How to Pass Numerical Reasoning (5 page)

Prime factorization =

2 × 3 × 3 = 18

The lowest common multiple of 6 and 9 = 18.

Lowest common multiple: practice drill 1

Set a stopwatch and aim to complete the following drill in four minutes. Find the lowest common multiple of the following sets of numbers:

Lowest common multiple: practice drill 2

Set a stopwatch and aim to complete the following drill in four minutes. Find the lowest common multiple of the following sets of numbers:

Working with large numbers

Test-writers sometimes set questions that ask you to perform an operation on very large or very small numbers. This is a cruel test trap, as it is easy to be confused by a large number of decimal places or zeros. Operations on small and large numbers are dealt with in
Chapter 2
. This section reminds you of some commonly used terms and their equivalents.

Millions, billions and trillions

The meaning of notations such as millions, billions and trillions is ambiguous. The terms vary and the UK definition of these terms is different from the US definition. If you are taking a test developed in the United States (such as the GMAT or GRE), make sure you know the difference.

US definition

Million: 1,000,000 or ‘a thousand thousand’ (same as UK definition)

Billion: 1,000,000,000 or ‘a thousand million’

Trillion: 1,000,000,000,000 or ‘a thousand billion’

UK definition

Million: 1,000,000 or ‘a thousand thousand’ (same as US definition)

Billion: 1,000,000,000,000 or ‘a million million’

Trillion: 1,000,000,000,000,000,000 or ‘a million million million’

The US definitions are more commonly used now. If you are in doubt and do not have the means to clarify which notation is being used, assume the US definition.

Multiplying large numbers

To multiply large numbers containing enough zeros to make you go cross-eyed, follow these three steps:

Step 1: Multiply the digits greater than 0 together.

Step 2: Count up the number of zeros in each number.

Step 3: Add that number of zeros to the result of Step 1.

Worked example

What is the result of 2,000,000 × 2,000?

Step 1: Multiply the digits greater than 0 together

2 × 2 = 4

Step 2: Count up the number of zeros in each number

2,000,000 × 2,000 = nine zeros

Step 3: Add that number of zeros to the right of the result of Step 1

Result of Step 1 = 4

nine zeros = 000,000,000

Answer = 4,000,000,000 (4 billion, US definition)

Multiplying large numbers: practice drill

Use the US definition of billion and trillion to complete this practice drill. Set a stopwatch and aim to complete the following drill in three minutes.

Dividing large numbers

Divide large numbers in exactly the same way as you would smaller numbers, but cancel out equivalent zeros before you start.

Worked example

4,000,000 ÷ 2,000

Cancel out equivalent zeros:

Now you are left with an easier calculation:

4,000 ÷ 2

Answer
= 2,000

Dividing large numbers: practice drill

Set a stopwatch and aim to complete the following drill in four minutes.

Working with signed numbers

Multiplication of signed numbers

There are a few simple rules to remember when multiplying signed numbers.

Tip:
note that it doesn’t matter which sign is presented first in a multiplication calculation.

The product of an odd number of negatives = negative

N × N × N = N

The product of an even number of negatives = positive

N × N × N × N = P

Worked example

Division of signed numbers

Worked example

Multiplication and division of signed numbers: practice drill I

Set a stopwatch and aim to complete each drill within five minutes.

Multiplication and division of signed numbers: practice drill 2

Averages

One way to compare sets of numbers presented in tables, graphs or charts is by working out the average. This is a technique used in statistical analysis to analyse data and to draw conclusions about the content of the data set. The three types of averages are the
arithmetic mean
, the
mode
and the
median
.

Arithmetic mean

The arithmetic mean (also known simply as the average) is a term you are probably familiar with. To find the mean, simply add up all the numbers in the set and divide by the number of terms.