4.1 Ratios, Simplification, Part-to-Part vs. Part-to-Whole, and Proportional Sharing

Understanding and Simplifying Ratios

A ratio compares the relative sizes of two or more quantities measured in the same unit. We write a two-part ratio as $a : b$ (read "a to b") and a three-part ratio as $a : b : c$. Because the quantities share a common unit, the unit cancels out and a simplified ratio is dimensionless—it never carries a unit symbol.

Order is critical in every ratio question. If a primary school library stocks fiction to non-fiction books in the ratio $5 : 3$, the first number ($5$) belongs to fiction and the second number ($3$) belongs to non-fiction. Reversing the ratio to $3 : 5$ changes the meaning completely.

Simplifying Integer, Decimal, Fraction, and Mixed-Unit Ratios

Like fractions, ratios are simplest when written using the smallest possible whole numbers whose highest common factor ($\text{HCF}$) is $1$. Four common simplification cases appear on the numeracy test:

1. Whole-number ratios: Divide every term by their $\text{HCF}$. For a three-part staff allocation of $24 : 36 : 60$, the $\text{HCF}$ is $12$, giving:

$$24 : 36 : 60 \xrightarrow{\div 12} 2 : 3 : 5$$

1. Mixed metric units: Convert both quantities to the smaller unit first so that decimals disappear, remove the unit labels, and divide by the $\text{HCF}$. For example, to simplify $400\text{ mL} : 1.2\text{ L}$, convert $1.2\text{ L}$ into $1200\text{ mL}$:

$$400\text{ mL} : 1.2\text{ L} = 400\text{ mL} : 1200\text{ mL} = 400 : 1200 \xrightarrow{\div 400} 1 : 3$$

Similarly, to simplify $45\text{ minutes} : 2\text{ hours}$, convert $2\text{ hours}$ into $120\text{ minutes}$ to get $45 : 120$, and divide by $15$ to obtain $3 : 8$. Never attempt to simplify $400 : 1.2$ or $45 : 2$ before matching the units. 3. Decimal ratios: Multiply every term by $10$, $100$, or $1000$ to clear all decimal places, then simplify. For $0.45 : 1.8$, multiply both terms by $100$ to obtain $45 : 180$, then divide by $45$ to get $1 : 4$. 4. Fractional ratios: Multiply every term by the lowest common denominator ($\text{LCD}$) of the fractions. To simplify $\frac{2}{3} : \frac{3}{4}$, multiply both terms by the $\text{LCD}$ of $12$:

$$\left(\frac{2}{3} \times 12\right) : \left(\frac{3}{4} \times 12\right) = 8 : 9$$

Combining Two Linked Ratios into a Three-Part Ratio

Occasionally a question gives two separate two-part ratios that share a common middle category—for example, in a literacy intervention program, the ratio of specialist teachers to learning support aides is $2 : 3$, and the ratio of learning support aides to students is $4 : 15$. To combine them into a single three-part ratio (teachers : aides : students), match the middle term (aides) using the lowest common multiple of $3$ and $4$, which is $12$:

• Scale $2 : 3$ by multiplying by $4 \implies 8 : 12$.
• Scale $4 : 15$ by multiplying by $3 \implies 12 : 45$.
• Combine to obtain $8 : 12 : 45$.

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Part-to-Part Ratios vs. Part-to-Whole Fractions

One of the most frequent errors in test questions is confusing a part-to-part ratio with a part-to-whole fraction. When a ratio is written as $a : b$, the total number of equal parts in the whole is $a + b$, not $b$.

$$\text{First Share Fraction} = \frac{a}{a + b}, \qquad \text{Second Share Fraction} = \frac{b}{a + b}$$

For a three-part ratio $a : b : c$, the total number of parts is $a + b + c$, so the first share represents $\frac{a}{a + b + c}$ of the total.

Worked Example: Science Solution Mixing

A school laboratory technician prepares a plant nutrient solution by mixing concentrate with water in the ratio $2 : 7$.

• Common Trap: Reading $2 : 7$ and assuming concentrate makes up $\frac{2}{7}$ of the mixture.
• Correct Method: Add the parts to find the whole: $2 + 7 = 9\text{ total parts}$. Concentrate makes up $\frac{2}{9}$ of the total solution, and water makes up $\frac{7}{9}$ of the total solution. If the technician needs $3.6\text{ L}$ ($3600\text{ mL}$) of diluted solution, the volume of concentrate required is $\frac{2}{9} \times 3600\text{ mL} = 800\text{ mL}$, and the volume of water is $\frac{7}{9} \times 3600\text{ mL} = 2800\text{ mL}$.

Conversely, if you are told that $\frac{3}{8}$ of a Year 6 cohort travel to school by bus and the remainder walk or are driven, the whole is $8\text{ parts}$. Since $3\text{ parts}$ catch the bus, the remaining $8 - 3 = 5\text{ parts}$ do not. Thus, the part-to-part ratio of bus travellers to other students is $3 : 5$, not $3 : 8$.

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Proportional Sharing: The Unitary (1-Part) Method

Proportional sharing questions always give you a ratio (such as $a : b : c$) alongside one known dollar or physical quantity. Before calculating, identify which of the three problem structures applies, then find the value of 1 part.

Worked Example: Mixing Acrylic Paint by One Known Share

An art teacher mixes blue, yellow, and white paint in the ratio $3 : 5 : 2$ to create a custom green tint. She uses $450\text{ mL}$ of yellow paint. How much total paint does she make?

1. Identify the given share: Yellow corresponds to the middle term, which is $5\text{ parts}$.
2. Find the value of $1\text{ part}$: $450\text{ mL} \div 5 = 90\text{ mL}$.
3. Find the total parts: $3 + 5 + 2 = 10\text{ parts}$.
4. Multiply to find the total volume: $10 \times 90\text{ mL} = 900\text{ mL}$ (consisting of $270\text{ mL}$ blue, $450\text{ mL}$ yellow, and $180\text{ mL}$ white).

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Australian School Excursion Supervision Ratios

Supervision ratios are set by each state or territory education department, school sector, or school policy, and they vary with students' ages and the risk of the activity—water-based activities usually require more adults per student than a museum visit. A LANTITE question always states the ratio to use, so treat the ratios in this section as illustrative.

When calculating the minimum number of supervising teachers for $N$ students at a ratio of $1 : k$, divide $N$ by $k$ and always round UP to the next whole integer (ceiling rounding), even when the decimal part is less than $0.5$.

Worked Example: Comparing Water vs. Museum Supervision Ratios

A secondary school plans two excursions for $148$ Year 9 students: a coastal marine water-safety day (mandatory ratio $1 : 12$) and a history museum visit (mandatory ratio $1 : 20$).

• Marine excursion ($1 : 12$):

$$\text{Teachers} = \frac{148}{12} = 12.333... \xrightarrow{\text{round UP}} 13\text{ teachers}$$

If you rounded down to $12\text{ teachers}$, those $12$ adults could legally supervise at most $12 \times 12 = 144\text{ students}$, leaving $4\text{ students}$ without compliant supervision.

• Museum visit ($1 : 20$):

$$\text{Teachers} = \frac{148}{20} = 7.4 \xrightarrow{\text{round UP}} 8\text{ teachers}$$

Thus, the water activity requires $13 - 8 = 5\text{ more supervising teachers}$ than the museum visit for the exact same cohort.