5.6 Ratios and Rate

Learning objectives

Learning objectives

• Write a ratio as a fraction

• Write a rate as a fraction

• Find unit rates

• Find unit price

• Translate phrases to expressions with fractions

Write a Ratio as a Fraction

When you apply for a mortgage, the loan officer will compare your total debt to your total income to decide if you qualify for the loan. This comparison is called the debt-to-income ratio. A ratio compares two quantities that are measured with the same unit. If we compare $a$ and $b$, the ratio is written as $a\,\mathrm{to}\,b,\,\frac{a}{b},\,\mathrm{or}\,a:b.$

In this section, we will use the fraction notation. When a ratio is written in fraction form, the fraction should be simplified. If it is an improper fraction, we do not change it to a mixed number. Because a ratio compares two quantities, we would leave a ratio as $\frac{4}{1}$ instead of simplifying it to $4$ so that we can see the two parts of the ratio.

Example 5.58

Solution

1. Form each ratio. Place the first stated quantity in the numerator and the second stated quantity in the denominator, preserving the order in the problem.
2. Remove common factors. Divide $15$ and $27$ by $3$ to obtain $\frac{5}{9}$, and divide $45$ and $18$ by $9$ to obtain $\frac{5}{2}$.
3. Check the reduced fractions. Confirm that $5$ and $9$ have no common factor greater than $1$, and that $5$ and $2$ also have no common factor greater than $1$.
4. Report both results. The ratio $15$ to $27$ is $\frac{5}{9}$, while $45$ to $18$ is the improper fraction $\frac{5}{2}$.

Takeaway: A ratio keeps the stated order when it becomes a fraction, so simplifying $15$ to $27$ gives $\frac{5}{9}$ and $45$ to $18$ gives $\frac{5}{2}$.

Ratios Involving Decimals

We will often work with ratios of decimals, especially when we have ratios involving money. In these cases, we can eliminate the decimals by using the Equivalent Fractions Property to convert the ratio to a fraction with whole numbers in the numerator and denominator.

For example, consider the ratio $0.8\,\mathrm{to}\,0.05.$ We can write it as a fraction with decimals and then multiply the numerator and denominator by $100$ to eliminate the decimals.

Do you see a shortcut to find the equivalent fraction? Notice that $0.8=\frac{8}{10}$ and $0.05=\frac{5}{100}.$ The least common denominator of $\frac{8}{10}$ and $\frac{5}{100}$ is $100.$ By multiplying the numerator and denominator of $\frac{0.8}{0.05}$ by $100,$ we ‘moved’ the decimal two places to the right to get the equivalent fraction with no decimals. Now that we understand the math behind the process, we can find the fraction with no decimals like this:

You do not have to write out every step when you multiply the numerator and denominator by powers of ten. As long as you move both decimal places the same number of places, the ratio will remain the same.

Example 5.59

Solution

1. Write the ratios as fractions. The ratio $4.8$ to $11.2$ becomes $\frac{4.8}{11.2}$, and the ratio $2.7$ to $0.54$ becomes $\frac{2.7}{0.54}$.
2. Clear the first decimal pair. Multiply the numerator and denominator of $\frac{4.8}{11.2}$ by $10$ to obtain the equivalent fraction $\frac{48}{112}$.
3. Clear the second decimal pair. Multiply the numerator and denominator of $\frac{2.7}{0.54}$ by $100$ to obtain the equivalent fraction $\frac{270}{54}$.
4. Simplify and compare. Reducing by greatest common factors gives $\frac{3}{7}$ and $\frac{5}{1}$, respectively, so these are the fractions of whole numbers.

Takeaway: Moving both decimal points the same number of places preserves a ratio and produces whole-number terms, giving $\frac{3}{7}$ and $\frac{5}{1}$ here.

Some ratios compare two mixed numbers. Remember that to divide mixed numbers, you first rewrite them as improper fractions.

Example 5.60

Solution

1. Write the ratio as a fraction. Place the mixed number $1\frac{1}{4}$ above the mixed number $2\frac{3}{8}$ to represent the ratio in its stated order.
2. Convert to improper fractions. Rewrite the numerator as $\frac{5}{4}$ and the denominator as $\frac{19}{8}$, giving $\frac{\frac{5}{4}}{\frac{19}{8}}$.
3. Divide by the reciprocal. Change the expression to $\frac{5}{4}\div\frac{19}{8}$, then multiply by the reciprocal to get $\frac{5}{4}\cdot\frac{8}{19}$.
4. Reduce the result. Cancel the common factor $4$ between the numerators and denominators, leaving the final ratio $\frac{10}{19}$.

