Percentages, ratio and proportional reasoning · Extension
הוצג| English | משמעות |
|---|---|
| multiplier/ˌmʌltɪˈplaɪə/ | מכפיל |
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.
The method
- 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.
A price falls 20% then rises 20%. Why is the final price not the starting price?
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
A 20% decrease multiplies by 0.8 and the later 20% increase multiplies that result by 1.2: 0.8×1.2=0.96. Each percentage uses a different base.
A worked case
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.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
Multiply by 1.15: 80×1.15=92.
Where it goes wrong
- A percentage uses a stated base.
- Subtracting the percentages loses that base.
- For compound change, multiply the multipliers; do not add the percentages.
Not true: A 20% decrease followed by a 20% increase restores the starting price. The 20% increase is taken of a smaller amount: 100×0.8×1.2=96, not 100.
A 25% discount leaves a price of 90. Find the original price.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
The sale price is 75% of the original: 90/0.75=120.
Multiplier means a factor that performs a percentage change in one multiplication.
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
True: A factor that performs a percentage change in one multiplication.
Applying it
- 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.
Share 84 in the ratio 2:5. Find the larger share.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
There are seven ratio parts. The larger share is 84×5/7=60.
Match each change to its multiplier.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
A multiplier performs the change in one step; reversing a change divides by its multiplier rather than applying the opposite percentage.
In your course
- OxfordAQA International GCSE Mathematics (9260), Extension tier, specification section 3.1.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Multiplier: A factor that performs a percentage change in one multiplication.
- A p% increase has multiplier 1+p/100; a decrease has multiplier 1-p/100.
- Watch for: A percentage uses a stated base.
Write from a final price of 162
- A price of 200 falls by 10 percent and then falls by 10 percent again.
- Find the final price. Two 10 percent falls are not one 20 percent fall.
A price of 200 falls by 10 percent and then falls by another 10 percent. What is the final price, and why is it not 160?
200 times 0.9 is 180, and 180 times 0.9 is 162.
What is the final price?
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
180 minus 18 is 162.