Percentages
Introduced| English | Meaning |
|---|---|
| multiplier/ˌmʌltɪˈplaɪə/ | A number by which another quantity is multiplied. |
| percentage change/pəˈsentɪdʒ tʃeɪndʒ/ | percentage change |
| reverse percentage/rɪˈvɜːs pəˈsentɪdʒ/ | Recovering the original amount before a percentage change by dividing by the multiplier. |
| compound interest/ˈkɒmpaʊnd ˈɪntrest/ | Interest calculated on the original principal together with previously accumulated interest. |
| simple interest/ˈsɪmpl ˈɪntrest/ | Interest calculated only on the original principal, without adding previous interest to it. |
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.
Percentage change lab
new value = old value x multiplier
Change the multiplier and see the final value change.
Percentage of an amount
- Convert the percentage to a decimal and multiply:
- $15\%$ of $80 = 0.15 \times 80 = 12$.
- $40\%$ of $250 = 0.40 \times 250 = 100$.
What is 15% of 80?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
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.
A price rises from 40 to 50. What is the percentage increase?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
change/original × 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.
A coat costs 60 after a 20% increase. What was the original price?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
60 is 120% of the original, so 60 ÷ 1.2 = 50.
A 20% decrease corresponds to which multiplier?
Difficulty 1/5 · 100 coins + a small bonus on first completion today
Losing 20% leaves 80%, so the multiplier is 0.80.
If a price increases by 20% and then decreases by 20%, it returns to the original price.
Difficulty 1/5 · 100 coins + a small bonus on first completion today
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.
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: $1000 \times 1.05^3 = 1157.63$.
1000 is invested at 5% compound interest. What is the value after 3 years (to the nearest whole number)?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
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 $= Prt/100$