Takeaway: A ratio of mixed numbers is simplified by converting both to improper fractions and dividing with a reciprocal, which produces $\frac{10}{19}$.

Applications of Ratios

One real-world application of ratios that affects many people involves measuring cholesterol in blood. The ratio of total cholesterol to HDL cholesterol is one way doctors assess a person's overall health. A ratio of less than $5$ to $1$ is considered good.

Example 5.61

Solution

1. Form the cholesterol ratio. Put total cholesterol in the numerator and HDL cholesterol in the denominator, giving $\frac{249}{39}$.
2. Simplify the fraction. Divide both terms by their greatest common factor $3$ to obtain $\frac{83}{13}$ without changing the ratio's value.
3. Estimate the ratio. Divide $83$ by $13$ to get approximately $6.4$, so the ratio is approximately $6.4$ to $1$.
4. Compare with the guideline. Because $6.4$ is greater than $5$, Hector should lower his total cholesterol or raise his HDL cholesterol.

Takeaway: Hector's ratio of total cholesterol to HDL cholesterol is $\frac{83}{13}$, approximately $6.4$ to $1$, which exceeds the guideline of $5$ to $1$.

Ratios of Two Measurements in Different Units

To find the ratio of two measurements, we must make sure the quantities have been measured with the same unit. If the measurements are not in the same units, we must first convert them to the same units.

We know that to simplify a fraction, we divide out common factors. Similarly in a ratio of measurements, we divide out the common unit.

Example 5.62

Solution

1. Choose matching units. The rise is measured in inches and the run is measured in feet, so convert the run from $1$ foot to $12$ inches.
2. Write the ratio. Place the vertical rise in the numerator and the horizontal run in the denominator to obtain $\frac{1\text{ inch}}{12\text{ inches}}$.
3. Cancel matching units. The unit inches appears once in the numerator and once in the denominator, so canceling them leaves the numerical fraction $\frac{1}{12}$.
4. Interpret the ratio. A rise-to-run ratio of $\frac{1}{12}$ means the ramp rises $1$ inch for every $12$ inches of horizontal run.

Takeaway: Converting both ramp measurements to inches makes their units cancel, producing the required rise-to-run ratio of $1$ to $12$.

Write a Rate as a Fraction

Frequently we want to compare two different types of measurements, such as miles to gallons. To make this comparison, we use a rate. Examples of rates are $120$ miles in $2$ hours, $160$ words in $4$ minutes, and $\text{\textdollar 5}$ dollars per $64$ ounces.

When writing a fraction as a rate, we put the first given amount with its units in the numerator and the second amount with its units in the denominator. When rates are simplified, the units remain in the numerator and denominator.

Example 5.63

Solution

1. Identify the rate quantities. The distance traveled is $525$ miles, and the time traveled is $9$ hours, so miles belong in the numerator.
2. Write the rate as a fraction. Place miles over hours to form $\frac{525\text{ miles}}{9\text{ hours}}$ with the quantities in their stated order.
3. Reduce the numerical terms. Divide both $525$ and $9$ by their greatest common factor $3$ while retaining the original measurement units.
4. State the equivalent rate. The simplified rate is $\frac{175\text{ miles}}{3\text{ hours}}$, which represents the same distance-to-time relationship.

Takeaway: Reducing a distance-to-time rate preserves its value, so $525$ miles in $9$ hours equals $\frac{175\text{ miles}}{3\text{ hours}}$.

Find Unit Rates

In the last example, we calculated that Bob was driving at a rate of $\frac{\mathrm{175 miles}}{\mathrm{3 hours}}.$ This tells us that every three hours, Bob will travel $175$ miles. This is correct, but not very useful. We usually want the rate to reflect the number of miles in one hour. A rate that has a denominator of $1$ unit is referred to as a unit rate.

