## Task 1 - 170811

The price of a product (100 M) was reduced by $5 \%$ each year over three consecutive years.

a) What percentage of the initial price would a single price reduction need to be to achieve the same final price?

b) What percentage of the final price does the initial price of the product represent?

The percentage values should be rounded to 2 decimal places.
a) An item that initially cost $100 \mathrm{M}$, after the

1. price reduction by $\frac{5 \cdot 100}{100} \mathrm{M}=5 \mathrm{M}$ less, thus $95,-\mathrm{M}$,
2. price reduction by $\frac{5 \cdot 95}{100} \mathrm{M}=4.75 \mathrm{M}$ less, thus $90.25 \mathrm{M}$,
3. price reduction by $\frac{5 \cdot 90.25}{100} \mathrm{M}=4.5125 \mathrm{M} \approx 4.51 \mathrm{M}$ less, thus $85.74 \mathrm{M}$

i.e., with a single price reduction to this final price, the item would have become $100 \mathrm{M}-85.74 \mathrm{M}=14.26 \mathrm{M}$ cheaper. Therefore, the single price reduction would have to be 14.26% of the initial price to achieve the same final price.

b) From the final price $(85.74 \mathrm{M})$ as the base value, the initial price (100 M) is the percentage value for the sought percentage $x$. Consequently, we have:

$$
\frac{x}{100 \%}=\frac{100}{85.74} \quad ; \quad x=\frac{10000}{85.74} \% \approx 116.63 \%
$$

The initial price thus amounts to $116.63 \%$ of the final price.