Percentages: Question 4

Worked solution

Part (a): Reverse percentage

The $29.90 is the price after a 15 per cent increase, so it represents $100\% + 15\% = 115\%$ of the original price.

Step 1: Write 115 per cent as a multiplier.

$115\% = \frac{115}{100} = 1.15$

Step 2: Divide the new price by the multiplier to undo the increase.

$\text{original price} = \frac{29.90}{1.15} = 26$

$\text{Price before the rise} = \boxed{26.00}\ \text{dollars}$

Check: $26 \times 1.15 = 29.90$. ✓ (This confirms we divided rather than subtracted 15 per cent.)

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Part (b): Repeated percentage decrease (depreciation)

The value falls by 12 per cent each year. A 12 per cent decrease leaves $100\% - 12\% = 88\%$ of the value, so the multiplier for one year is:

$88\% = 0.88$

Because the same percentage change is applied year after year, we raise the multiplier to the power of the number of years:

$\text{value after 3 years} = 24000 \times (0.88)^{3}$

Step 1: Evaluate the multiplier.

$(0.88)^{3} = 0.88 \times 0.88 \times 0.88 = 0.681472$

Step 2: Multiply by the starting value.

$24000 \times 0.681472 = 16355.328$

Step 3: Round to the nearest dollar.

$16355.328 \approx 16355$

$\text{Value after 3 years} = \boxed{16355}\ \text{dollars}$

Note: doing three separate steps gives the same result. Year 1: $24000 \times 0.88 = 21120$; year 2: $21120 \times 0.88 = 18585.60$; year 3: $18585.60 \times 0.88 = 16355.328$. ✓

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Part (c): Compound interest

For compound interest at 4 per cent per year, each year’s value is multiplied by:

$100\% + 4\% = 1.04$

Over 5 years the amount is:

$A = 3500 \times (1.04)^{5}$

Step 1: Evaluate the multiplier.

$(1.04)^{5} = 1.2166529024$

Step 2: Find the total amount.

$A = 3500 \times 1.2166529024 = 4258.2851\ldots$

Rounded to 2 decimal places, the amount is $4258.29.

Step 3: Subtract the original investment to find the interest earned.

The question asks for the interest, not the total amount, so subtract the principal:

$\text{interest} = 4258.2851\ldots - 3500 = 758.2851\ldots$

$\text{Interest earned} = \boxed{758.29}\ \text{dollars}$

Check: this is larger than the simple-interest amount $3500 \times 0.04 \times 5 = 700$ (that is, $700), as expected for compound interest. ✓