Unit rates are very common in our lives. For example, when we say that we are driving at a speed of $68$ miles per hour we mean that we travel $68$ miles in $1$ hour. We would write this rate as $68$ miles/hour (read $68$ miles per hour). The common abbreviation for this is $68$ mph. Note that when no number is written before a unit, it is assumed to be $1.$

So $68$ miles/hour really means $\mathrm{68 miles/1 hour.}$

Two rates we often use when driving can be written in different forms, as shown:

Another example of unit rate that you may already know about is hourly pay rate. It is usually expressed as the amount of money earned for one hour of work. For example, if you are paid $\text{\textdollar 12.50}$ for each hour you work, you could write that your hourly (unit) pay rate is $\text{\textdollar 12.50/hour}$ (read $\text{\textdollar 12.50}$ per hour.)

To convert a rate to a unit rate, we divide the numerator by the denominator. This gives us a denominator of $1.$

Example 5.64

Solution

1. Choose the rate quantities. Anita earned $384$ dollars during $32$ hours, so dollars belong in the numerator and hours belong in the denominator.
2. Write the pay rate. Form the fraction $\frac{384\text{ dollars}}{32\text{ hours}}$ to represent total earnings for total working time.
3. Divide by the number of hours. Compute $384\div32=12$, which changes the denominator into $1$ hour and gives $\frac{12\text{ dollars}}{1\text{ hour}}$.
4. Label the unit rate. Anita's hourly pay rate is $12$ dollars per hour because the denominator now represents exactly one hour.

Takeaway: Total pay divided by total hours gives the hourly rate, so Anita earns $12$ dollars for each of the $32$ hours she worked.

Example 5.65

Solution

1. Write the distance-to-fuel rate. Place the $455$ miles traveled in the numerator and the $14$ gallons used in the denominator.
2. Divide for one gallon. Compute $455\div14=32.5$, so the rate becomes $\frac{32.5\text{ miles}}{1\text{ gallon}}$.
3. Check the division. Multiplying $14$ gallons by $32.5$ miles per gallon gives $455$ miles, confirming that the unit rate matches the distance traveled.
4. State the fuel efficiency. Sven's car travels $32.5$ miles for each gallon of gasoline used, so it gets $32.5$ miles per gallon.

Takeaway: Miles per gallon compares distance traveled with fuel consumed, so dividing $455$ miles by $14$ gallons gives $32.5$ miles per gallon.

Find Unit Price

Sometimes we buy common household items ‘in bulk’, where several items are packaged together and sold for one price. To compare the prices of different sized packages, we need to find the unit price. To find the unit price, divide the total price by the number of items. A unit price is a unit rate for one item.

Example 5.66

Solution

1. Identify the unit needed. The question asks for the price of one bottle, so the denominator must represent $1$ bottle.
2. Write the cost rate. Place the total cost $3.99$ dollars in the numerator and the $24$ bottles in the denominator to form $\frac{3.99\text{ dollars}}{24\text{ bottles}}$.
3. Calculate the exact division. Dividing $3.99$ by $24$ gives $0.16625$ dollars for each bottle before rounding.
4. Round to a penny. Round $0.16625$ dollars to the nearest hundredth, obtaining a unit price of approximately $0.17$ dollars per bottle.

Takeaway: The unit price of the water is about $0.17$ dollars per bottle because the total cost of $3.99$ dollars is divided among $24$ bottles.

Unit prices are very useful if you comparison shop. The better buy is the item with the lower unit price. Most grocery stores list the unit price of each item on the shelves.

Example 5.67

Solution

1. Form both unit-price rates. Use $\frac{14.99\text{ dollars}}{64\text{ loads}}$ for liquid detergent and $\frac{15.99\text{ dollars}}{80\text{ loads}}$ for powder detergent.
2. Divide each cost by its loads. The liquid rate is approximately $0.234$ dollars per load, while the powder rate is approximately $0.199$ dollars per load.
3. Round to the nearest cent. These unit prices become $0.23$ dollars per load for liquid and $0.20$ dollars per load for powder.
4. Compare the unit prices. Since $0.20$ dollars is less than $0.23$ dollars, the powder detergent has the lowest cost per load.

Takeaway: The powder detergent costs about $0.20$ dollars per load, less than the liquid detergent's about $0.23$ dollars per load.

Notice in Example 5.67 that we rounded the unit price to the nearest cent. Sometimes we may need to carry the division to one more place to see the difference between the unit prices.

