# Problem of the Week Problem B and Solution When is This Deal a Deal?

## Problem

Danielle uses a battery-powered magnifying headlamp when creating silver jewellery. The headlamp requires one AA battery.

Instead of buying a $$10$$-pack of non-rechargeable AA batteries for $$\17.50$$, she decides to buy one rechargeable battery and a charger for $$\40$$.

Suppose each non-rechargeable battery is used until it no longer works and the rechargeable battery is used until it needs to be recharged. Also suppose that the length of time until a non-rechargeable battery no longer works is the same as the length of time until a rechargeable battery needs to be recharged.

After how many rechargeable battery uses will Danielle’s choice be a better deal than buying $$10$$-packs?

You may find the table below to be useful. For example, after $$5$$ uses of the rechargeable battery, the price per use for the rechargeable battery will be $$\40\div 5 = \8.00$$.

Number of Rechargeable Battery Uses Price Per Rechargeable Battery Use
$$5$$ $$\8.00$$
$$10$$
$$15$$
$$20$$
$$\vdots$$

## Solution

The cost of a single battery in a $$10$$-pack is $$\17.50\div 10 = \1.75$$. Therefore, the price per non-rechargeable battery use is $$\1.75$$.

We will use the following completed table to answer the question.

Number of Rechargeable Battery Uses Price Per Rechargeable Battery Use
$$5$$ $$\8.00$$
$$10$$ $$\4.00$$
$$15$$ $$\2.67$$
$$20$$ $$\2.00$$
$$25$$ $$\1.60$$
$$22$$ $$\1.82$$
$$23$$ $$\1.74$$

Examining the completed table above, we see that when we look at increasing the number of rechargeable battery uses by $$5$$, Danielle’s purchase becomes a better deal after $$25$$ uses. We then calculate the price per battery use for $$22$$ and $$23$$ uses. We notice that $$23$$ uses is the smallest number of uses where the price per battery use is less than $$\1.75$$.

Therefore, on the $$23^\text{rd}$$ use the rechargeable battery is a better deal.