9.1 Ratios, Rates & Proportional Reasoning

Ratios, Rates & Proportional Reasoning

Core Concept: Proportional reasoning connects the numbers and measurement strands Edutest lists for its Mathematics test. With 40 to 60 questions in 30 minutes and no calculator, success relies on replacing cumbersome algebraic manipulation with rapid mental unit scaling, the unitary method, and cross-multiplication shortcuts.

Whether an item describes dividing prize money, scaling chemical solutions, calculating machine productivity, or converting aerial map distances, the underlying mathematics reduces to either direct variation (constant quotient) or inverse variation (constant product). Mastering the mechanical distinctions between these systems allows candidates to solve multi-step problems in under 20 seconds.

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Ratio Simplification & Equivalent Ratios

A ratio is a mathematical comparison of two or more quantities of the same physical kind, expressed without units. Because a ratio represents relative magnitude rather than absolute amount, multiplying or dividing all terms in a ratio by the same non-zero constant produces an equivalent ratio.

1. The Uniform Dimension Rule

Before a ratio can be simplified, all quantities must be converted to the same unit of measurement. Failing to convert units is one of the highest-frequency traps planted by exam authors.

$\text{Simplify: } 450\text{ grams to } 1.2\text{ kilograms}$

• Step 1 (Convert to common units): $1.2\text{ kg} = 1.2 \times 1000\text{ g} = 1200\text{ g}$.
• Step 2 (Set up ratio): $450 : 1200$.
• Step 3 (Divide by greatest common divisor, 150): $\frac{450}{150} : \frac{1200}{150} = 3 : 8$

2. Clearing Fractions and Decimals in Ratios

Ratios must always be presented as integers in simplest form.

• Decimal Ratios: Multiply all terms by powers of $10$ until all decimals vanish (e.g., $1.75 : 2.5 \implies 175 : 250 \implies 7 : 10$).
• Fractional Ratios: Multiply every term by the Lowest Common Multiple (LCM) of all denominators: $1\frac{1}{2} : 2\frac{1}{4} = \frac{3}{2} : \frac{9}{4}$ $\text{Multiply by LCM (4): } \left(\frac{3}{2} \times 4\right) : \left(\frac{9}{4} \times 4\right) = 6 : 9 = 2 : 3$

3. Combining Linked Two-Part Ratios ($A:B$ and $B:C$)

Scholarship exams frequently provide two separate ratios linking three variables: $A : B$ and $B : C$. To find the compound ratio $A : B : C$, find the LCM of the shared variable ($B$) and scale both ratios accordingly.

Worked Example: Linking Ratios

$\text{If } A : B = 4 : 5 \quad \text{and} \quad B : C = 3 : 7, \quad \text{find } A : B : C.$

• Step 1 (Find LCM of the common term $B$): The values for $B$ are $5$ and $3$. $\text{LCM}(5, 3) = 15$.
• Step 2 (Scale both ratios to match $B = 15$): $A : B = (4 \times 3) : (5 \times 3) = 12 : 15$ $B : C = (3 \times 5) : (7 \times 5) = 15 : 35$
• Step 3 (Unify): $A : B : C = 12 : 15 : 35$

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The Total Parts Method ($A : B : C$)

The Total Parts Method is the standard algorithmic tool for dividing an aggregate quantity into a multi-part ratio. It converts abstract ratio parts into concrete unitary values.

[Total Quantity to be Divided]
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[Sum All Ratio Parts: Total Parts = A + B + C]
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[Compute Unitary Value: 1 Part = Total Quantity ÷ Total Parts]
               │
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[Multiply 1 Part Value by Respective Ratio Terms: A × (1 Part), B × (1 Part), etc.]

Worked Example: Dividing Profits with Discrepancy Analysis

Three business partners—Anya, Ben, and Chloe—share an annual profit of $140,000 in the ratio $2 : 3 : 5$. How much more money does Chloe receive than Anya?

