How To Find Unit Rate With Decimals

Decimals

33 Ratios and Rate

Learning Objectives

Past the end of this department, you lot will exist able to:

Write a ratio every bit a fraction
Write a rate as a fraction
Notice unit rates
Find unit cost
Interpret phrases to expressions with fractions

Write a Ratio as a Fraction

When you employ for a mortgage, the loan officeholder volition compare your full debt to your full 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 $a\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}b,\phantom{\rule{0.2em}{0ex}}\frac{a}{b},\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}\mathit{\text{a}}\text{:}\mathit{\text{b}}\text{.}$

Ratios

A ratio compares two numbers or two quantities that are measured with the same unit. The ratio of $a$ to $b$ is written $a\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}b,\phantom{\rule{0.2em}{0ex}}\frac{a}{b},\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}\mathit{\text{a}}\text{:}\mathit{\text{b}}\text{.}$

In this department, nosotros will use the fraction note. When a ratio is written in fraction grade, the fraction should be simplified. If it is an improper fraction, we do not change information technology to a mixed number. Because a ratio compares two quantities, we would leave a ratio every bit $\frac{4}{1}$ instead of simplifying it to $4$ so that we can run into the 2 parts of the ratio.

Write each ratio every bit a fraction: ⓐ $\phantom{\rule{0.2em}{0ex}}15\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}27\phantom{\rule{0.2em}{0ex}}$ ⓑ $\phantom{\rule{0.2em}{0ex}}45\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}18.$

Write each ratio as a fraction: ⓐ $\phantom{\rule{0.2em}{0ex}}21\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}56\phantom{\rule{0.2em}{0ex}}$ ⓑ $\phantom{\rule{0.2em}{0ex}}48\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}32.$

Write each ratio as a fraction: ⓐ $\phantom{\rule{0.2em}{0ex}}27\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}72\phantom{\rule{0.2em}{0ex}}$ ⓑ $\phantom{\rule{0.2em}{0ex}}51\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}34.$

Ratios Involving Decimals

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

For example, consider the ratio $0.8\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}0.05.$ We can write it as a fraction with decimals and then multiply the numerator and denominator past $100$ to eliminate the decimals.

$A fraction is shown with 0.8 in the numerator and 0.05 in the denominator. Below it is the same fraction with both the numerator and denominator multiplied by 100. Below that is a fraction with 80 in the numerator and 5 in the denominator.$

Do you see a shortcut to detect 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 ii places to the correct to go the equivalent fraction with no decimals. At present that we understand the math behind the process, we can observe the fraction with no decimals like this:


"Motility" the decimal 2 places.	$\frac{80}{5}$
Simplify.	$\frac{16}{1}$

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

Write each ratio as a fraction of whole numbers:

ⓐ $\phantom{\rule{0.2em}{0ex}}4.8\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}11.2$

ⓑ $\phantom{\rule{0.2em}{0ex}}2.7\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}0.54$

Write each ratio every bit a fraction: ⓐ $\phantom{\rule{0.2em}{0ex}}4.6\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}11.5\phantom{\rule{0.2em}{0ex}}$ ⓑ $\phantom{\rule{0.2em}{0ex}}2.3\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}0.69.$

Write each ratio equally a fraction: ⓐ $\phantom{\rule{0.2em}{0ex}}3.4\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}15.3\phantom{\rule{0.2em}{0ex}}$ ⓑ $\phantom{\rule{0.2em}{0ex}}3.4\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}0.68.$

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

Write the ratio of $1\frac{1}{4}\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}2\frac{3}{8}$ as a fraction.

Write each ratio as a fraction: $1\frac{3}{4}\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}2\frac{5}{8}.$

$\frac{2}{3}$

Write each ratio as a fraction: $1\frac{1}{8}\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}2\frac{3}{4}.$

$\frac{9}{22}$

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 ane mode doctors assess a person's overall health. A ratio of less than $5$ to $1$ is considered good.

Detect the patient's ratio of full cholesterol to HDL cholesterol using the given information.

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

$\frac{37}{8}$

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

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

$\frac{102}{19}$

Ratios of 2 Measurements in Dissimilar Units

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

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

The Americans with Disabilities Act (ADA) Guidelines for bicycle chair ramps require a maximum vertical rise of $1$ inch for every $1$ human foot of horizontal run. What is the ratio of the rising to the run?

Solution

In a ratio, the measurements must be in the same units. Nosotros can modify feet to inches, or inches to feet. It is commonly easier to convert to the smaller unit, since this avoids introducing more fractions into the problem.

Write the words that limited the ratio.

	Ratio of the ascent to the run
Write the ratio every bit a fraction.	$\frac{\text{rise}}{\text{run}}$
Substitute in the given values.	$\frac{\text{1 inch}}{\text{1 foot}}$
Convert i foot to inches.	$\frac{\text{1 inch}}{\text{12 inches}}$
Simplify, dividing out common factors and units.	$\frac{1}{12}$

So the ratio of rising to run is $1$ to $12.$ This means that the ramp should rising $1$ inch for every $12$ inches of horizontal run to comply with the guidelines.

