Percentages, ratio and proportional reasoning · Extension · 百分数、比率和比例推理 · 拓展
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| multiplier/ˌmʌltɪˈplaɪə/ | 乘数 | chéng shù |
Can we recover the original price?
- A coat is reduced by 20% to ¥240. The discount applies to the original price, not to the sale price.
- This lesson studies multiplier 乘数: A factor that performs a percentage change in one multiplication.
Choose the mathematical structure
- A p% increase has multiplier 1+p/100; a decrease has multiplier 1-p/100. Reverse a percentage by dividing by the multiplier. In a ratio, first find the total number of parts.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines multiplier? · 哪项描述正确定义了“倍数”？
A factor that performs a percentage change in one multiplication. · 在一次乘法运算中实现百分比变化的因子。
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Let the original price be P. The model is sale price=0.8P. Hence P=240/0.8=300. A later 20% increase gives 240×1.2=288, so the two changes do not cancel.
Percentages, ratio and proportional reasoning · 百分数、比例与成数推理
A p% increase has multiplier 1+p/100; a decrease has multiplier 1-p/100 · 增加p%的倍数为1+p/100；减少的倍数为1-p/100
Explain why a 20% decrease followed by a 20% increase gives 288 rather than 300. · 解释为何先下降20%再上升20%的结果为288而非300。
A price of 80 rises by 15%. Find the new price. · 80的价格上涨15%。求新价格。
Multiply by 1.15: 80×1.15=92. · 乘以1.15：80×1.15=92。
Test a tempting shortcut
- A percentage uses a stated base. Subtracting the percentages loses that base. For compound change, multiply the multipliers; do not add the percentages.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A 20% decrease followed by a 20% increase restores the starting price. This claim is false. Explain which definition or assumption it violates.
A 25% discount leaves a price of 90. Find the original price. · 打25%折扣后价格为90。求原价。
The sale price is 75% of the original: 90/0.75=120. · 折后价是原价的75%：90/0.75=120。
A 20% decrease followed by a 20% increase restores the starting price. · 先下降20%，再上升20%，恢复初始价格。
A percentage uses a stated base. Subtracting the percentages loses that base. For compound change, multiply the multipliers; do not add the percentages. · 百分数基于指定的基数。直接相减百分比会丢失该基数。对于复合变化，应相乘倍数；不可相加百分比。
Interpret a new situation
- For direct proportion use y=kx; for inverse proportion use y=k/x. Calculate k from a known pair before using a new value. State what you held constant.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Share 84 in the ratio 2:5. Find the larger share. · 将84按2:5的比例分配。求较大的份额。
There are seven ratio parts. The larger share is 84×5/7=60. · 共有七份比例。较大份额为84×5/7=60。
Match each part of a complete solution to its purpose. · 将完整解答的每个部分与其目的相匹配。
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · 假设用于论证模型的合理性；检查用于验证结果；解释用于将其与问题建立联系。
Use this in your course
- 9260 · Extension · 3.1. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A factor that performs a percentage change in one multiplication. Choose the relationship, show the method, check its assumptions and interpret the result.