• Step 1 (Sum total parts): $2 + 3 + 5 = 10\text{ total parts}$.
• Step 2 (Determine value of 1 part): $\text{Value of } 1\text{ part} = \frac{\$140,000}{10} = \$14,000$
• Step 3 (Calculate target difference directly): $\text{Difference in parts between Chloe and Anya} = 5 - 2 = 3\text{ parts}$ $\text{Monetary difference} = 3 \times \$14,000 = \$42,000$

Exam Speed Tip: Do not waste precious seconds calculating Anya's full share ($28,000) and Chloe's full share ($70,000) separately to subtract them. Subtract their ratio parts first ($5 - 2 = 3$) and multiply once by the unitary value. This saves 10 seconds of calculation.

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Solving Unknowns via Cross-Multiplication

A proportion is an equation stating that two ratios are equal: $\frac{a}{b} = \frac{c}{d}$. The fundamental property of proportions is that cross-products are equal:

$\frac{a}{b} = \frac{c}{d} \iff a \cdot d = b \cdot c$

To isolate an unknown variable in any position: $x = \frac{b \cdot c}{a}$

Mental Arithmetic Shortcut: Simplify Before Multiplying

Never multiply large numbers before dividing on non-calculator tests. Always express terms as a single rational expression and cancel common factors first:

$\frac{14}{35} = \frac{x}{85} \implies x = \frac{14 \times 85}{35}$ $\text{Cancel factor of 7 from 14 and 35: } x = \frac{2 \times 85}{5}$ $\text{Divide 85 by 5: } x = 2 \times 17 = 34$

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Direct vs. Inverse Variation (Proportion)

One of the most consequential conceptual hurdles in competitive mathematics is correctly identifying whether a relationship is directly or inversely proportional.

1. Direct Proportion ($y = kx$)

In direct proportion, increasing one variable causes a strictly proportional increase in the second variable. The constant $k = \frac{y}{x}$ is the constant of proportionality or the unitary rate.

2. Inverse Proportion ($xy = k$)

In inverse proportion, increasing one variable forces the other variable to decrease so that their product remains fixed. The classic manifestation is workforce productivity or speed across a fixed distance.

Worked Example: Inverse Workforce Calculation

A construction contractor estimates that 8 builders can construct a modular security wall in 15 days. How many additional builders must be hired to complete the identical wall in just 10 days?

• Step 1 (Identify invariant product): Total work is measured in "builder-days". $\text{Total Work} = 8\text{ builders} \times 15\text{ days} = 120\text{ builder-days}$
• Step 2 (Calculate required workforce for 10 days): $\text{Total Builders Needed} = \frac{120\text{ builder-days}}{10\text{ days}} = 12\text{ builders}$
• Step 3 (Solve for additional workers): $\text{Additional Builders} = 12 - 8 = 4\text{ additional builders}$

Exam Trap: Candidates who rush this question often select $12$, the total workforce, and lose the mark. Always re-read the stem to verify whether the examiner asked for the total workforce or the additional workers required.

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Compound Unitary Rates (Multi-Dimensional Work)

Complex selective entry questions feature compound rates where work depends on multiple independent inputs (e.g., workers, machines, hours per day, total output).

$\text{The Compound Invariance Principle: } \frac{W_1}{N_1 \times T_1} = \frac{W_2}{N_2 \times T_2}$

Where:

• $W$ is the total work accomplished (items produced, books printed, fields mowed)
• $N$ is the number of producing units (workers, printers, pumps)
• $T$ is the total elapsed time (hours, days)

Worked Example: Industrial Machine Production

If 5 identical industrial 3D printers manufacture 1,500 plastic components in 4 hours, how many components can 7 such printers manufacture in 6 hours?

