3.1 Word Problems: Rates, Ratios, Proportions & Percentages

3.1 Word Problems: Rates, Ratios, Proportions & Percentages

The Arithmetic Reasoning (AR) subtest on the ASVAB-style classification practice evaluates your ability to analyze practical word problems, translate verbal statements into mathematical expressions, and solve them accurately. Rates, ratios, proportions, and percentages are useful arithmetic-reasoning practice topics.

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1. Ratios and Ratio Scaling

A ratio is a comparison of two quantities by division. Ratios can be written in three forms: as a fraction ($\frac{a}{b}$), with a colon ($a:b$), or using the word "to" ($a\text{ to }b$).

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

Understanding whether a ratio represents a part-to-part relationship or a part-to-whole relationship is essential:

• Part-to-Part: Compares one subgroup to another subgroup. For example, if a unit has 15 officers and 45 enlisted personnel, the ratio of officers to enlisted is $15:45$, which simplifies to $1:3$.
• Part-to-Whole: Compares a subgroup to the total. In the same unit, the total personnel is $15 + 45 = 60$. The ratio of officers to total personnel is $15:60 = 1:4$.

The Ratio Box / Total Parts Method

When a total quantity must be divided according to a given ratio $a : b : c$:

1. Add the ratio terms together to find the total number of equal parts: $\text{Total Parts} = a + b + c$.
2. Divide the total quantity by the total number of parts to determine the value of one part: $\text{Value per Part} = \frac{\text{Total Quantity}}{\text{Total Parts}}$.
3. Multiply each ratio term by the value of one part to find the specific quantity for each category.

Step-by-Step Worked Example 1: A tactical supply unit receives a shipment of 450 cases of ammunition distributed among caliber types Alpha, Bravo, and Charlie in the ratio $2 : 4 : 3$. How many cases of Bravo ammunition were received?

• Step 1: Calculate total ratio parts: $2 + 4 + 3 = 9\text{ parts}$.
• Step 2: Find value per part: $\frac{450\text{ cases}}{9\text{ parts}} = 50\text{ cases per part}$.
• Step 3: Calculate Bravo cases ($4\text{ parts}$): $4 \times 50 = 200\text{ cases}$.

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2. Rates, Unit Rates & Dimensional Analysis

A rate is a ratio that compares two quantities measured in different units (such as miles per hour, dollars per gallon, or fuel consumption per engine hour). A unit rate is a rate expressed with a denominator of 1.

Dimensional Analysis (Unit Conversion)

To convert rates across different measurement units, multiply by conversion factors structured as fractions equal to 1, ensuring unwanted units cancel out diagonally.

$$\text{Target Rate} = \text{Given Rate} \times \left( \frac{\text{Unit } B}{\text{Unit } A} \right) \times \left( \frac{\text{Unit } C}{\text{Unit } B} \right)$$

Step-by-Step Worked Example 2: A generator consumes diesel fuel at a rate of 12 quarts per hour. If fuel costs $4.00 per gallon, what is the operational fuel cost of running the generator for an 8-hour shift?

• Step 1: Find total quarts consumed in 8 hours: $12\text{ quarts/hr} \times 8\text{ hrs} = 96\text{ quarts}$.
• Step 2: Convert quarts to gallons ($4\text{ quarts} = 1\text{ gallon}$): $\frac{96\text{ quarts}}{4\text{ quarts/gal}} = 24\text{ gallons}$.
• Step 3: Multiply by cost per gallon: $24\text{ gallons} \times \text{\textdollar}4.00 = \text{\textdollar}96.00$.

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3. Direct vs. Inverse Proportions

A proportion is an equation stating that two ratios are equal: $\frac{a}{b} = \frac{c}{d}$. Proportions fall into two distinct categories: direct and inverse.

Direct Proportions ($y = kx$)

In a direct proportion, as one variable increases, the other variable increases at the same multiplier rate. The ratio $\frac{y}{x} = k$ remains constant.

$$\frac{a_1}{b_1} = \frac{a_2}{b_2} \implies a_1 \cdot b_2 = b_1 \cdot a_2$$

Inverse Proportions ($x \cdot y = k$)

In an inverse proportion, as one variable increases, the other variable decreases proportionally. The product of the two variables $x \cdot y = k$ remains constant. Common classification practice inverse proportion scenarios include workers vs. completion time or travel speed vs. travel time.

$$a_1 \cdot b_1 = a_2 \cdot b_2$$

Step-by-Step Worked Example 3 (Inverse Proportion): If 8 soldiers can dig a trench system in 15 hours, how many hours will it take 12 soldiers working at the exact same rate to dig the identical trench system?

• Step 1: Identify relationship: More soldiers means fewer hours (inverse proportion).
• Step 2: Set up product equality: $\text{Soldiers}_1 \times \text{Hours}_1 = \text{Soldiers}_2 \times \text{Hours}_2$.
• Step 3: Substitute known values: $8 \times 15 = 12 \times H_2$.
• Step 4: Solve for $H_2$: $120 = 12 \cdot H_2 \implies H_2 = 10\text{ hours}$.

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4. Percentages, Percent Change & Multi-Step Discounts

A percentage is a fraction expressed out of 100. The foundational percentage relationship is:

$$\text{Part} = \text{Percent (in decimal form)} \times \text{Whole}$$

Percent Increase and Decrease

When calculating percentage increase or decrease, the change is always evaluated relative to the original starting amount:

$$\text{Percent Change} = \frac{|\text{New Value} - \text{Original Value}|}{\text{Original Value}} \times 100\%$$

classification practice Common Pitfall: Never divide the change by the new value! Always divide by the starting (original) value.

Successive (Multi-Step) Percentages

When multiple percentage changes occur sequentially (e.g., a $20\%$ discount followed by an additional $10\%$ discount), do not add the percentages together.

Step-by-Step Worked Example 4: A piece of tactical equipment originally priced at $500 is placed on sale for $20\%$ off. A military personnel discount reduces the sale price by an additional $10\%$. What is the final price?

• Step 1: Calculate price after first discount ($20\%$ off means paying $80\%$): $\text{\textdollar}500 \times 0.80 = \text{\textdollar}400$.
• Step 2: Apply second discount to the new price ($10\%$ off of $400 means paying $90\%$): $\text{\textdollar}400 \times 0.90 = \text{\textdollar}360$.
• Step 3: Note total discount is $\text{\textdollar}500 - \text{\textdollar}360 = \text{\textdollar}140$, which is $\frac{140}{500} = 28\%$, not $30\%$.

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5. classification practice Word Problem Translation Strategy

To translate word problems quickly into solvable mathematical equations, use this translation matrix: