3.2 Ratios, Proportions & Averages

Ratios, Proportions & Averages

Understanding how different quantities relate to one another is a vital skill tested in the Defence Aptitude Assessment. This section covers ratios, proportions, and statistical averages—tools used to compare and analyze data sets.

Understanding Ratios

A ratio is a mathematical way of comparing the sizes of two or more quantities. It tells you how much of one thing there is compared to another. Ratios are typically written with a colon, such as 3:2.

Simplifying Ratios

Just like fractions, ratios should usually be simplified to their lowest terms. To simplify a ratio, find the greatest common divisor (GCD) of all the numbers in the ratio and divide each by that number.

Worked Example: Simplify the ratio 24:36.

1. Identify the largest number that divides evenly into both 24 and 36. This is 12.
2. Divide both sides by 12: 24 ÷ 12 = 2, and 36 ÷ 12 = 3.
3. The simplified ratio is 2:3.

Sharing in a Ratio

A common exam question involves splitting a total amount according to a specific ratio. The method involves finding the total number of "parts" and then determining the value of a single part.

Worked Example: A budget of £4,500 is split between training, equipment, and travel in the ratio 4:3:2. How much is allocated to equipment?

1. Find the total number of parts: 4 + 3 + 2 = 9 parts.
2. Find the value of one part by dividing the total budget by the total parts: £4,500 / 9 = £500.
3. The equipment allocation corresponds to 3 parts. Multiply the value of one part by 3: 3 × £500 = £1,500.

Proportions

Proportion problems involve relationships where two variables change relative to each other. There are two main types: direct proportion and inverse proportion.

Direct Proportion

Two quantities are in direct proportion if an increase in one leads to a proportional increase in the other. If you double one, you double the other.

Worked Example: If 5 identical combat boots weigh 6 kg, how much do 12 boots weigh?

1. Find the weight of 1 boot (the unit rate): 6 kg / 5 boots = 1.2 kg per boot.
2. Multiply the unit rate by the new quantity: 12 boots × 1.2 kg = 14.4 kg.

Inverse Proportion

Two quantities are in inverse proportion if an increase in one leads to a proportional decrease in the other. For instance, if you double the number of workers, the time taken to complete a task is halved.

Worked Example: If 4 engineers can complete a repair task in 15 hours, how long would it take 6 engineers working at the same rate?

1. Find the total "engineer-hours" required for the task: 4 engineers × 15 hours = 60 engineer-hours.
2. Divide the total effort by the new number of engineers: 60 / 6 = 10 hours.

Averages and Range

Data analysis often requires summarizing a set of numbers using a single representative value, known as an average. The DAA tests four key concepts: Mean, Median, Mode, and Range.

The Mean

The mean is the most common type of average. It is calculated by adding all the values together and dividing by the total number of values.

Worked Example: The fuel consumptions (in liters) for a vehicle over 5 trips were: 45, 52, 48, 50, and 45.

1. Find the sum: 45 + 52 + 48 + 50 + 45 = 240.
2. Divide by the number of trips (5): 240 / 5 = 48 liters.

The Median

The median is the middle value when the data set is arranged in ascending order. If there is an even number of values, the median is the mean of the two middle numbers.

Worked Example: Find the median of the following test scores: 67, 82, 55, 91, 74, 88.

1. Order the numbers: 55, 67, 74, 82, 88, 91.
2. Since there are 6 values (an even number), find the mean of the two middle values (74 and 82).
3. Median = (74 + 82) / 2 = 156 / 2 = 78.

The Mode

The mode is the number that appears most frequently in a data set. A set can have one mode, more than one mode (bimodal/multimodal), or no mode if all numbers appear with equal frequency.

Worked Example: Find the mode of: 3, 7, 5, 7, 2, 9, 3, 7. The number 7 appears three times, which is more frequent than any other number. Thus, the mode is 7.

The Range

The range is a measure of dispersion, showing how spread out the data is. It is calculated by subtracting the lowest value from the highest value.

Worked Example: Find the range of temperatures: -2, 5, 14, 8, 22, 10.

1. Identify the highest value: 22.
2. Identify the lowest value: -2.
3. Calculate the difference: 22 - (-2) = 22 + 2 = 24.

Weighted Averages

Sometimes, different components contribute differently to an overall average. This is known as a weighted average.

Worked Example: A candidate's final score is based on a written test (weighted 60%) and a physical test (weighted 40%). If the candidate scored 80 on the written test and 90 on the physical test, what is the overall average?

1. Multiply each score by its respective weight (as a decimal): Written: 80 × 0.60 = 48 Physical: 90 × 0.40 = 36
2. Add the results together: 48 + 36 = 84.
3. The weighted average score is 84.