Section 3.2: Percentages

Section 3.2: Percentages

In Victoria Police operations and administrative functions, percentages are used to express parts of a whole in a standardized format out of 100. Crime analysts, station managers, and road safety officers use percentages to report crime rates, evaluate performance targets, and allocate budgets. On the ACER numeracy subtest, percentage calculations are highly frequent and assess your ability to interpret quantitative data at the Australian Core Skills Framework (ACSF) Level 3 standard. Understanding how to quickly perform percentage calculations, find percentage change, and interpret percentage-based tables is key to achieving the required scale score of 111.

Core Percentage Calculations

To calculate a percentage of a given quantity, convert the percentage to a decimal or fraction and multiply it by the total amount. The mathematical formula is:

$\text{Amount} = \frac{\text{Percentage}}{100} \times \text{Total}$

For example, if the Victoria Police Highway Patrol conducts a targeted road safety blitz in Geelong and performs 1,200 random breath tests, and statistics show that historically $1.5%$ of drivers test positive, the expected number of positive tests can be calculated as:

$\text{Amount} = \frac{1.5}{100} \times 1,200 = 0.015 \times 1,200 = 18 \text{ drivers}$

Conversely, to find what percentage one number is of another, divide the part by the whole and multiply the result by 100:

$\text{Percentage} = \left(\frac{\text{Part}}{\text{Whole}}\right) \times 100$

For instance, if a station receives 80 calls for assistance during a weekend shift, and 24 of those calls are classified as domestic or family violence incidents, the percentage of family-violence-related calls is:

$\text{Percentage} = \left(\frac{24}{80}\right) \times 100 = 0.3 \times 100 = 30\%$

Percentage Increase and Decrease

In law enforcement, measuring trends over time is vital for determining whether crime prevention strategies are working. This is achieved using percentage change, which measures the relative difference between an old value and a new value.

The general formula for percentage change is:

$\text{Percentage Change} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100$

If the result is positive, it represents a percentage increase. If the result is negative, it represents a percentage decrease.

Calculating Percentage Increase

Suppose the number of residential burglaries in a suburb of Melbourne increases from 40 in one quarter to 50 in the next. To find the percentage increase:

$\text{Percentage Increase} = \frac{50 - 40}{40} \times 100 = \frac{10}{40} imes 100 = 0.25 \times 100 = 25\%$

This indicates a 25% increase in residential burglaries. Note a common mistake: candidates often incorrectly divide by the new value (50) instead of the original value (40), which would yield an incorrect result of 20%. Always divide by the base (original) value.

Calculating Percentage Decrease

Now, suppose a targeted policing initiative in Ballarat successfully reduces retail thefts from 120 incidents in month one to 102 incidents in month two. The percentage decrease is:

$\text{Percentage Decrease} = \frac{120 - 102}{120} \times 100 = \frac{18}{120} \times 100 = 0.15 \times 100 = 15\%$

The volume of retail thefts decreased by 15%. Understanding this formula is critical because exam questions will often ask you to calculate the success of a police program by measuring percentage drops in crime.

Percentage-Based Data Analysis in Policing

During the ACER exam, you will encounter data interpretation questions where crime statistics are presented in tables, graphs, or pie charts. Analyzing this data requires a careful approach.

Interpreting Tables and Charts

A common exam format presents a table of crime categories across different Local Government Areas (LGAs) in Victoria. You might be asked to find which LGA experienced the greatest percentage decrease in a particular crime type. When comparing percentages, always distinguish between a percentage point change and a percentage change. For example, if the proportion of positive drug tests among drivers in a region drops from 8% to 6%, this is a drop of 2 percentage points, but it represents a:

$\frac{8 - 6}{8} \times 100 = 25\% \text{ decrease in the rate of positive tests}$

Compounding Percentages

Another critical concept is that percentages cannot simply be added together if they apply to different bases. For example, if a station's staff increases by 10% in year one and then decreases by 10% in year two, the staff does not return to its original number.

Let the original staff be 100 officers:

• Year 1 (10% increase): $100 \times 1.10 = 110$ officers.
• Year 2 (10% decrease on the new base of 110): $110 \times 0.90 = 99$ officers.

The net result is a 1% decrease overall from the original 100. This compounding effect is a frequent trap on the numeracy exam.

Mental Math Strategies for Percentages

Under exam conditions, you can save time by breaking percentages down into manageable parts:

• To find 10% of a number, shift the decimal point one place to the left (e.g., 10% of 350 is 35).
• To find 5%, find 10% first and divide it by 2 (e.g., 10% of 350 is 35, so 5% is 17.5).
• To find 1%, shift the decimal point two places to the left (e.g., 1% of 350 is 3.5).
• To find 20%, find 10% and multiply by 2 (e.g., 10% of 350 is 35, so 20% is 70).

Using these benchmarks allows you to quickly solve multi-step percentage problems mentally or with minimal scratchpad work, leaving more time for complex questions.