50。
A 20% decrease corresponds to which multiplier? · 减少 20% 对应哪个乘数？
Losing 20% leaves 80%, so the multiplier is 0.80. · 减少 20% 剩下 80%，因此乘数为 0.80。
If a price increases by 20% and then decreases by 20%, it returns to the original price. · 如果价格上涨 20% 然后下降 20%，它会回到原价。
After +20% (×1.2) then −20% (×0.8): 1.2 × 0.8 = 0.96, so you end up at 96% of the original — a 4% loss. · 经过 +20% (×1.2) 然后 −20% (×0.8)：1.2 × 0.8 = 0.96，因此最终值为原价的 96% —— 损失了 4%。
Simple and compound interest 复利
- Simple interest 单利: $I = \dfrac{Prt}{100}$ (the same amount each year).
- Compound interest: final value $= P \times \left(1 + \dfrac{r}{100}\right)^t$ (interest earns interest).
- 1000 at 5% compound for 3 years: $1000 \times 1.05^3 = 1157.63$.
单利和复利
- 单利(simple interest):$I = \dfrac{Prt}{100}$(每年相同的量)。
- 复利(compound interest):最终值 $= P \times \left(1 + \dfrac{r}{100}\right)^t$(利息赚取利息)。
- 1000 以 5% 复利 3 年:$1000 \times 1.05^3 = 1157.63$。
1000 is invested at 5% compound interest. What is the value after 3 years (to the nearest whole number)? · 投资 1000 元，年利率 5% 复利。3 年后的价值是多少（四舍五入到整数）？
1000 × 1.05³ = 1000 × 1.157625 ≈ 1158.
You've got it
- % of amount = decimal × amount ($15\%$ of $80 = 12$)
- % change $= \dfrac{\text{change}}{\text{original}} \times 100\%$
- reverse %: divide by the multiplier (don't subtract the %)
- compound value $= P(1 + r/100)^t$; simple interest $= Prt/100$
你掌握了
- 一个量的 % = 小数 × 量($80$ 的 $15\%$ $= 12$)
- % 变化 $= \dfrac{\text{change}}{\text{original}} \times 100\%$
- 逆 %:除以乘数(不要减去 %)
- 复利值 $= P(1 + r/100)^t$;单利 $= Prt/100$