Hello there and welcome to week one of Beyond’s daily maths problems! Specifically designed to keep your maths brain in tip-top shape throughout the year, this series of blogs will walk you through a selection of revision questions and problems in order to hone your skillset in preparation for your exams.

This week, we’re exploring:

1. Adding and Subtracting Fractions
2. Finding a Percentage of an Amount
3. Expanding Single Brackets
4. The Product of Prime Factors
5. Estimating

Are you ready?

Question 1: Adding and Subtracting Fractions

When adding or subtracting fractions, you need to have the denominators (the bottom numbers) the same. We do this by finding equivalent fractions.

Let’s calculate:

First, we need to find a common multiple of 7 and 3 – ideally the lowest common multiple. In this case, the lowest common multiple is 21. So, we need to find equivalent fractions with the denominator 21. We do this by multiplying the top (numerator) and bottom (denominator) by the same number.

	(multiply the top and bottom by 3)
	(multiply the top and bottom by 7)

 

Subtracting is just the same; simply subtract the numerators instead of adding them.

Question 2: Finding a Percentage of an Amount

The key to finding a percentage of an amount is remembering the original amount is equivalent to 100%. Then, you can multiply or divide the percentage and the original amount by the same thing to find the required percentage. You can also add amounts together!

Let’s find:

17% of 240

We want to make 17% using small percentages that are easier to find, such as 10%, 5% and 1%. In this case, 17% = 10% + 5% + 1% + 1%

	100% = 240
First, divide both sides by 10 to get 10%.	10% = 24
Now, divide this by 2 to get 5%.	5% = 12
Next, divide 10% by 10 to get 1%.	1% = 2.4
Finally, write another 1%	1% = 2.4

Now, we add together 10%, 5%, 1% and 1% to find 17%.

Question 3: Expanding Single Brackets

The trick to expanding brackets is to be systematic. Multiply each term in the bracket in turn by the term on the outside.

Let’s expand:

The easiest way to expand single brackets is using the grid method. This method easily translates to harder expansions, so it makes sense to get used to it now. We write the expression inside the bracket across the top and the term outside the brackets on the left-hand side. Remember to take the signs with the terms.

Now, we multiply the term on each row by the term on each column.

Our answer is simply the terms in the white boxes added together:

Question 4: The Product of Prime Factors

Writing a number as a product of prime factors is like finding the building blocks of that number. Every number can be written as a unique product of prime factors. The easiest way to find the prime factors is by drawing a factor tree.

Let’s write 460 as a product of primes.

If you find it difficult to recognise prime numbers, start by writing down all the primes below 30. You might need larger primes, but you can always add to them as you go.

2

3

5

7

11

13

17

19

23

29

Now, you divide by different prime numbers in turn. It’s best to be systematic here – start with 2 and divide by this as many times as you can before moving onto a higher number. We draw this in the form of a factor tree to help keep the numbers organised. We also circle any prime numbers to make them stand out.

The factor tree is only part of our answer – the product of prime factors means the prime factors need to be multiplied by each other.

Question 5: Estimating

Estimating is essentially rounding the numbers to make the calculation easier. It won’t give you the exact answer, but it gives you a rough idea.

Let’s estimate:

In this case, we will round 3.9 and 19.7 to one significant figure. So, 3.9 becomes 4 and 19.7 becomes 20.

However, if we round 24 to one significant figure, we get 20. The square root of 20 isn’t easy to find so we round to 25 instead. 25 is a square number so our calculation is much easier.

 

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