# Decimals

## Ratios and Rate


### 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  and , the ratio is written as 
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  instead of simplifying it to  so that we can see the two parts of the ratio.

### 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  We can write it as a fraction with decimals and then multiply the numerator and denominator by  to eliminate the decimals.
Do you see a shortcut to find the equivalent fraction? Notice that  and  The least common denominator of  and  is  By multiplying the numerator and denominator of  by  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.
Some ratios compare two mixed numbers. Remember that to divide mixed numbers, you first rewrite them as improper fractions.

### 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  to  is considered good.

### 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.

### 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  miles in  hours,  words in  minutes, and  dollars per  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.

### Find Unit Rates

In the last example, we calculated that Bob was driving at a rate of  This tells us that every three hours, Bob will travel  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  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  miles per hour we mean that we travel  miles in  hour. We would write this rate as  miles/hour (read  miles per hour). The common abbreviation for this is  mph. Note that when no number is written before a unit, it is assumed to be 
So  miles/hour really means 
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  for each hour you work, you could write that your hourly (unit) pay rate is  (read  per hour.) 
To convert a rate to a unit rate, we divide the numerator by the denominator. This gives us a denominator of 

### 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.
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.
Notice in  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.

### Practice Makes Perfect

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?
