Ratios and proportions

1. What a Ratio Represents#

A ratio is a mathematical comparison of the relative magnitudes of two or more quantities of the same kind. Ratios express how many times one number contains another.

A ratio between quantity $a$ and quantity $b$ is formally written in three interchangeable notations:

1. Colon Notation: $a : b$ (read aloud as "a to b")
2. Fractional Notation: $\frac{a}{b}$
3. Verbal Form: "a to b"

In the ratio $a : b$, the first term $a$ is traditionally termed the antecedent, and the second term $b$ is the consequent.

Order and Meaning

Order is critically significant in ratio mathematics:

• The ratio of boys to girls in a classroom of 12 boys and 16 girls is $12 : 16 = 3 : 4$.
• The ratio of girls to boys is $16 : 12 = 4 : 3$.
• Inverting the terms fundamentally alters the comparison.

Part-to-Part versus Part-to-Whole Comparisons

A mixture containing 3 parts red paint and 5 parts yellow paint:
[ Red ] [ Red ] [ Red ] | [ Yel ] [ Yel ] [ Yel ] [ Yel ] [ Yel ]
<----- 3 parts ----->   | <------------- 5 parts ------------->
<----------------------- Total: 8 parts ----------------------->

• Part-to-Part Ratio: Compares one component directly to another: $$\text{Red} : \text{Yellow} = 3 : 5$$
• Part-to-Whole Ratio: Compares a single component to the aggregate sum: $$\text{Red} : \text{Total} = \frac{3}{3 + 5} = \frac{3}{8}$$ $$\text{Yellow} : \text{Total} = \frac{5}{3 + 5} = \frac{5}{8}$$

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2. Unit Homogeneity and Simplification#

A fundamental axiom of ratios is that quantities compared must be measured in identical, compatible units. Because a ratio represents a dimensionless scalar factor, units must cancel completely.

The Conversion Requirement

Problem: Find the simplified ratio of $75\text{ cm}$ to $1.5\text{ metres}$.

1. Notice that units differ (centimetres vs metres).
2. Convert both measurements into the smaller unit: $$1.5\text{ m} = 1.5 \times 100\text{ cm} = 150\text{ cm}$$
3. Express the ratio without units and reduce by dividing by $\gcd(75, 150) = 75$: $$\frac{75\text{ cm}}{150\text{ cm}} = \frac{75}{150} = \frac{1}{2} = 1 : 2$$

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Caution: Calculating $75 : 1.5 = 50 : 1$ without unit conversion produces an erroneous mathematical relationship by a factor of 100.

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3. Dividing a Total Quantity in a Given Ratio#

A frequent examination and practical task involves dividing a known total $T$ into parts defined by a ratio $a : b : c$.

General Formula:

1. Compute the sum of all ratio terms: $$\text{Sum of parts} = a + b + c$$
2. Determine the value of one single part: $$\text{Unit Value} = \frac{T}{a + b + c}$$
3. Multiply the unit value by each individual ratio share: $$\text{Share}_A = a \times \text{Unit Value}, \quad \text{Share}_B = b \times \text{Unit Value}, \quad \dots$$

Worked Example: Partnership Profit Allocation

Three entrepreneurs invest capital into a startup venture in the ratio $4 : 5 : 7$. At year's end, the business distributes a net profit pool of $$80,000$. Determine each partner's monetary share.

1. Calculate total parts: $$4 + 5 + 7 = 16\text{ parts}$$
2. Find the value of one part: $$\text{Value of 1 part} = \frac{$80,000}{16} = $5,000$$
3. Distribute shares:
   • Partner 1: $4 \times $5,000 = $20,000$
   • Partner 2: $5 \times $5,000 = $25,000$
   • Partner 3: $7 \times $5,000 = $35,000$
4. Verification: $$$20,000 + $25,000 + $35,000 = $80,000 \quad \text{(Confirmed)}$$

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4. Proportions: Direct versus Inverse Relationships#

A proportion is an equation affirming that two distinct ratios are equal:

$$\frac{a}{b} = \frac{c}{d} \quad \text{or} \quad a : b :: c : d$$

The outer terms $a$ and $d$ are called extremes; the inner terms $b$ and $c$ are called means.

