2.1 Ratios, Proportions & Scale Conversions

2.1 Ratios, Proportions & Scale Conversions

Arithmetic Reasoning (AR) on the Armed Services Vocational Aptitude Battery (ASVAB) / Armed Forces Qualification Test (AFQT) heavily emphasizes word problems involving ratios, proportions, and scale conversions. These concepts test your ability to convert written real-world relationships into mathematical equations and solve them under strict time constraints.

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Understanding Ratios: Part-to-Part vs. Part-to-Whole

A ratio is a mathematical comparison of two numbers or quantities showing how many times one value contains another. Ratios can be written in three distinct formats:

• Fraction form: $\frac{a}{b}$
• Colon notation: $a : b$
• Verbal statement: "$a$ to $b$"

Distinguishing Part-to-Part and Part-to-Whole Ratios

The most common error on the AFQT is confusing a part-to-part ratio with a part-to-whole ratio.

• Part-to-Part Ratio: Compares one subgroup to another subgroup within a set. Example: In a motor pool with $12$ Humvees and $8$ trucks, the ratio of Humvees to trucks is $12:8$, which simplifies to $3:2$.
• Part-to-Whole Ratio: Compares one subgroup to the total set. Example: In the same motor pool of $20$ total vehicles ($12 + 8 = 20$), the ratio of Humvees to total vehicles is $12:20$, which simplifies to $3:5$.

Key Exam Tip: Always check whether the question asks for a ratio comparing two components against each other or a component against the total quantity!

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The Total-Shares Method for Multi-Part Ratios

When a word problem distributes a total quantity among multiple entities based on a given ratio (such as $2:3:5$), use the Total-Shares Method.

4-Step Total-Shares Algorithm

1. Add the ratio numbers to find the total number of equal shares. $$\text{Total Shares} = a + b + c$$
2. Divide the total quantity by the total number of shares to calculate the value of one single share ($x$). $$\text{Value per Share } (x) = \frac{\text{Total Quantity}}{\text{Total Shares}}$$
3. Multiply the value of one share by each individual ratio part to find the specific amounts. $$\text{Part } A = a \cdot x, \quad \text{Part } B = b \cdot x, \quad \text{Part } C = c \cdot x$$
4. Verify your results by adding the calculated parts together to ensure they sum to the original total.

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Setting Up and Solving Proportions

A proportion is an equation stating that two ratios are equal:

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

The Cross-Multiplication Rule

For any valid proportion, the product of the means equals the product of the extremes:

$$a \cdot d = b \cdot c$$

To solve for an unknown quantity $x$ when three values are known:

$$\frac{a}{b} = \frac{x}{d} \implies x = \frac{a \cdot d}{b}$$

Direct Proportions in Word Problems

In direct proportions, as one quantity increases, the other increases at a constant rate. Example: If $4$ tactical radios require $14$ batteries, how many batteries are required for $10$ tactical radios?

$$\frac{4 \text{ radios}}{14 \text{ batteries}} = \frac{10 \text{ radios}}{x \text{ batteries}}$$ $$4x = 140 \implies x = 35 \text{ batteries}$$

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Scale Conversions for Maps and Blueprints

Scale problems present a ratio between map/blueprint measurements and real-world physical distances. These problems test your ability to convert units before setting up proportions.

Unit Conversion Reference Table

Solving Scale Problems Step-by-Step

When given a scale such as $\frac{1}{4} \text{ inch} = 5 \text{ miles}$:

1. Express the scale as a ratio: $\frac{\text{Scale Inches}}{\text{Actual Miles}} = \frac{0.25}{5}$.
2. Set up a proportion matching the scale ratio to the problem parameters.
3. Solve for the missing variable using cross-multiplication.

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Step-by-Step Worked AFQT Examples

Worked Example 1: Multi-Part Ratio Distribution

Problem: A supply sergeant needs to distribute $1,400$ rounds of ammunition among three squads in the ratio $2 : 3 : 5$. How many rounds does the squad with the largest share receive?

• Step 1: Find total shares. $$\text{Total Shares} = 2 + 3 + 5 = 10$$
• Step 2: Calculate value per share. $$\text{Value per Share} = \frac{1,400 \text{ rounds}}{10 \text{ shares}} = 140 \text{ rounds/share}$$
• Step 3: Identify largest share ratio part ($5$) and multiply. $$\text{Largest Share} = 5 \times 140 = 700 \text{ rounds}$$
• Step 4: Verify all shares sum to $1,400$. $$2(140) + 3(140) + 5(140) = 280 + 420 + 700 = 1,400 \text{ rounds. (Correct)}$$

Worked Example 2: Part-to-Part to Part-to-Whole Conversion

Problem: In an infantry platoon, the ratio of officers to enlisted personnel is $2 : 15$. If the total platoon strength is $68$ personnel, how many enlisted personnel are in the platoon?

• Step 1: Identify the ratio components. Ratio of officers to enlisted is $2:15$.
• Step 2: Determine the part-to-whole relationship for enlisted personnel. $$\text{Total Ratio Parts} = 2 + 15 = 17$$ $$\text{Enlisted Fraction} = \frac{15}{17}$$
• Step 3: Multiply total personnel by the enlisted fraction. $$\text{Enlisted Personnel} = \frac{15}{17} \times 68 = 15 \times 4 = 60$$

Worked Example 3: Map Scale Conversion

Problem: On a military tactical map, a scale of $\frac{1}{2} \text{ inch} = 15 \text{ miles}$ is used. If two outposts are $3.5 \text{ inches}$ apart on the map, what is the actual physical distance between them in miles?

• Step 1: Set up a proportion comparing map inches to actual miles. $$\frac{0.5 \text{ in.}}{15 \text{ miles}} = \frac{3.5 \text{ in.}}{x \text{ miles}}$$
• Step 2: Cross-multiply to solve for $x$. $$0.5 \cdot x = 15 \cdot 3.5$$ $$0.5x = 52.5$$
• Step 3: Divide by $0.5$ (or multiply by $2$). $$x = \frac{52.5}{0.5} = 105 \text{ miles}$$

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Common AFQT Traps & Shortcut Strategies

1. The Inverted Ratio Trap: Always double-check which order the ratio asks for! If a question asks for the ratio of rejections to acceptances, writing acceptances to rejections will lead to a distractor choice.
2. The Fractional Scale Shortcut: When multiplying by scale fractions like $\frac{1}{4} \text{ in} = 10 \text{ miles}$, remember that $1 \text{ in} = 40 \text{ miles}$. Multiplying the scale factor by $4$ immediately simplifies your arithmetic.