# Blog

## Brainteaser: Batteries

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Difficulty: Tricky

A team of engineers have invented a new, stretchy battery that is clear. Another team have invented a jelly-like battery that is rainbow coloured. Both teams want to compare how long their batteries last to a regular, silver AA battery.

Both kinds of batteries have been invented in real life recently! For the sake of this problem, let’s say that one silver AA battery lasts as long as three clear batteries. And let’s say it takes two clear batteries to last as long as six rainbow batteries.

How many rainbow batteries does it take to last as long as one silver AA battery?

## Need a hint?

We don’t have any direct comparison between silver batteries and rainbow ones – so we’ll need to find a way to combine the comparisons we already have.

We know how many rainbow batteries are needed to match two clear batteries, but can you work out how many are needed to match just one clear battery?

It takes nine rainbow batteries to last as long as one silver AA battery.

Let’s convert the word problem into equations. First, we learn that one silver AA battery lasts as long as three clear batteries. This is a ratio that can be written as 1 silver = 3 clear

Next, we learn that it takes two clear batteries to last as long as six rainbow batteries. 2 clear = 6 rainbow

You can make changes to an equation, and as long as you do the same thing on both sides of the equals, it will remain true. In this case, we can divide by 2:

2 clear ÷ 2 = 6 rainbow ÷ 2

1 clear = 3 rainbow

Here are the equations we have so far:
• 1 silver = 3 clear
• 1 clear = 3 rainbow

Notice that both equations include clear batteries. Is there a way we could combine them? We’d need the second equation to start with 3 clear. To do that, we’d need to multiply both sides of that equation by 3.

1 clear x 3 = 3 rainbow x 3
3 clear = 9 rainbow

Now we can combine this with the first equation:
1 silver = 3 clear = 9 rainbow
It takes 9 rainbow batteries to last as long as 1 silver AA battery!

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