Use Ratio and Rate Vocabulary Study Notes

Key Terminology in Ratios and Rates

Understanding ratio and rate problems requires mastery of specific mathematical vocabulary used to describe comparisons between quantities. A ratio is a mathematical comparison of two quantities, often written in forms such as $a:b$, $\frac{a}{b}$, or "$a$ to $b$". When a ratio compares two quantities measured in different units—such as dollars and feet—it is called a rate.

A unit rate is a simplified rate in which the second quantity (the denominator) is equal to $1$ unit. Examples of unit rates include $3\text{ feet per yard}$ or $\$1.40\text{ per foot}$. The word "per" denotes division or the relationship between one unit of measurement and another. Equivalent ratios are two or more ratios that express the exact same relationship between numbers, obtained by multiplying or dividing both terms of a ratio by the same non-zero number.

To convert measurements from one unit to another (for instance, converting yards to feet), one must use a known unit rate as a conversion factor. Converting ensures that all quantities being compared share identical units of measurement, allowing for direct mathematical evaluation.

Unit Conversion and Rate Comparison Workflow

When evaluating real-world purchasing decisions to identify the better buy, prices must be compared on an equal basis by finding the unit cost in identical units. In the scenario where Harper buys fabric to make pillows, flannel fabric is sold at $3\text{ yards}$ for $\$12.60$ (or $\$13.60$), while fleece fabric is sold at $5\text{ feet}$ for $\$8.75$. Because the two fabrics are measured in different units of length (yards versus feet), the units must first be made equal before comparing prices.

To convert the length of flannel fabric from yards into feet, use the known conversion factor between yards and feet. The ratio of feet to yards is $3\text{ feet}$ to $1\text{ yard}$ ($3:1$). The unit rate is $3\text{ feet per yard}$, making the conversion rate equal to $3$. Multiplying the number of yards ($3\text{ yards}$) by the unit rate ($3\text{ feet/yard}$) gives the total length in feet:

$3\text{ yards} \times 3\text{ feet per yard} = 9\text{ feet}$

Thus, the flannel fabric has a total length of $9\text{ feet}$.

Calculating Unit Costs and Determining the Better Buy

With both fabrics measured in feet, the next step is finding the unit cost of each fabric in dollars per foot. Finding a unit cost involves dividing the total price by the total number of feet to establish an equivalent ratio where the denominator is $1\text{ foot}$.

For the flannel fabric, dividing the total cost of $\$12.60$ by the total length of $9\text{ feet}$ yields the unit price:

\frac{\12.60}{9\text{ feet}} = \$1.40\text{ per foot}$

For the fleece fabric, dividing the total cost of $\$8.75$ by the total length of $5\text{ feet}$ yields its unit price:

\frac{\8.75}{5\text{ feet}} = \$1.75\text{ per foot}$

Comparing the two unit costs shows that $\$1.40\text{ per foot}$ is less than $\$1.75\text{ per foot}$. Because the flannel fabric costs less per foot than the fleece fabric, the flannel fabric represents the better buy.

Reference Tables and Bank Values

The equivalent ratio tables for each fabric demonstrate the relationship between total cost and unit cost across equivalent rates.

For Flannel Fabric:

• Total Rate: $\$12.60$ for $9\text{ feet}$
• Unit Rate: $\$1.40$ for $1\text{ foot}$

For Fleece Fabric:

• Total Rate: $\$8.75$ for $5\text{ feet}$
• Unit Rate: $\$1.75$ for $1\text{ foot}$

The complete list of terms available in the Word Bank includes: convert, divide, dollars, equivalent ratio, feet, less, more, multiply, per, rate, ratio, and unit rate.

The complete list of values in the Number Bank includes: $1$, $1.40$, $1.75$, $3$, $5$, $8.75$, $9$, and $12.60$ (or $13.60$).