5.3 Ratios, Unit Rates & Proportional Reasoning

5.3 Ratios, Unit Rates & Proportional Reasoning

Proportional reasoning is a cornerstone of quantitative literacy and a heavily tested topic on the ACCUPLACER Arithmetic test. Ratios allow us to compare quantities, unit rates simplify comparisons to a single baseline unit, and proportions provide powerful equations for solving real-world problems involving scaling, pricing, distance, and mixtures.

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

A ratio is a mathematical comparison of two numerical quantities by division. Ratios can be expressed in three standard formats:

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

Distinguishing Ratio Types

It is vital to distinguish between part-to-part ratios and part-to-whole ratios when setting up mathematical models.

• Part-to-Part Ratio: Compares one subgroup to another subgroup within a total collection.
  • Example: In a computer lab with $12$ desktop computers and $18$ laptops, the ratio of desktops to laptops is $12:18$, which simplifies to $2:3$.
• Part-to-Whole Ratio: Compares a single subgroup to the total combined quantity of all groups.
  • Example: The total number of computers is $12 + 18 = 30$. The ratio of desktops to total computers is $12:30$, which simplifies to $2:5$.

Key Difference: Only part-to-whole ratios can be directly converted into standard fractions, percentages, or probabilities of the total group. If given a part-to-part ratio $a:b$, the part-to-whole ratio for quantity $a$ is $\frac{a}{a+b}$.

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Unit Rates & Comparison Pricing

A rate is a special type of ratio that compares two quantities measured in different units (such as miles to hours, or dollars to ounces).

A unit rate is a rate in which the denominator quantity is simplified to exactly $1$ unit.

$\text{Unit Rate} = \frac{\text{Total Quantity}}{\text{Total Number of Units}}$

Common Unit Rates

• Speed: $\frac{\text{Miles}}{\text{Hours}} = \text{Miles per Hour (mph)}$
• Unit Price: $\frac{\text{Total Cost}}{\text{Number of Units}} = \text{Price per Unit}$
• Pay Rate: $\frac{\text{Total Earnings}}{\text{Hours Worked}} = \text{Dollars per Hour}$
• Fuel Efficiency: $\frac{\text{Total Miles}}{\text{Gallons of Gas}} = \text{Miles per Gallon (mpg)}$

Calculating "Best Buy" Unit Prices

To determine which product package offers the lower unit cost, calculate the price per single unit for each package and compare the decimal values.

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Proportions and the Cross-Products Property

A proportion is a mathematical statement asserting that two ratios or rates are equal:

$\frac{a}{b} = \frac{c}{d} \quad (b \neq 0, d \neq 0)$

The Fundamental Cross-Products Property

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

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

This cross-multiplication equation allows you to solve for any single unknown variable ($x$) when three terms of the proportion are known.

Mandatory Unit Alignment Rule

When setting up a proportion equation to solve word problems, corresponding units must occupy matching positions in both ratios:

$\frac{\text{Unit A (Numerator 1)}}{\text{Unit B (Denominator 1)}} = \frac{\text{Unit A (Numerator 2)}}{\text{Unit B (Denominator 2)}}$

Alternatively, you may align by scenario, provided both numerators share scenario 1 and both denominators share scenario 2. Mixing unit positions (such as placing miles in the numerator on the left but in the denominator on the right) produces incorrect equations.

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Proportional Problem Types Reference Table

The table below summarizes common proportional reasoning applications tested on the ACCUPLACER:

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Step-by-Step Worked Math Examples

Example 1: Unit Price Comparison ("Best Buy")

Problem: A grocery store sells a 16-ounce container of yogurt for $3.20 and a 24-ounce container of the same yogurt for $4.56. Which container is the better buy, and what is the savings per ounce?

Solution:

• Step 1: Calculate the unit price for the 16-ounce container: $\text{Unit Price}_{16} = \frac{\$3.20}{16\text{ oz}} = \$0.20\text{ per ounce}$

• Step 2: Calculate the unit price for the 24-ounce container: $\text{Unit Price}_{24} = \frac{\$4.56}{24\text{ oz}} = \$0.19\text{ per ounce}$

• Step 3: Compare unit prices and calculate savings:

  • Since $\text{\textdollar}0.19 < \text{\textdollar}0.20$, the 24-ounce container has a lower cost per unit volume.
  • Savings $= \text{\textdollar}0.20 - \text{\textdollar}0.19 = \text{\textdollar}0.01\text{ per ounce}$.

• Final Answer: The 24-ounce container is the better buy, saving $\text{\textdollar}0.01$ per ounce.

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Example 2: Solving Proportions with Map Scales

Problem: On a map, a scale of $\frac{1}{2}\text{ inch}$ represents $15\text{ miles}$. If the distance between two towns on the map is $3.5\text{ inches}$, what is the actual distance between the towns in miles?

Solution:

• Step 1: Set up a proportion with aligned units ($\frac{\text{inches}}{\text{miles}}$): $\frac{0.5\text{ inches}}{15\text{ miles}} = \frac{3.5\text{ inches}}{x\text{ miles}}$

• Step 2: Apply the cross-products property: $0.5 \cdot x = 15 \cdot 3.5$

• Step 3: Perform multiplication on the right side: $15 \cdot 3.5 = 52.5$ $0.5 x = 52.5$

• Step 4: Solve for $x$ by dividing both sides by $0.5$: $x = \frac{52.5}{0.5} = 105$

• Final Answer: The actual distance between the two towns is $105\text{ miles}$.

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Example 3: Part-to-Part to Total Group Reasoning

Problem: The ratio of red marbles to blue marbles in a bag is $4:7$. If there are $28$ blue marbles in the bag, what is the total number of marbles (red and blue combined) in the bag?

Solution:

• Step 1: Set up a part-to-part proportion to find the number of red marbles: $\frac{\text{Red Marbles}}{\text{Blue Marbles}} = \frac{4}{7}$ $\frac{R}{28} = \frac{4}{7}$

• Step 2: Cross-multiply and solve for $R$: $7 \cdot R = 4 \cdot 28$ $7 R = 112 \implies R = \frac{112}{7} = 16\text{ red marbles}$

• Step 3: Calculate total marbles by adding red and blue marbles: $\text{Total Marbles} = \text{Red} + \text{Blue} = 16 + 28 = 44\text{ marbles}$

• Alternative Part-to-Whole Method: Total ratio parts $= 4 + 7 = 11\text{ parts}$. Since $7\text{ parts} = 28\text{ blue marbles}$, $1\text{ part} = 28 \div 7 = 4\text{ marbles}$. Total marbles $= 11\text{ parts} \times 4 = 44\text{ marbles}$.

• Final Answer: The total number of marbles in the bag is $44$.

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ACCUPLACER Exam Traps & Common Errors