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 [latex]a[/latex] and [latex]b[/latex] , the ratio is written as [latex]a\text{ to }b,\frac{a}{b},\text{or}\mathit{\text{a}}\text{:}\mathit{\text{b}}\text{.}[/latex]
example
Write each ratio as a fraction: ⓐ [latex]15\text{ to }27[/latex] ⓑ [latex]45\text{ to }18[/latex].
Solution
| ⓐ |
|
|
[latex]\text{15 to 27}[/latex] |
| Write as a fraction with the first number in the numerator and the second in the denominator. |
[latex]\frac{15}{27}[/latex] |
| Simplify the fraction. |
[latex]\frac{5}{9}[/latex] |
| ⓑ |
|
|
[latex]\text{45 to 18}[/latex] |
| Write as a fraction with the first number in the numerator and the second in the denominator. |
[latex]\frac{45}{18}[/latex] |
| Simplify. |
[latex]\frac{5}{2}[/latex] |
We leave the ratio in ⓑ as an improper fraction.
example
Write each ratio as a fraction of whole numbers:
ⓐ [latex]4.8\text{ to }11.2[/latex]
ⓑ [latex]2.7\text{ to }0.54[/latex]
Answer:
Solution
| ⓐ [latex]\text{4.8 to 11.2}[/latex] |
|
| Write as a fraction. |
[latex]\frac{4.8}{11.2}[/latex] |
| Rewrite as an equivalent fraction without decimals, by moving both decimal points [latex]1[/latex] place to the right. |
[latex]\frac{48}{112}[/latex] |
| Simplify. |
[latex]\frac{3}{7}[/latex] |
So [latex]4.8\text{ to }11.2[/latex] is equivalent to [latex]\frac{3}{7}[/latex].
| ⓑ
The numerator has one decimal place and the denominator has [latex]2[/latex]. To clear both decimals we need to move the decimal [latex]2[/latex] places to the right.
[latex]2.7\text{ to }0.54[/latex] |
|
| Write as a fraction. |
[latex]\frac{2.7}{0.54}[/latex] |
| Move both decimals right two places. |
[latex]\frac{270}{54}[/latex] |
| Simplify. |
[latex]\frac{5}{1}[/latex] |
So [latex]2.7\text{ to }0.54[/latex] is equivalent to [latex]\frac{5}{1}[/latex].
example
Write the ratio of [latex]1\frac{1}{4}\text{ to }2\frac{3}{8}[/latex] as a fraction.
Answer:
Solution
|
[latex]1\frac{1}{4}\text{ to }2\frac{3}{8}[/latex] |
| Write as a fraction. |
[latex]\frac{1\frac{1}{4}}{2\frac{3}{8}}[/latex] |
| Convert the numerator and denominator to improper fractions. |
[latex]\frac{\frac{5}{4}}{\frac{19}{8}}[/latex] |
| Rewrite as a division of fractions. |
[latex]\frac{5}{4}\div \frac{19}{8}[/latex] |
| Invert the divisor and multiply. |
[latex]\frac{5}{4}\cdot \frac{8}{19}[/latex] |
| Simplify. |
[latex]\frac{10}{19}[/latex] |
Do you see a shortcut to find the equivalent fraction? Notice that [latex]0.8=\frac{8}{10}[/latex] and [latex]0.05=\frac{5}{100}[/latex]. The least common denominator of [latex]\frac{8}{10}[/latex] and [latex]\frac{5}{100}[/latex] is [latex]100[/latex]. By multiplying the numerator and denominator of [latex]\frac{0.8}{0.05}[/latex] by [latex]100[/latex], 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:
