Writing a Ratio as a Fraction
Writing 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 \(a\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}b,\phantom{\rule{0.2em}{0ex}}\frac{a}{b},\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}\mathit{\text{a}}\text{:}\mathit{\text{b}}\text{.}\)
Definition: 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 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.
Optional Video: Ratios
Example
Write each ratio as a fraction:
- \(\phantom{\rule{0.2em}{0ex}}15\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}27\phantom{\rule{0.2em}{0ex}}\)
- \(\phantom{\rule{0.2em}{0ex}}45\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}18.\)
Solution
| \(\text{15 to 27}\) | |
| Write as a fraction with the first number in the numerator and the second in the denominator. | \(\frac{15}{27}\) |
| Simplify the fraction. | \(\frac{5}{9}\) |
| \(\text{45 to 18}\) | |
| Write as a fraction with the first number in the numerator and the second in the denominator. | \(\frac{45}{18}\) |
| Simplify. | \(\frac{5}{2}\) |
We leave the ratio in as an improper fraction.
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 \(0.8\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}0.05.\) We can write it as a fraction with decimals and then multiply the numerator and denominator 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:
| "Move" the decimal 2 places. | \(\frac{80}{5}\) |
| Simplify. | \(\frac{16}{1}\) |
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
Write each ratio as a fraction of whole numbers:
- \(\phantom{\rule{0.2em}{0ex}}4.8\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}11.2\)
- \(\phantom{\rule{0.2em}{0ex}}2.7\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}0.54\)
Solution
| \(\phantom{\rule{0.2em}{0ex}}\text{4.8 to 11.2}\) | |
| Write as a fraction. | \(\frac{4.8}{11.2}\) |
| Rewrite as an equivalent fraction without decimals, by moving both decimal points 1 place to the right. | \(\frac{48}{112}\) |
| Simplify. | \(\frac{3}{7}\) |
So \(4.8\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}11.2\) is equivalent to \(\frac{3}{7}.\)
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| Write as a fraction. | \(\frac{2.7}{0.54}\) |
| Move both decimals right two places. | \(\frac{270}{54}\) |
| Simplify. | \(\frac{5}{1}\) |
So \(2.7\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}0.54\) is equivalent to \(\frac{5}{1}.\)
Some ratios compare two mixed numbers. Remember that to divide mixed numbers, you first rewrite them as improper fractions.
Optional Video: Write Ratios as a Simplified Fractions Involving Decimals and Fractions
Example
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.
Solution
| \(1\frac{1}{4}\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}2\frac{3}{8}\) | |
| Write as a fraction. | \(\frac{1\frac{1}{4}}{2\frac{3}{8}}\) |
| Convert the numerator and denominator to improper fractions. | \(\frac{\frac{5}{4}}{\frac{19}{8}}\) |
| Rewrite as a division of fractions. | \(\frac{5}{4}÷\frac{19}{8}\) |
| Invert the divisor and multiply. | \(\frac{5}{4}·\frac{8}{19}\) |
| Simplify. | \(\frac{10}{19}\) |
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.
Optional Video: Write a Ratio as a Simplified Fraction
Example
Hector's total cholesterol is \(249\) mg/dl and his HDL cholesterol is \(39\) mg/dl. Find the ratio of his total cholesterol to his HDL cholesterol. Assuming that a ratio less than \(5\) to \(1\) is considered good, what would you suggest to Hector?
Solution
First, write the words that express the ratio. We want to know the ratio of Hector's total cholesterol to his HDL cholesterol.
| Write as a fraction. | \(\frac{\text{total cholesterol}}{\text{HDL cholesterol}}\) |
| Substitute the values. | \(\frac{249}{39}\) |
| Simplify. | \(\frac{83}{13}\) |
Is Hector's cholesterol ratio ok? If we divide \(83\) by \(13\) we obtain approximately \(6.4,\) so \(\frac{83}{13}\approx \frac{6.4}{1}.\) Hector's cholesterol ratio is high! Hector should either lower his total cholesterol or raise his HDL cholesterol.
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
The Americans with Disabilities Act (ADA) Guidelines for wheel chair ramps require a maximum vertical rise of \(1\) inch for every \(1\) foot of horizontal run. What is the ratio of the rise to the run?
Solution
In a ratio, the measurements must be in the same units. We can change feet to inches, or inches to feet. It is usually easier to convert to the smaller unit, since this avoids introducing more fractions into the problem.
Write the words that express the ratio.
| Ratio of the rise to the run | |
| Write the ratio as a fraction. | \(\frac{\text{rise}}{\text{run}}\) |
| Substitute in the given values. | \(\frac{\text{1 inch}}{\text{1 foot}}\) |
| Convert 1 foot to inches. | \(\frac{\text{1 inch}}{\text{12 inches}}\) |
| Simplify, dividing out common factors and units. | \(\frac{1}{12}\) |
So the ratio of rise to run is \(1\) to \(12.\) This means that the ramp should rise \(1\) inch for every \(12\) inches of horizontal run to comply with the guidelines.
This lesson is part of:
Introducing Decimals