Section 3.1: Whole Numbers, Fractions, and Decimals

Section 3.1: Whole Numbers, Fractions, and Decimals

To succeed on the Victoria Police Entrance Examination, candidates must demonstrate high proficiency in numeracy. The exam, administered by the Australian Council for Educational Research (ACER), aligns with the Australian Core Skills Framework (ACSF) Level 3. A scale score of 111 or greater is the required standard to pass the Numeracy subtest. While an online calculator is provided for the Numeracy exam, having a strong grasp of mental arithmetic and manual calculations is a major advantage. In a fast-paced environment where you must answer 30 questions in 30 minutes, relying too heavily on a calculator can drain valuable time. Furthermore, as a Police Officer or Protective Services Officer (PSO), you will routinely perform basic calculations under pressure—such as counting seized property, estimating travel distances, or measuring evidence.

Arithmetic Operations with Whole Numbers

At the foundation of numeracy are the four basic arithmetic operations: addition, subtraction, multiplication, and division. In policing, these operations are applied to discrete objects and counts. For instance, officers may need to compile the total quantity of items recovered during a multi-agency search warrant.

When performing addition and subtraction, the concept of place value is essential. Aligning digits by their columns (units, tens, hundreds, thousands) prevents alignment errors, particularly when working with numbers of varying lengths. For multiplication, using the grid method or traditional long multiplication is crucial when calculators are unavailable. In division, particularly long division, you partition a larger number into equal groups. For example, if a division command has 2,416 patrol hours to allocate equally among 8 local police stations, the calculation is:

$2,416 \div 8 = 302 \text{ hours per station}$

In traffic management, officers frequently sum the total number of vehicles passing a checkpoint or add up demerit points on a driver's record. Understanding how to perform these calculations quickly and verify them using estimation (e.g., rounding numbers to the nearest ten or hundred to check if the sum is reasonable) is a core skill tested on the exam.

Working with Fractions in Policing

Fractions represent parts of a whole, defined by a numerator (the top number indicating how many parts we have) and a denominator (the bottom number indicating the total number of equal parts). Fractions are common in police rosters, crime statistics, and resource management.

Proper, Improper, and Mixed Fractions

• A proper fraction has a numerator smaller than the denominator (e.g., $3/4$).
• An improper fraction has a numerator greater than or equal to the denominator (e.g., $5/4$).
• A mixed number combines a whole number and a fraction (e.g., $1 \frac{1}{4}$).

Addition and Subtraction

To add or subtract fractions, they must have a common denominator. For example, if a patrol van uses $1/4$ of a tank of fuel on the morning shift and $2/5$ of a tank on the afternoon shift, the total fuel used is found by finding the lowest common multiple of 4 and 5, which is 20:

$\frac{1}{4} + \frac{2}{5} = \frac{5}{20} + \frac{8}{20} = \frac{13}{20} \text{ of a tank}$

To find the remaining fuel, subtract this total from a full tank (represented by 1, or $20/20$):

$\frac{20}{20} - \frac{13}{20} = \frac{7}{20} \text{ of a tank remaining}$

Multiplication and Division

When multiplying fractions, multiply the numerators together and the denominators together. For example, if $2/3$ of a station's active officers are assigned to general duties, and $1/4$ of those officers are on night shift, the fraction of the total station staff on night shift general duties is:

$\frac{2}{3} \times \frac{1}{4} = \frac{2}{12} = \frac{1}{6}$

Dividing fractions involves multiplying by the reciprocal of the divisor (the "keep, change, flip" method). If an investigator has $\frac{3}{4}$ of a shift remaining and needs to divide it equally among 3 cases, the calculation is:

$\frac{3}{4} \div 3 = \frac{3}{4} \times \frac{1}{3} = \frac{3}{12} = \frac{1}{4} \text{ of a shift per case}$

Decimals in Police Reporting and Evidence

Decimals are another way of representing parts of a whole and are crucial for recording precise measurements. Police officers use decimals when documenting evidence at a crime scene, writing incident reports, and reading measurement instruments.

Aligning the Decimal Point

For addition and subtraction of decimals, the decimal point must be vertically aligned. For example, if an officer measures a three-part skid mark at an accident scene on the Nepean Highway:

• Segment 1: $12.45 \text{ m}$
• Segment 2: $8.80 \text{ m}$
• Segment 3: $15.35 \text{ m}$
• Total Skid Mark: $12.45 + 8.80 + 15.35 = 36.60 \text{ m}$

Without aligning the decimal point, adding these numbers would result in a significant error.

Decimals in Specific Police Contexts

One of the most critical decimal applications in Victorian law enforcement is blood alcohol concentration (BAC). BAC is measured as the weight of alcohol in a volume of blood (grams per 100 millilitres). A BAC of 0.05 is the legal limit for full licence holders in Victoria. This decimal representation ($0.05$) represents a fraction of $5/100$ of a gram per 100 mL, or a decimal fraction of $0.0005$ of the total blood volume. Officers must understand decimal place values to interpret breathalyser readings accurately (e.g., distinguishing between 0.048 and 0.052).

Converting Between Fractions, Decimals, and Percentages

The ability to convert numbers between fractions, decimals, and percentages is a highly tested skill. The steps for these conversions are straightforward but require precision:

1. Fraction to Decimal: Divide the numerator by the denominator. For example, $3/8 = 3 \div 8 = 0.375$.
2. Decimal to Fraction: Write the decimal over its place value (e.g., $0.65 = 65/100$) and simplify by dividing the numerator and denominator by their greatest common divisor ($65/100 = 13/20$).
3. Decimal to Percentage: Multiply the decimal by 100 (move the decimal point two places to the right) and add the "%" symbol. For example, $0.05 \times 100 = 5%$.
4. Percentage to Decimal: Divide the percentage by 100 (move the decimal point two places to the left). For example, $12.5% = 0.125$.
5. Fraction to Percentage: Convert the fraction to a decimal first, then multiply by 100. For example, $1/5 = 0.2$, and $0.2 \times 100 = 20%$.

Reference Table

Below is a reference table of common conversions that candidates should memorise for speed:

Rounding Decimals in Police Reports

In many mathematical contexts, including the ACER numeracy exam, you will be required to round decimals to a specified number of decimal places or significant figures. In policing, rounding is frequently applied to measurements of distance, time, weight, or speed.

The general rule for rounding is to look at the digit to the right of the rounding digit:

• If that digit is 5 or greater, round up (increase the rounding digit by 1).
• If that digit is less than 5, round down (leave the rounding digit unchanged).

For example, if a speed detection radar records a vehicle traveling at $104.76 \text{ km/h}$ on the EastLink tollway, and the reporting policy requires speed to be logged to the nearest whole number, we look at the tenths place ($7$). Since 7 is greater than or equal to 5, we round up to $105 \text{ km/h}$.

Similarly, if forensic chemists measure the weight of an illicit substance as $45.342 \text{ grams}$ and are asked to report it to the nearest tenth of a gram, they examine the hundredths digit ($4$). Since 4 is less than 5, they round down, resulting in a reported weight of $45.3 \text{ grams}$. Mastering these rounding principles prevents calculation discrepancies and ensures reports remain consistent with legal guidelines.