• Step 1 (Find the unitary rate per printer-hour): $\text{Total Printer-Hours in Setup 1} = 5\text{ printers} \times 4\text{ hours} = 20\text{ printer-hours}$ $\text{Unitary Production Rate} = \frac{1500\text{ components}}{20\text{ printer-hours}} = 75\text{ components per printer-hour}$
• Step 2 (Calculate total printer-hours in Setup 2): $\text{Printer-Hours in Setup 2} = 7\text{ printers} \times 6\text{ hours} = 42\text{ printer-hours}$
• Step 3 (Compute total output): $\text{Total Output} = 42\text{ printer-hours} \times 75\text{ components/hr} = 42 \times \frac{300}{4} = \frac{12,600}{4} = 3,150\text{ components}$

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Map Scales & The Areal Dimensional Trap

A representative fraction (RF) expresses the ratio between a distance on a map and the corresponding distance on the Earth's surface in the form $1 : n$. Because it is a pure ratio, both numbers share the same units.

$\text{Scale } 1 : 50,000 \implies 1\text{ cm on map} = 50,000\text{ cm in reality}$ $50,000\text{ cm} = 500\text{ m} = 0.5\text{ km}$

The Linear vs. Areal Scaling Rule

While linear measurements (length, perimeter, distance) scale by the factor $n$, two-dimensional measurements (area, land surface) scale by $n^2$.

$\text{Linear Scale Ratio: } 1 : n$ $\text{Surface Area Ratio: } 1 : n^2$

Worked Example: Map Area Conversion

On a regional topographic map with a representative scale of $1 : 40,000$, a rectangular nature reserve measures $2.5\text{ cm}$ by $4.0\text{ cm}$. What is the actual land area of the reserve in square kilometres ($\text{km}^2$)?

• Method A: Convert Linear Dimensions First (Recommended):
  • Linear scale: $1\text{ cm} = 40,000\text{ cm} = 400\text{ m} = 0.4\text{ km}$.
  • Actual width: $2.5\text{ cm} \times 0.4\text{ km/cm} = 1.0\text{ km}$.
  • Actual length: $4.0\text{ cm} \times 0.4\text{ km/cm} = 1.6\text{ km}$.
  • Actual area: $1.0\text{ km} \times 1.6\text{ km} = 1.6\text{ km}^2$.
• Method B: Area Scale Factor:
  • Map area: $2.5\text{ cm} \times 4.0\text{ cm} = 10.0\text{ cm}^2$.
  • Area scale factor: $1\text{ cm}^2 = (0.4\text{ km})^2 = 0.16\text{ km}^2$.
  • Actual area: $10.0\text{ cm}^2 \times 0.16\text{ km}^2/\text{cm}^2 = 1.6\text{ km}^2$.

Exam Trap: Candidates who multiply map area ($10\text{ cm}^2$) directly by the linear distance factor ($0.4\text{ km}$) calculate $4.0\text{ km}^2$. Expect this incorrect answer to appear as a distractor.

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Proportional Reasoning Reference Table

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High-Yield Traps in Proportional Reasoning

1. The Inverse Proportion Default Error

When a word problem involves workers, pumps, or travel time, untrained students instinctively apply direct cross-multiplication. For instance, if 4 men build a wall in 6 days, an untrained student writes $\frac{4}{6} = \frac{8}{x} \implies x = 12\text{ days}$. More men must take less time. Always conduct a common-sense sanity check: when the workforce doubles, completion time must halve ($4 \times 6 = 8 \times x \implies x = 3\text{ days}$).

2. Centimetre-to-Kilometre Zero-Count Confusion

Converting centimetres to kilometres involves two distinct divisions: divide by $100$ (centimetres to metres), then divide by $1,000$ (metres to kilometres). That is a total division by $100,000$ ($5$ decimal places). For example, $40,000\text{ cm} = 400\text{ m} = 0.4\text{ km}$, not $4\text{ km}$.

3. The Unadjusted Base Quantity in Recipe Scaling

When scaling recipes, exam stems frequently state: "A recipe for 6 people requires 250 g of flour. How much flour is needed for 15 people?" Students often attempt to double and add fractions messily. Use the unitary method directly: $\text{Per person} = \frac{250}{6}\text{ g}$, so for $15$ people: $\frac{250}{6} \times 15 = 250 \times \frac{5}{2} = 125 \times 5 = 625\text{ g}$.