The Cross-Product Property

For any true proportion, the product of the extremes is strictly equal to the product of the means:

$$a \times d = b \times c$$

| Characteristic | Direct Proportion | Inverse (Indirect) Proportion | | :--- | :--- | :--- | | Definition | As one variable multiplies by factor $k$, the other multiplies by the same factor. | As one variable multiplies by factor $k$, the other divides by factor $k$. | | Mathematical Equation | $\frac{y}{x} = k \implies y = kx$ | $x \times y = k \implies y = \frac{k}{x}$ | | Cross-Relationship | $\frac{x_1}{y_1} = \frac{x_2}{y_2} \implies x_1 y_2 = x_2 y_1$ | $x_1 y_1 = x_2 y_2$ | | Typical Scenarios | Distance travelled vs time (at constant speed); weight vs price. | Workers needed vs days to finish; speed vs travel time for fixed distance. |

Worked Example 1: Direct Proportion (Map Scale)

On a topographical map, a distance of $3\text{ cm}$ represents an actual ground distance of $45\text{ km}$. How many kilometres apart are two mountain peaks separated by $7.4\text{ cm}$ on the map?

1. Set up the direct proportion: $$\frac{3\text{ cm}}{45\text{ km}} = \frac{7.4\text{ cm}}{x\text{ km}}$$
2. Apply cross-multiplication: $$3 \times x = 45 \times 7.4$$ $$3x = 333 \implies x = \frac{333}{3} = 111\text{ km}$$

Worked Example 2: Inverse Proportion (Workforce Scheduling)

A construction project requires 12 workers working at identical pace to assemble a prefabricated footbridge in 15 days. If the contractor must complete the project in only 9 days, how many total workers are required?

1. Identify the relationship: Fewer days require more workers, so workforce size and duration are inversely proportional: $$\text{Workers}_1 \times \text{Days}_1 = \text{Workers}_2 \times \text{Days}_2 = \text{Constant Total Man-Days}$$
2. Substitute known quantities: $$12 \times 15 = W_2 \times 9$$ $$180 = 9 W_2 \implies W_2 = \frac{180}{9} = 20\text{ workers}$$
3. The contractor requires 20 total workers (an addition of 8 workers).

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5. Continued and Compounded Ratios#

Combining Ratios with a Common Term

Frequently, two separate pairwise ratios are provided, such as $A : B = 2 : 3$ and $B : C = 4 : 5$. To find the unified tripartite ratio $A : B : C$:

1. Identify the common linking term ($B$).
2. Find the $\text{LCM}$ of the two values of $B$ ($3$ and $4$, so $\text{LCM} = 12$).
3. Scale both ratios so that $B$ equals $12$: $$A : B = (2 \times 4) : (3 \times 4) = 8 : 12$$ $$B : C = (4 \times 3) : (5 \times 3) = 12 : 15$$
4. Combine into a continuous sequence: $$A : B : C = 8 : 12 : 15$$

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6. Common Pitfalls and Diagnostic Traps#

• Trap 1: Comparing Incompatible Units Without Conversion. Never write a ratio comparing litres to millilitres or hours to minutes directly without converting both terms to the shared base unit.
• Trap 2: Assuming All Proportion Problems Are Direct. Always ask: "If the first quantity increases, must the second quantity increase (direct) or decrease (inverse)?" Speed and travel time for a fixed destination are always inverse.
• Trap 3: Confusing Part-to-Part with Part-to-Whole. If the ratio of passing students to failing students is $7 : 2$, the fraction of students who passed is $\frac{7}{7+2} = \frac{7}{9}$, not $\frac{7}{2}$.
• Trap 4: Incorrect Cross-Multiplication in Inverse Scenarios. In inverse proportions, multiply across the rows ($x_1 \cdot y_1 = x_2 \cdot y_2$) rather than diagonally.