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

$\frac{8}{3}$

Observe the ratio of the first length to the 2d length: $1$ foot to $54$ inches.

$\frac{2}{9}$

Write a Rate as a Fraction

Often we want to compare two different types of measurements, such equally miles to gallons. To brand this comparing, we utilize a rate. Examples of rates are $120$ miles in $2$ hours, $160$ words in $4$ minutes, and $\text{?5}$ dollars per $64$ ounces.

Rate

A rate compares 2 quantities of different units. A rate is usually written every bit a fraction.

When writing a fraction as a rate, we put the first given corporeality with its units in the numerator and the 2d amount with its units in the denominator. When rates are simplified, the units remain in the numerator and denominator.

Bob drove his car $525$ miles in $9$ hours. Write this rate every bit a fraction.

Write the charge per unit as a fraction: $492$ miles in $8$ hours.

$\frac{\text{123 miles}}{\text{2 hours}}$

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

$\frac{\text{121 miles}}{\text{3 hours}}$

Find Unit Rates

In the final example, we calculated that Bob was driving at a rate of $\frac{\text{175 miles}}{\text{3 hours}}.$ This tells us that every iii hours, Bob will travel $175$ miles. This is correct, merely not very useful. We unremarkably 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 Rate

A unit rate is a rate with denominator of $1$ unit.

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 nosotros travel $68$ miles in $1$ hour. We would write this rate every bit $68$ miles/hr (read $68$ miles per hour). The mutual abbreviation for this is $68$ mph. Note that when no number is written before a unit of measurement, it is assumed to be $1.$

Then $68$ miles/hour really means $\text{68 miles/1 hour.}$

Ii rates we oftentimes utilize when driving can be written in dissimilar forms, as shown:

Instance	Rate	Write	Abridge	Read
$68$ miles in $1$ hr	$\frac{\text{68 miles}}{\text{1 hour}}$	$68$ miles/hr	$68$ mph	$\text{68 miles per hour}$
$36$ miles to $1$ gallon	$\frac{\text{36 miles}}{\text{1 gallon}}$	$36$ miles/gallon	$36$ mpg	$\text{36 miles per gallon}$

Another instance 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 60 minutes of work. For example, if you are paid $\text{?12.50}$ for each hour you lot work, you could write that your hourly (unit) pay rate is $\text{?12.50/hour}$ (read $\text{?12.50}$ per hour.)

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

Anita was paid $\text{?384}$ final week for working $\text{32 hours}.$ What is Anita'southward hourly pay rate?

Notice the unit charge per unit: $\text{?630}$ for $35$ hours.

?18.00/hour

Observe the unit rate: $\text{?684}$ for $36$ hours.

?19.00/hour

Sven drives his car $455$ miles, using $14$ gallons of gasoline. How many miles per gallon does his car become?

Solution

Offset with a rate of miles to gallons. Then divide.

	$\text{455 miles to 14 gallons of gas}$
Write equally a rate.	$\frac{\text{455 miles}}{\text{14 gallons}}$
Carve up 455 past xiv to get the unit rate.	$\frac{\text{32.5 miles}}{\text{1 gallon}}$

Sven'due south motorcar gets $32.5$ miles/gallon, or $32.5$ mpg.

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

23.5 mpg

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

28 mpg

Find Unit Price

Sometimes we buy common household items 'in bulk', where several items are packaged together and sold for one cost. To compare the prices of unlike sized packages, we demand to find the unit price. To find the unit cost, dissever the total toll by the number of items. A unit price is a unit rate for ane item.

Unit price

A unit price is a unit rate that gives the price of ane particular.

The grocery store charges $\text{?3.99}$ for a case of $24$ bottles of water. What is the unit of measurement price?

Solution

What are we asked to find? We are asked to find the unit of measurement cost, which is the toll per bottle.

Write as a rate.	$\frac{?3.99}{\text{24 bottles}}$
Carve up to find the unit toll.	$\frac{?0.16625}{\text{1 bottle}}$
Circular the event to the nearest penny.	$\frac{?0.17}{\text{1 bottle}}$

The unit of measurement price is approximately $\text{?0.17}$ per bottle. Each bottle costs about $\text{?0.17}.$

Find the unit of measurement toll. Round your answer to the nearest cent if necessary.

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

?0.29/box

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

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

?0.53/bottle

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

Paul is shopping for laundry detergent. At the grocery store, the liquid detergent is priced at $\text{?14.99}$ for $64$ loads of laundry and the same make of powder detergent is priced at $\text{?15.99}$ for $80$ loads.

Which is the better buy, the liquid or the pulverisation detergent?

Find each unit of measurement price and so make up one's mind the better buy. Round to the nearest cent if necessary.

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

Brand A costs ?0.12 per bag. Make B costs ?0.thirteen per bag. Brand A is the better purchase.

Find each unit price then determine the better purchase. Round to the nearest cent if necessary.

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

Brand C costs ?0.07 per ounce. Brand D costs ?0.09 per ounce. Brand C is the better purchase.

Detect in (Effigy) that we rounded the unit of measurement cost to the nearest cent. Sometimes we may need to bear the division to one more identify to meet the departure between the unit prices.