Translate Phrases to Expressions with Fractions

Have you noticed that the examples in this section used the comparison words ratio of, to, per, in, for, on, and from? When you translate phrases that include these words, you should think either ratio or rate. If the units measure the same quantity (length, time, etc.), you have a ratio. If the units are different, you have a rate. In both cases, you write a fraction.

Example 5.68

Solution

1. Locate the first quantity. In $427$ miles per $h$ hours, miles are named first and hours second, so the rate begins with $427$ miles in the numerator.
2. Locate the other quantities. In $x$ students to $3$ teachers, students belong above teachers, while $y$ dollars for $18$ hours places dollars above hours.
3. Form the three fractions. Write the expressions as $\frac{427\text{ miles}}{h\text{ hours}}$, $\frac{x\text{ students}}{3\text{ teachers}}$, and $\frac{y\text{ dollars}}{18\text{ hours}}$.
4. Check the order. In each expression, the quantity named first appears above the quantity named second, matching the meaning of each original phrase.

Takeaway: The phrases translate to $\frac{427}{h}$, $\frac{x}{3}$, and $\frac{y}{18}$ because each first-named quantity belongs in the numerator.

Section 5.6 Exercises

Section 5.6 Exercises

Write a Ratio as a Fraction

In the following exercises, write each ratio as a fraction.

Write a Rate as a Fraction

In the following exercises, write each rate as a fraction.

Find Unit Rates

In the following exercises, find the unit rate. Round to two decimal places, if necessary.

Find Unit Price

In the following exercises, find the unit price. Round to the nearest cent.

In the following exercises, find each unit price and then identify the better buy. Round to three decimal places.

Translate Phrases to Expressions with Fractions

In the following exercises, translate the English phrase into an algebraic expression.

Everyday Math

Writing Exercises

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

1. Write each ratio as a fraction: ⓐ $\,21\,\mathrm{to}\,56\,$ⓑ $\,48\,\mathrm{to}\,32.$

2. Write each ratio as a fraction: ⓐ $\,27\,\mathrm{to}\,72\,$ⓑ $\,51\,\mathrm{to}\,34.$

3. Write each ratio as a fraction: ⓐ $\,4.6\,\mathrm{to}\,11.5\,$ⓑ $\,2.3\,\mathrm{to}\,0.69.$

4. Write each ratio as a fraction: ⓐ $\,3.4\,\mathrm{to}\,15.3\,$ⓑ $\,3.4\,\mathrm{to}\,0.68.$

5. Write each ratio as a fraction: $1\frac{3}{4}\,\mathrm{to}\,2\frac{5}{8}.$

1. Write each ratio as a fraction: $1\frac{1}{8}\,\mathrm{to}\,2\frac{3}{4}.$

2. Find the patient's ratio of total cholesterol to HDL cholesterol using the given information.

Total cholesterol is $185$ mg/dL and HDL cholesterol is $40$ mg/dL.

3. Find the patient’s ratio of total cholesterol to HDL cholesterol using the given information.

Total cholesterol is $204$ mg/dL and HDL cholesterol is $38$ mg/dL.

4. Find the ratio of the first length to the second length: $32$ inches to $1$ foot.

5. Find the ratio of the first length to the second length: $1$ foot to $54$ inches.

1. Write the rate as a fraction: $492$ miles in $8$ hours.

2. Write the rate as a fraction: $242$ miles in $6$ hours.

3. Find the unit rate: $\text{\textdollar 630}$ for $35$ hours.

4. Find the unit rate: $\text{\textdollar 684}$ for $36$ hours.

5. Find the unit rate: $423$ miles to $18$ gallons of gas.

1. Translate the word phrase into an algebraic expression.

ⓐ $\,689$ miles per $h$ hours ⓑ $y$ parents to $22$ students ⓒ $d$ dollars for $9$ minutes

2. Translate the word phrase into an algebraic expression.

ⓐ $m$ miles per $9$ hours ⓑ $x$ students to $8$ buses ⓒ $y$ dollars for $40$ hours

