3.1 Mental Arithmetic, Percentages & Fractions

Mental Arithmetic, Percentages & Fractions

Numerical reasoning forms a significant component of the Defence Aptitude Assessment (DAA). Unlike advanced mathematics, the numerical reasoning section tests your ability to quickly and accurately process basic quantitative information. This section focuses on mental arithmetic, percentages, fractions, and the relationships between them.

Mental Arithmetic and BODMAS

At the core of numerical reasoning is basic arithmetic: addition, subtraction, multiplication, and division. Because you are working under strict time constraints, relying purely on written calculations will consume precious time. Developing mental math shortcuts is crucial.

The Order of Operations

When faced with a calculation involving multiple operations, you must follow the correct order. This is universally remembered by the acronym BODMAS (or PEMDAS in some regions):

• Brackets (Parentheses)
• Orders (Powers, roots, etc.)
• Division and Multiplication (Left to right)
• Addition and Subtraction (Left to right)

Worked Example: Calculate: 12 + 4 × (6 - 2) ÷ 2

1. Brackets first: (6 - 2) = 4. The equation becomes 12 + 4 × 4 ÷ 2.
2. Multiplication and Division next (left to right): 4 × 4 = 16. Then 16 ÷ 2 = 8.
3. Addition last: 12 + 8 = 20.

Working with Fractions

A fraction represents a part of a whole. It consists of a numerator (the top number) and a denominator (the bottom number). You must be comfortable performing all four fundamental operations with fractions.

Adding and Subtracting Fractions

To add or subtract fractions, they must have a common denominator. If they do not, you must find the lowest common multiple (LCM) of the denominators.

Worked Example: Calculate: 1/4 + 2/5

1. Find the LCM of 4 and 5, which is 20.
2. Convert 1/4: Multiply top and bottom by 5 to get 5/20.
3. Convert 2/5: Multiply top and bottom by 4 to get 8/20.
4. Add the numerators: 5/20 + 8/20 = 13/20.

Multiplying Fractions

Multiplying fractions is straightforward: simply multiply the numerators together and the denominators together.

Worked Example: Calculate: 3/7 × 2/5 Numerator: 3 × 2 = 6 Denominator: 7 × 5 = 35 Result: 6/35 (This cannot be simplified further).

Dividing Fractions

To divide fractions, remember the rule "Keep, Change, Flip." Keep the first fraction, change the division sign to multiplication, and flip the second fraction (find its reciprocal).

Worked Example: Calculate: 3/4 ÷ 1/2

1. Keep 3/4, change ÷ to ×, flip 1/2 to 2/1.
2. Multiply: 3/4 × 2/1 = 6/4.
3. Simplify: Divide top and bottom by 2 to get 3/2, or 1.5.

Mastering Percentages

Percentages are simply fractions with a denominator of 100. The word "percent" literally translates to "per hundred." A solid grasp of percentages is essential for interpreting financial data, technical specifications, and general statistics in the DAA.

Finding a Percentage of an Amount

To find a percentage of a number, convert the percentage to a decimal (divide by 100) and multiply by the number.

Worked Example: Find 35% of 240.

1. Convert 35% to 0.35.
2. Calculate 0.35 × 240.
3. A mental shortcut: 10% of 240 is 24. So, 30% is 3 × 24 = 72. 5% is half of 10%, which is 12. 72 + 12 = 84.

Calculating Percentage Change

Questions asking for the percentage increase or decrease are highly common. The formula is:

Percentage Change = (Difference / Original Value) × 100

Worked Example: A squadron's fuel reserve drops from 800 gallons to 640 gallons. What is the percentage decrease?

1. Find the difference: 800 - 640 = 160.
2. Divide by the original value (800): 160 / 800 = 0.20.
3. Multiply by 100: 0.20 × 100 = 20%. The fuel reserve decreased by 20%.

Reverse Percentages

Sometimes you are given the final amount after a percentage change and must find the original amount. To do this, divide the final amount by the decimal multiplier representing the percentage change.

Worked Example: After a 15% pay raise, an engineer's salary is £46,000. What was the original salary?

1. A 15% increase means the new salary is 115% of the original, or 1.15 as a decimal multiplier.
2. Original Salary = New Salary / Multiplier.
3. Original Salary = £46,000 / 1.15 = £40,000.

Conversions: Fractions, Decimals, and Percentages

You must be able to seamlessly convert between fractions, decimals, and percentages. Here is a table of common conversions you should memorize to save time during the exam:

Memorizing these will allow you to quickly approximate and calculate exact figures without resorting to long division. For example, if asked to find 12.5% of 64, knowing that 12.5% is 1/8 allows you to simply calculate 64 / 8 = 8, saving valuable seconds.