Translate Phrases to Expressions with Fractions

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

Translate the give-and-take phrase into an algebraic expression:

ⓐ $\phantom{\rule{0.2em}{0ex}}427$ miles per $h$ hours

ⓑ $\phantom{\rule{0.2em}{0ex}}x$ students to $3$ teachers

ⓐ 689 mi/h hours
ⓑ y parents/22 students
ⓒ ?d/nine min

ⓐ thousand mi/9 h
ⓑ x students/8 buses
ⓒ ?y/40 h

Practice Makes Perfect

Write a Ratio as a Fraction

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

$20$ to $32$

$45$ to $54$

$56$ to $16$

$6.4$ to $0.8$

$1.26$ to $4.2$

$1\frac{3}{4}$ to $2\frac{5}{8}$

$5\frac{3}{5}$ to $3\frac{3}{5}$

$\text{?16}$ to $\text{?72}$

$\text{?1.38}$ to $\text{?0.69}$

$32$ ounces to $128$ ounces

$15$ anxiety to $57$ anxiety

$304$ milligrams to $48$ milligrams

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

$\frac{35}{9}$

total cholesterol of $215$ to HDL cholesterol of $55$

$28$ inches to $1$ foot

Write a Charge per unit every bit a Fraction

In the post-obit exercises, write each rate equally a fraction.

$180$ calories per $16$ ounces

$9.5$ pounds per $4$ square inches

$527$ miles in $9$ hours

$\text{?798}$ for $40$ hours

Discover Unit Rates

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

$140$ calories per $12$ ounces

11.67 calories/ounce

$180$ calories per $16$ ounces

$8.2$ pounds per $3$ square inches

two.73 lbs./sq. in.

$9.5$ pounds per $4$ foursquare inches

$488$ miles in $7$ hours

69.71 mph

$527$ miles in $9$ hours

$\text{?595}$ for $40$ hours

?14.88/hour

$\text{?798}$ for $40$ hours

$576$ miles on $18$ gallons of gas

32 mpg

$435$ miles on $15$ gallons of gas

$43$ pounds in $16$ weeks

ii.69 lbs./week

$57$ pounds in $24$ weeks

$46$ beats in $0.5$ minute

92 beats/minute

$54$ beats in $0.5$ minute

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

8,000

The pressroom at a printing plant prints $540,000$ sections in $12$ hours. How many sections are printed per hour?

Detect Unit Cost

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

Soap bars at $8$ for $\text{?8.69}$

?1.09/bar

Soap bars at $4$ for $\text{?3.39}$

Women'southward sports socks at $6$ pairs for $\text{?7.99}$

?one.33/pair

Men's dress socks at $3$ pairs for $\text{?8.49}$

Snack packs of cookies at $12$ for $\text{?5.79}$

?0.48/pack

Granola bars at $5$ for $\text{?3.69}$

CD-RW discs at $25$ for $\text{?14.99}$

?0.60/disc

CDs at $50$ for $\text{?4.49}$

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

?1.29/box

The pet store has a special on cat food. The price is $\text{?4.32}$ for $12$ cans. How much does each can toll?

In the following exercises, find each unit price then identify the meliorate purchase. Round to iii decimal places.

Mouthwash, $\text{50.7-ounce}$ size for $\text{?6.99}$ or $\text{33.8-ounce}$ size for $\text{?4.79}$

The fifty.7-ounce size costs ?0.138 per ounce. The 33.viii-ounce size costs ?0.142 per ounce. The 50.7-ounce size is the better buy.

Breakfast cereal, $18$ ounces for $\text{?3.99}$ or $14$ ounces for $\text{?3.29}$

The 18-ounce size costs ?0.222 per ounce. The 14-ounce size costs ?0.235 per ounce. The 18-ounce size is a ameliorate buy.

Ketchup, $\text{40-ounce}$ regular canteen for $\text{?2.99}$ or $\text{64-ounce}$ clasp canteen for $\text{?4.39}$

The regular bottle costs ?0.075 per ounce. The squeeze canteen costs ?0.069 per ounce. The squeeze bottle is a better buy.

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

The one-half-pound block costs ?vi.78/lb, so the ane-lb. block is a better purchase.

Translate Phrases to Expressions with Fractions

In the post-obit exercises, translate the English phrase into an algebraic expression.

$78$ feet per $r$ seconds

$j$ beats in $0.5$ minutes

$400$ minutes for $m$ dollars

the ratio of $12x$ and $y$

Everyday Math

One elementary schoolhouse in Ohio has $684$ students and $45$ teachers. Write the student-to-teacher ratio as a unit charge per unit.

15.2 students per teacher

The average American produces most $1,600$ pounds of newspaper trash per yr $\text{(365 days).}$ How many pounds of paper trash does the average American produce each day? (Round to the nearest tenth of a pound.)

Writing Exercises

Would yous prefer the ratio of your income to your friend'southward income to exist $\text{3/1}$ or $1/3?$ Explicate your reasoning.

Answers will vary.

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 down off the calories in the frozen yogurt? Explain your reasoning.