Соотношения, перевод единиц и последовательные проценты
Введено| English | Значение |
|---|---|
| proportion/prəˈpɔːʃn/ | A part's size relative to a whole, or the relative sizes of quantities being compared. |
| inverse/ɪnˈvɜːs/ | A function that undoes another function. |
Miles, gallons and kilometres
- A car uses 6 litres of fuel per 100 km. How much fuel does a 350 km trip need?
- Ratios, rates and units questions are about keeping the units straight and using the right base for comparisons.
Proportions 比例 with units
- Write the rate as a fraction with units and multiply so units cancel: $350 \text{ km} \times \dfrac{6 \text{ L}}{100 \text{ km}} = 21 \text{ L}$.
- For unit conversion, multiply by a fraction equal to 1: $3 \text{ m} \times \dfrac{100 \text{ cm}}{1 \text{ m}} = 300 \text{ cm}$.
Kilometre and litre scales in step: 350 km needs 21 litres at 6 L per 100 km
A printer prints 18 pages per minute. How many minutes does it take to print 270 pages?
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
$270 \div 18 = 15$.
Direct versus inverse 反比
- Direct: $y = kx$ — double $x$, double $y$ (cost and number of items).
- Inverse: $y = \dfrac{k}{x}$ — double $x$, halve $y$ (workers and time to finish a fixed job: 4 workers take 6 days, so 8 workers take 3 days).
Direct or inverse?
Recognise direct and inverse proportion from a situation.
6 workers can paint a hall in 10 hours. Working at the same rate, how many hours would 4 workers take?
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Worker-hours stay 60: $60 \div 4 = 15$.
If doubling $x$ halves $y$, then $y$ is ____ proportional to $x$.
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
$y = k/x$ is an inverse relationship.
Percentages and the base
- Percentage change $= \dfrac{\text{new} - \text{old}}{\text{old}} \times 100\%$. From 40 to 50 is $+25\%$; from 50 to 40 is $-20\%$.
- "A is what percent of B?" divides by B. "A is what percent more than B?" uses B as the base too.
A price falls from 80 to 60: change $= \dfrac{60 - 80}{80} = -25\%$. To get back from 60 to 80 needs $+\dfrac{20}{60} \approx 33.3\%$.
The number of club members rose from 25 to 35. What is the percentage increase?
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
$\dfrac{35 - 25}{25} = 0.4 = 40\%$.
Traps
- Wrong base: the base is the original value (or the one after "than" / "of"), not the bigger number.
- Area and volume units: $1 \text{ m}^2 = 10\,000 \text{ cm}^2$, not 100.
$1 \text{ m}^2$ equals $10\,000 \text{ cm}^2$.
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
True. 1 m = 100 cm, so 1 m² = 100 × 100 = 10 000 cm².
Recap
- Write rates with units and let units cancel; convert with fractions equal to 1.
- Direct: $y = kx$; inverse: $y = \dfrac{k}{x}$. Percentage change divides by the original value.
- inverse: A function that undoes another function.