6.4 Ratios, Rates & Unit Rates

6.4 Ratios, Rates & Unit Rates

Quick Answer: A ratio is a multiplicative comparison of two quantities, expressed as $a:b$, "$a$ to $b$", or $\frac{a}{b}$. A part-to-part ratio compares two individual subsets, whereas a part-to-whole ratio compares one subset to the entire collection. A rate compares two quantities with distinct units of measurement, and a unit rate simplifies that comparison to a denominator of $1$ unit (e.g., miles per hour, dollars per ounce). Percent represents a rate per hundred ($\frac{\text{Part}}{\text{Whole}} = \frac{\text{Percent}}{100}$). In percent of change calculations, the denominator is always the original starting value.

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Understanding Ratio Concepts & Representations

A ratio expresses the quantitative relationship between two numbers, indicating how many times one value contains or is contained within the other. Ratios can be recorded in three standard mathematical forms:

1. Colon Notation: $a:b$
2. Verbal Notation: "$a$ to $b$"
3. Fractional Notation: $\frac{a}{b}$

Part-to-Part vs. Part-to-Whole Comparisons

Under Florida B.E.S.T. benchmark MA.6.AR.3.1, students must clearly distinguish between part-to-part and part-to-whole relationships:

• Part-to-Part: Compares one component of a group to another component. For example, in an art studio with $12$ tubes of blue paint and $16$ tubes of yellow paint, the ratio of blue paint to yellow paint is $12:16 = 3:4$.
• Part-to-Whole: Compares one component to the entire combined total. In the same studio, the total number of paint tubes is $12 + 16 = 28$. The ratio of blue paint to total paint is $12:28 = 3:7$.

[!IMPORTANT] Fraction and Percent Conversion Rule: Only part-to-whole ratios can be directly interpreted as fractions of the entire set or converted into percentages. The part-to-part ratio $3:4$ does not mean that $\frac{3}{4}$ ($75%$) of the paint is blue; rather, blue paint represents $\frac{3}{7} \approx 42.9%$ of the total collection.

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Visual Models for Equivalent Ratios

Equivalent ratios represent identical multiplicative proportions across different scales. Three foundational visual models help students solve complex proportional problems:

1. Ratio Tables

A ratio table organizes scaled rows or columns by multiplying or dividing all terms by common positive scale factors:

Every pair of values shares the identical constant ratio: $\frac{3}{2} = 1.5$ cups of flour per cup of sugar.

2. Tape Diagrams (Bar Models)

Tape diagrams depict quantities as partitioned bars composed of identical, equal-sized units. They are especially powerful when the total sum or difference between quantities is provided.

• Worked Problem: The ratio of fiction books to non-fiction books checked out of a school library is $5:3$. If there are $96$ total books checked out, how many are fiction books?
  • Step 1: Model the ratio blocks:
    • Fiction: $[;u;][;u;][;u;][;u;][;u;]$ ($5$ units)
    • Non-Fiction: $[;u;][;u;][;u;]$ ($3$ units)
  • Step 2: Determine the total number of equal units: $5 + 3 = 8$ units.
  • Step 3: Calculate the value of one unit: $\text{Value of } 1 \text{ unit} = \frac{96}{8} = 12$ books.
  • Step 4: Multiply by the target quantity: $\text{Fiction} = 5 \times 12 = 60$ books.

3. Double Number Lines

A double number line plots two co-varying quantities along parallel axes pinned to an origin of $0$. They allow students to visualize rates that change continuously, such as distance over time.

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Rates, Unit Rates, and Constant Unit Pricing

While a ratio often compares quantities with the same units, a rate is a specialized ratio comparing two measurements that possess different units (e.g., $240\text{ miles}$ per $4\text{ hours}$, or $$18.00$ per $3\text{ pounds}$).

The Unit Rate Concept

A unit rate is a rate simplified so that the second quantity (the denominator) equals $1$ unit of measurement:

$\text{Unit Rate} = \frac{\text{Quantity } A}{\text{Quantity } B} = \frac{240\text{ miles}}{4\text{ hours}} = 60\text{ miles per hour (mph)}$

Constant Unit Pricing (The "Better Buy")

In consumer economics, the unit price identifies the cost per unit of product, allowing direct comparison between packages of different sizes:

$\text{Unit Price} = \frac{\text{Total Cost}}{\text{Total Units (ounces, pounds, grams)}}$

• Comparison Example: A shopper must choose between two containers of laundry detergent:
  • Container A: $64\text{ fl oz}$ for $$9.60 \implies \frac{$9.60}{64} = $0.15\text{ per fl oz}$.
  • Container B: $100\text{ fl oz}$ for $$14.00 \implies \frac{$14.00}{100} = $0.14\text{ per fl oz}$.
  • Container B is the more economical selection because its unit price is $$0.01$ lower per fluid ounce.

