Ratios, unit conversion and successive percentages
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| unit rate/ˈjuːnɪt reɪt/ | 单位率 | dān wèi lǜ |
| percentage point/pəˈsentɪdʒ pɔɪnt/ | 百分点 | bǎi fēn diǎn |
A decision before an answer
- A twenty-percent reduction followed by a twenty-percent increase does not return to the original price.
- Your goal: Use proportional relationships while preserving units.
Read the relationship
- A ratio compares quantities in a chosen order. If red:blue=3:5 and the total is 64, the eight equal parts each contain eight, so red is 24 and blue 40. The ratio 3/5 describes red relative to blue; 3/8 describes red relative to the total. Read the requested denominator rather than using the first visible fraction.
- Convert rates and distinguish direct from inverse relationships.
A ratio A:B=2:3 with total 45 gives A:
Two of five equal parts gives 45·2/5=18.
Use the defining rule
- A unit rate divides quantity by its corresponding time, distance or count. Write conversion factors so unwanted units cancel: 72 km/hour times 1000 m/km divided by 3600 s/hour is 20 m/s. Area and volume conversions square and cube the length factor. A one-centimetre square is not one metre squared divided by one hundred.
- Calculate successive changes and percentage comparisons using the correct base.
A price rises from 80 to 100. Its percentage increase is:
The change 20 is divided by original 80.
Check the conditions
- In direct proportion y=kx, doubling x doubles y; in inverse proportion xy=k, doubling x halves y. Constant work shared by identical workers gives an inverse time model only when productivity and conditions are fixed. A fixed initial fee added to a rate makes an affine linear model, not direct proportion through the origin.
- Calculate successive changes and percentage comparisons using the correct base.
A 250-unit price discounted 20% becomes 200; increasing that result 20% gives 240, four percent below 250. A pump delivers 18 litres in 3 minutes, so its rate is 6 L/min or 0.1 L/s. At a fixed rate, 45 litres takes 7.5 minutes. If a share rises from 40% to 50%, the rise is 10 percentage points and 25% relative to the old share.
36 km/hour equals ____ m/s.
36·1000/3600=10.
Apply the task format
- A percentage change divides the change by the original comparison base. Successive percentage changes multiply their factors: a 20% discount followed by 20% increase gives 0.8·1.2=0.96 of the original. Distinguish percentage points from relative percent: 30% to 36% is six points, but a 20% relative increase.
- Calculate successive changes and percentage comparisons using the correct base.
State which amount is the percentage base and cancel units visibly. Equal positive and negative percentage changes use different bases.
Which answer fits this case?
Use proportional relationships while preserving units
y=4x+7 is directly proportional to x.
The nonzero intercept prevents y/x being constant.
Keep the distinctions
- unit rate 单位率 — A quantity per one unit of the comparison measure.
- percentage point 百分点 — An absolute difference between two percentage values.
- Use proportional relationships while preserving units.
- Convert rates and distinguish direct from inverse relationships.
- Calculate successive changes and percentage comparisons using the correct base.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.