Sometimes, you’ll be given a problem-solving question and a letter will be introduced to represent a number. It’s obvious then that you’ll be using algebra to solve the problem. In other questions though, algebra can be a great tool to help you to solve the problem, but you’ll need to make that realisation. Here are some tips for recognising when algebra can help and when to use it.
When you’re asked to show or find something, but you’ve not been given any numbers:
- Example 1: show that the shaded area is more than three-quarters of the entire area of the square.
Tip: It’s hard to proceed with this without further information. We know how to find the area of a square and of a circle, but only if we know side lengths and radius. We can introduce the radius of the circle, say as r, which gives a side length of 2r. The shaded area as a fraction of the entire area =
- Example 2: prove that the sum of three consecutive numbers is always equal to three times the middle of those numbers.
Tip: Again, we have no numbers to work with. We could come up with three consecutive whole numbers, but then we’d just be illustrating it for one example, not proving it. Instead, we introduce letters to represent our numbers: Let the lowest number be n, an integer. The next two are n + 1 and n + 2. The sum of those numbers is n + n + 1 + n + 2 = 3n + 3, which is n + 1 multiplied by 3.
The problem suggests a calculation where the answer is given, but at least some of the numbers which go into the calculation are missing:
- Example: Mike buys 5 identical multipacks of crisps and 3 individual packs. Bert buys 2 of the same multipacks and 16 individual packs. Mike has two more bags than Bert. How many bags are there in a multipack?
Tip: If we had a letter to represent the number of bags in a multipack, we could form some equations. Let’s say that’s m.
5m + 3 = 2m + 16 + 2
3m = 15
m = 5; there are 5 packs in a multipack.
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