Unit Rates with Complex Fractions

Florida B.E.S.T. benchmark MA.7.AR.3.1 tests computing unit rates from fractions divided by fractions:

$\text{Rate} = \frac{\frac{3}{4}\text{ mile}}{\frac{2}{5}\text{ hour}} = \frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8} = 1\frac{7}{8}\text{ mph}$

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Percent Problems as Rates per 100

The word percent derives from the Latin per centum, meaning "per hundred." A percent is simply a ratio whose denominator is fixed at $100$:

$p\% = \frac{p}{100} = 0.01p$

The Proportional Percent Model

Every single-step percent problem can be solved using the universal proportional proportion:

$\frac{\text{Part}}{\text{Whole}} = \frac{\text{Percent}}{100} \iff \text{Part} = (\text{Decimal Percent}) \times \text{Whole}$

$\begin{array}{|l|l|l|l|} \hline \textbf{Problem Type} & \textbf{Unknown} & \textbf{Formula Setup} & \textbf{Concrete Example} \\ \hline \textbf{Find the Part} & \text{Part} & \text{Part} = \frac{\%}{100} \times \text{Whole} & \text{Find } 35\% \text{ of } 140: \; 0.35 \times 140 = 49 \\ \hline \textbf{Find the Percent} & \text{Percent} & \% = \frac{\text{Part}}{\text{Whole}} \times 100 & 27 \text{ is what } \% \text{ of } 90?: \; \frac{27}{90} \times 100 = 30\% \\ \hline \textbf{Find the Whole} & \text{Whole} & \text{Whole} = \frac{\text{Part}}{\text{Decimal } \%} & 42 \text{ is } 60\% \text{ of what}?: \; \frac{42}{0.60} = 70 \\ \hline \end{array}$

Percent of Change: Increase and Decrease

Percent of change measures the relative change between an initial state and a final state:

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

[!IMPORTANT] The Original Base Rule: The denominator of the percent change formula is always the original starting value, never the final or new value.

Multi-Step Real-World Applications: Markups, Discounts, and Taxes

1. Markups: Retailers buy goods at wholesale cost and add a markup percentage to determine retail price: $\text{Retail Price} = \text{Wholesale} \times (1 + \text{Markup Rate})$
2. Discounts (Sales): A discount reduces the retail price by a specified percentage: $\text{Sale Price} = \text{Retail Price} \times (1 - \text{Discount Rate})$
3. Sales Tax and Tip: Taxes and tips are calculated on the negotiated sale price and added to the subtotal: $\text{Total Cost} = \text{Subtotal} \times (1 + \text{Tax Rate})$

• Comprehensive Multi-Step Example: A video game console with a wholesale cost of $$300$ is marked up by $30%$. During a holiday sale, it is discounted by $20%$. What is the final checkout price with $7%$ sales tax?
  • Retail Price: $$300 \times 1.30 = $390.00$.
  • Sale Price: $$390.00 \times (1 - 0.20) = $390.00 \times 0.80 = $312.00$.
  • Final Total with Tax: $$312.00 \times 1.07 = $333.84$.

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Common Exam Traps & Misconceptions

[!WARNING]

Exam Trap 1: Inverting the Unit Price Calculation

A widespread student error when finding unit price is dividing the quantity by the dollar cost rather than dividing cost by quantity. For example, if $12$ sodas cost $$6$, computing $\frac{12}{6} = 2$ indicates $2$ sodas per dollar, not the cost per soda. Cost per unit requires $\frac{$6.00}{12} = $0.50$ per soda.

[!WARNING]

Exam Trap 2: Dividing by the New Value in Percent of Change

If an item priced at $$50$ increases to $$75$, the increase is $$25$. Students frequently calculate $\frac{25}{75} = 33.3%$. The formula mandates dividing by the original value: $\frac{25}{50} = 50%$ increase. Always identify the chronologically original value before computing.

[!WARNING]

Exam Trap 3: Additively Combining Successive Discounts

If a store offers a $20%$ off sale and a coupon grants an additional $10%$ off, students often add $20% + 10% = 30%$ total discount. This is mathematically incorrect because the second discount applies to the already reduced price, not the original price: $\text{Final Price} = P \times (1 - 0.20) \times (1 - 0.10) = P \times 0.80 \times 0.90 = P \times 0.72$ A multiplier of $0.72$ represents an effective net discount of $28%$, not $30%$.