Percentages: Question 5

Worked solution

Part (a): the marked price

A $45\%$ increase means the marked price is $100\% + 45\% = 145\%$ of the cost price, a multiplier of $1.45$:

$\text{marked price} = 40 \times 1.45 = 58$

$\text{Marked price} = \boxed{58}\ \text{dollars}$

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Part (b): the sale price after the reduction

A $20\%$ reduction leaves $100\% - 20\% = 80\%$ of the marked price, a multiplier of $0.8$:

$\text{sale price} = 58 \times 0.8 = 46.4$

$\text{Sale price} = \boxed{46.40}\ \text{dollars}$

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Part (c): the overall percentage change

Compare the sale price with the original cost price the shop paid ($40), not with the marked price.

Step 1: Find the change.

$\text{change} = 46.40 - 40 = 6.40$

Since the sale price is greater than the cost price, this is an increase.

Step 2: Express the change as a percentage of the original amount.

$\text{percentage change} = \frac{\text{change}}{\text{original}} \times 100\% = \frac{6.40}{40} \times 100\% = 16\%$

$\boxed{16\%\ \text{increase}}$

Check using combined multipliers: the two changes together give a single multiplier

$1.45 \times 0.8 = 1.16$

which is $116\%$ of the original amount, i.e. a $16\%$ increase overall, confirming the answer without needing the individual dollar values. Note this is not simply $45\% - 20\% = 25\%$; the second percentage change is applied to the already-increased marked price, not to the original cost price, so the changes cannot be subtracted directly.

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Final answers

• (a) Marked price $=$ $58
• (b) Sale price $=$ $46.40
• (c) Overall change $=$ $16\%$ increase