Percentages, ratio and proportional reasoning
| English | Español |
|---|---|
| multiplier/ˌmʌltɪˈplaɪə/ | multiplicador |
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?
A factor that performs a percentage change in one multiplication.
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
A p% increase has multiplier 1+p/100; a decrease has multiplier 1-p/100
Explain why a 20% decrease followed by a 20% increase gives 288 rather than 300.
A price of 80 rises by 15%. Find the new price.
Multiply by 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.
The sale price is 75% of the original: 90/0.75=120.
A 20% decrease followed by a 20% increase restores the starting price.
A percentage uses a stated base. Subtracting the percentages loses that base. For compound change, multiply the multipliers; do not add the percentages.
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.
There are seven ratio parts. The larger share is 84×5/7=60.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.