3. $20$ to $36$

4. $42$ to $48$

5. $49$ to $21$

1. $84$ to $36$

2. $0.56$ to $2.8$

3. $1\frac{2}{3}$ to $2\frac{5}{6}$

4. $4\frac{1}{6}$ to $3\frac{1}{3}$

5. $\text{\textdollar 18}$ to $\text{\textdollar 63}$

1. $\text{\textdollar 1.21}$ to $\text{\textdollar 0.44}$

2. $28$ ounces to $84$ ounces

3. $12$ feet to $46$ feet

4. $246$ milligrams to $45$ milligrams

5. total cholesterol of $175$ to HDL cholesterol of $45$

1. $27$ inches to $1$ foot

2. $140$ calories per $12$ ounces

3. $8.2$ pounds per $3$ square inches

4. $488$ miles in $7$ hours

5. $\text{\textdollar 595}$ for $40$ hours

1. $140$ calories per $12$ ounces

2. $8.2$ pounds per $3$ square inches

3. $488$ miles in $7$ hours

4. $\text{\textdollar 595}$ for $40$ hours

5. $576$ miles on $18$ gallons of gas

1. $43$ pounds in $16$ weeks

2. $46$ beats in $0.5$ minute

3. The bindery at a printing plant assembles $96,000$ magazines in $12$ hours. How many magazines are assembled in one hour?

4. Soap bars at $8$ for $\text{\textdollar 8.69}$

5. Women’s sports socks at $6$ pairs for $\text{\textdollar 7.99}$

1. Snack packs of cookies at $12$ for $\text{\textdollar 5.79}$

2. CD-RW discs at $25$ for $\text{\textdollar 14.99}$

3. The grocery store has a special on macaroni and cheese. The price is $\text{\textdollar 3.87}$ for $3$ boxes. How much does each box cost?

4. Mouthwash, $\mathrm{50.7-ounce}$ size for $\text{\textdollar 6.99}$ or $\mathrm{33.8-ounce}$ size for $\text{\textdollar 4.79}$

5. Breakfast cereal, $18$ ounces for $\text{\textdollar 3.99}$ or $14$ ounces for $\text{\textdollar 3.29}$

1. Ketchup, $\mathrm{40-ounce}$ regular bottle for $\text{\textdollar 2.99}$ or $\mathrm{64-ounce}$ squeeze bottle for $\text{\textdollar 4.39}$

2. Cheese $\text{\textdollar 6.49}$ for $1$ lb. block or $\text{\textdollar 3.39}$ for $\frac{1}{2}$ lb. block

3. $793$ miles per $p$ hours

4. $\text{\textdollar 3}$ for $0.5$ lbs.

5. $105$ calories in $x$ ounces

1. Find the unit rate: $406$ miles to $14.5$ gallons of gas.

2. Find the unit price. Round your answer to the nearest cent if necessary.

$\mathrm{24-pack}$ of juice boxes for $\text{\textdollar 6.99}$

3. Find the unit price. Round your answer to the nearest cent if necessary.

$\mathrm{24-pack}$ of bottles of ice tea for $\text{\textdollar 12.72}$

4. Find each unit price and then determine the better buy. Round to the nearest cent if necessary.

Brand A Storage Bags, $\text{\textdollar 4.59}$ for $40$ count, or Brand B Storage Bags, $\text{\textdollar 3.99}$ for $30$ count

5. Find each unit price and then determine the better buy. Round to the nearest cent if necessary.

Brand C Chicken Noodle Soup, $\text{\textdollar 1.89}$ for $26$ ounces, or Brand D Chicken Noodle Soup, $\text{\textdollar 0.95}$ for $10.75$ ounces

1. the ratio of $y$ and $5x$

2. One elementary school in Ohio has $684$ students and $45$ teachers. Write the student-to-teacher ratio as a unit rate.

3. A popular fast food burger weighs $7.5$ ounces and contains $540$ calories, $29$ grams of fat, $43$ grams of carbohydrates, and $25$ grams of protein. Find the unit rate of ⓐ calories per ounce ⓑ grams of fat per ounce ⓒ grams of carbohydrates per ounce ⓓ grams of protein per ounce. Round to two decimal places.

4. Would you prefer the ratio of your income to your friend’s income to be $\mathrm{3/1}$ or $1/3?$ Explain your reasoning.

5. Kathryn ate a $4-ounce$ cup of frozen yogurt and then went for a swim. The frozen yogurt had $115$ calories. Swimming burns $422$ calories per hour. For how many minutes should Kathryn swim to burn off the calories in the frozen yogurt? Explain your reasoning.