Tỷ lệ, tỷ số và tốc độ biến thiên
AQA · GCSE · Mathematics Chủ đề 3 5:13 Giọng đọc tiếng Anh · phụ đề tiếng Anh + 中文 được ghi trực tiếp
Chương
Biên bản ghi âm
Nothing for this topic is on the formulae sheet.
这个主题的公式都不在公式表上。
Percentage changes are multipliers, and an original value divides by the multiplier.
百分比变化是乘数，求原值就除以乘数。
Speed, density and pressure are each one quantity over another.
速度、密度、压强都是一个量除以另一个量。
Proportion needs k from one pair of values.
正比例和反比例要用一组值求出 k。
And compound growth repeats the multiplier n times.
复利增长就是把乘数重复 n 次。
Share 350 pounds in the ratio 2 to 5.
把 350 英镑按 2:5 分。
Add the parts: seven parts, so one part is 50.
先把份数相加：共七份，所以一份是 50。
The shares are 100 and 250, and they add back to 350.
两份分别是 100 和 250，加起来回到 350。
A ratio is a part to part comparison: the first share is 2 sevenths of the whole.
比是部分与部分的比较：第一份是总数的 2/7。
A 2.5 kilogram bag costs 6.40, and a 400 gram box costs 1.12.
2.5 千克一袋 6.40 英镑，400 克一盒 1.12 英镑。
Put both per kilogram: 6.40 over 2.5 is 2.56 per kilogram; 1.12 over 0.4 is 2.80 per kilogram.
都换成每千克：6.40 ÷ 2.5 = 每千克 2.56；1.12 ÷ 0.4 = 每千克 2.80。
The bag is better value.
袋装更划算。
The examiner wants the divisions shown, not just the choice.
阅卷人要看到除法过程，而不只是结论。
A percentage change is a multiplier.
百分比变化就是一个乘数。
A 12 percent rise is times 1.12; a 17 percent fall is times 0.83.
增加百分之十二就乘以 1.12；减少百分之十七就乘以 0.83。
To find an original value, divide: 2200 after a 12 percent fall means 2200 divided by 0.88, which is 2500.
求原值要用除法：减少百分之十二后是 2200，原值是 2200 除以 0.88，等于 2500。
Simple interest is linear: 500 at 4 percent for 3 years earns 60.
单利是线性的：500 按百分之四存 3 年，得利息 60。
Sequential changes multiply, never add: a 20 percent rise then a 10 percent fall takes 100 to 120 and then to 108, up 8 percent overall, not 10.
连续变化要相乘，不能相加：先增加百分之二十再减少百分之十，100 先变成 120，再变成 108，总共增加百分之八，而不是百分之十。
Direct proportion is y equals k x to the n; inverse proportion is y equals k over x to the n.
正比例是 y 等于 k 乘 x 的 n 次方；反比例是 y 等于 k 除以 x 的 n 次方。
Find k from one pair of values.
用一组对应值求出 k。
If y is proportional to x cubed and y is 24 when x is 2, then k is 3, so y equals 3 x cubed, a curve rising from the origin.
若 y 与 x 的三次方成正比，且 x 等于 2 时 y 等于 24，则 k 等于 3，所以 y 等于 3x 的三次方，是一条从原点上升的曲线。
If p is inversely proportional to x squared and p is 8 when x is 2, then k is 32, so p equals 32 over x squared, a curve falling towards the axis.
若 p 与 x 的平方成反比，且 x 等于 2 时 p 等于 8，则 k 等于 32，所以 p 等于 32 除以 x 的平方，是一条向轴下降的曲线。
A journey of 234 kilometres takes 2 hours 36 minutes, which is 2.6 hours, not 2.36.
234 千米的路程用时 2 小时 36 分，即 2.6 小时，不是 2.36。
On a distance-time graph the journey is a straight line, and every 1 hour across climbs 90 kilometres up: the gradient is the speed, 90 kilometres per hour.
在距离—时间图上，这段旅程是一条直线，每向右 1 小时就上升 90 千米：斜率就是速度，90 千米/时。
Pay works the same way: 9.20 an hour for 7.5 hours is 69 pounds.
工资也一样：每小时 9.20 英镑，7.5 小时就是 69 英镑。
Lengths scale by k, areas by k squared and volumes by k cubed.
长度按 k 缩放，面积按 k 的平方，体积按 k 的三次方。
Statues with heights in the ratio 3 to 5 have volumes in the ratio 27 to 125, so 216 cubic centimetres scales to 216 times 125 over 27, which is 1000.
高度比为 3 比 5 的雕像，体积比是 27 比 125，所以 216 立方厘米变成 216 乘 125 除以 27， 等于 1000。
Areas would scale by 9 to 25: square the length ratio for an area, cube it for a volume.
面积则按 9 比 25 缩放：面积用长度比的平方，体积用长度比的三次方。
On a curved graph the rate keeps changing.
在曲线图上，变化率一直在变。
The chord from 1 second to 3 seconds gives the average speed: 4 metres in 2 seconds, 2 metres per second.
从 1 秒到 3 秒的弦给出平均速度：2 秒走 4 米，即 2 米/秒。
Draw the tangent at 3 seconds and measure its rise over run: 3 metres per second, the instantaneous rate.
在 3 秒处画切线，量它的升高除以水平距离：3 米/秒，这是瞬时变化率。
Put 1000 yuan in at 10 percent for four years.
把 1000 元按 10% 存四年。
Simple interest adds 100 every year, a straight line to 1400.
单利每年加 100，是一条直线，到 1400。
Compound interest multiplies by 1.1 every year, so each year earns interest on the interest: 1000 times 1.1 to the 4 is 1464.10.
复利每年乘 1.1，利息也会生利息：1000 × 1.1⁴ = 1464.10。
Depreciation multiplies by less than 1.
折旧则乘以小于 1 的数。
A house worth 90 000 pounds rises 3 percent, then 8 percent.
一栋价值 90 000 英镑的房子先涨 3%，再涨 8%。
Multiply the multipliers: 1.03 times 1.08 is 1.1124, not 1.11.
把乘数相乘：1.03 × 1.08 = 1.1124，而不是 1.11。
The value is 100 116 pounds, so yes, it passes 100 000.
价值为 100 116 英镑，所以超过了 100 000。
Iteration repeats a step like this: x n plus 1 equals 1.05 x n.
迭代就是重复这样的一步：xₙ₊₁ = 1.05xₙ。
Check yourself.
自我检测。
45 to 60 simplifies to 3 to 4.
45:60 化简为 3:4。
A population that fell 12 percent to 2200 was 2200 divided by 0.88, which is 2500.
人口下降 12% 后为 2200，原来是 2200 ÷ 0.88 = 2500。
p inversely proportional to x squared, with p equal to 8 at x equal to 2, has k equal to 32.
p 与 x² 成反比，x = 2 时 p = 8，k = 32。
Statues in the ratio 3 to 5 scale volume by 125 over 27: 1000 cubic centimetres.
比例 3:5 的雕像体积乘以 125/27：1000 立方厘米。
And 72 kilometres per hour is 20 metres per second.
72 千米/时是 20 米/秒。