# Numerical Reasoning Test for Training Contract Applications

## Introduction

A percentage is a fraction whose denominator (bottom) is 100. So if we say 50%, we mean 50/100 = 1/2 (after cancelling). So 50% means ½. If want to find 10% of something, 'of' just means 'times'. So 10% of 150 = 10/100 × 150 = 15.

If you have to turn a percentage into a decimal, just divide by 100. For example, 25% = 25/100 = 0.25. To change a decimal into a percentage, multiply by 100. So 0.3 = 0.3 × 100 =30% .

### Example

Find 25% of 10 (remember 'of' means 'times').

25/100 × 10

= 2.5

Percentage Change

% change = new value - original value × 100

original value

Example

The price of some apples is increased from 48p to 67p. By how much percent has the price increased by?

% change = 67 - 48 × 100

48

= 39.58%

Percentage Error

% error = error × 100

real value

Example

Nicola measures the length of her textbook as 20cm. If the length is actually 17.6cm, what is the percentage error in Nicola's calculation?

% error = 20 - 17.6 × 100 = 13.64%

17.6

Original value

Original value = New value × 100

100 + %change

Example

Amish buys a stamp collection and makes a 35% profit by selling it for £2700. Find the cost of the collection. It is the original value we wish to find, so the above formula is used.

2700 × 100 = £2000

100 + 35

Percentage Increases and Interest

New value = 100 + percentage increase × original value

100

Example

£500 is put in a bank where there is 6% per annum interest. Work out the amount in the bank after 1 year.

In other words, the old value is £500 and it has been increased by 6%.

Therefore, new value = 106/100 × 500 = £530 .

Compound Interest

If in this example, the money was left in the bank for another year, the £530 would increase by 6%. The interest, therefore, will be higher than the previous year (6% of £530 is more than 6% of £500). Every year, if the money is left sitting in the bank account, the amount of interest paid would increase each year. This phenomenon is known as compound interest.

The simple way to work out compound interest is to multiply the money that was put in the bank by nm, where n is (100 + percentage increase)/100 and m is the number of years the money is in the bank for, i.e:

(100 + %change)no of years × original value

So if the £500 had been left in the bank for 9 years, the amount would have increased to:

500 × (1.06)9 = £845

Percentage decreases

New value = 100 - percentage decrease × original value

100

Example

At the end of 2003 there were 5000 members of a certain rare breed of animal remaining in the world. It is predicted that their number will decrease by 12% each year. How many will be left at the end of 2005?

At the end of 2004, there will be (100 - 12)/100 × 5000 = 4400

At the end of 2005, there will be 88/100 × 4400 = 3872

The compound interest formula above can also be used for percentage decreases. So after 4 years, the number of animals left would be:

5000 x [(100-12)/100]4 = 2998

Ratios

Introduction

If the ratio of one length to another is 1 : 2, this means that the second length is twice as large as the first. If a boy has 5 sweets and a girl has 3, the ratio of the boy's sweets to the girl's sweets is 5 : 3 . The boy has 5/3 times more sweets as the girl, and the girl has 3/5 as many sweets as the boy. Ratios behave like fractions and can be simplified.

Example

Simon made a scale model of a car on a scale of 1 to 12.5 . The height of the model car is 10cm.

(a) Work out the height of the real car.

The ratio of the lengths is 1 : 12.5 .

So for every 1 unit of length the small car is, the real car is 12.5 units. So if the small car is 10 units long, the real car is 125 units long. If the small car is 10cm high, the real car is 125cm high.

(b) The length of the real car is 500cm. Work out the length of the model car.

We know that model : real = 1 : 12.5 . However, the real car is 500cm, so 1 : 12.5 = x : 500 (the ratios have to remain the same). x is the length of the model car.

To work out the answer, we convert the ratios into fractions:

1     = x

12.5  500

multiply both sides by 500:

500/12.5 = x

so x = 40cm

Example

Alix and Chloe divide £40 in the ratio 3 : 5. How much do they each get?

First, add up the two numbers in the ratio to get 8. Next divide the total amount by 8, i.e. divide £40 by 8 to get £5. £5 is the amount of each 'unit' in the ratio.

To find out how much Alix gets, multiply £5 by 3 ('units') = £15. To find out how much Chloe gets, multiply £5 by 5 = £25.

Map Scales

If a map has a scale of 1 : 50 000, this means that 1 unit on the map is actually 50 000 units across the land. So 1cm on the map is 50 000cm along the ground (= 0.5km).

So 1cm on the map is equivalent to half a kilometre in real life.

For 1 : 25 000, 1 unit on the map is the same as 25 000 units on the land. So 1 inch on the map is 25 000 inches across the land, or 1cm on the map is 25 000 cm in real life.

You can manipulate these ratios if necessary.

Fractions

½ means 1 divided by 2. If you try this on a calculator, you will get an answer of 0.5 . 3/6 means 3 divided by 6. Using a calculator, you will find that this too gives an answer of 0.5 . That is because 1/2 = 3/6 = 0.5 . Fractions such as 3/6 can be cancelled. You can divide the top and bottom of the fraction by 3 to get 1/2 .

With fractions, you are allowed to multiply or divide the top and bottom of the fraction by some number, as long as you multiply (or divide) everything on the top and everything on the bottom by that number.
So 5/12 = 10/24 (multiplying top and bottom by 2).

To add two fractions, the bottom (denominator) of the two fractions must be the same. 1/2 + 3/2 = 4/2 ; 1/10 + 3/10 + 5/10 = 9/10 . If the denominators are not the same, multiply the top and bottom of one (or more) of the fractions by a number to make the denominators the same.

Example

5+2=5+4=9=3
6 3 6 6 6 2

The same is true when subtracting fractions.

This video shows you how to add and subtract fractions with a common denominator

Multiplying Fractions

This is simple: just multiply the two numerators (top bits) together, and the two denominators together:

2×5=10= 5
3 8 24 12

These videos show you how to multiply fractions

Dividing Fractions

If A, B, C and D are any numbers,

A divided byC=A multiplied byD
B D B C

So:

1÷2=1×3=3
2 3 2 2 4

Harder examples

These rules work even when the fractions involve algebra.

2x÷x=2x×3=6x=6 (the x's cancel)
5 3 5 x 5x 5