Math Help for Section 3.4, Page 128
Setting
up Ratios
Because a ratio is a comparison of one number with another by division,
a class of 29 students made up of 16 women and 13 men can have the
following ratios.
| men to women: $\displaystyle {13 \over 16}$ | men to students: $\displaystyle {13 \over 29}$ | students to women: $\displaystyle {29 \over 16}$ |
Example
1: Tip
a.
The numbers 7 and 5 are prime so they have no
common factors other than 1. So, $7 \over 5$ is in simplest form. Prime
numbers and factors are defined on page 22 of the
textbook.
b. The
numbers 12 and 8
have a greatest common factor of 4, so you can obtain the simplest form
of $12 \over 8$ as follows.
| $\displaystyle{{12} \over 8} = {{12 \div \color{red}4\color{black}} \over {8 \div\color{red}4\color{black}}} = {3 \over 2}$ |
Divide numerator and denominator by 4. |
c. The
numbers 10 and 2
have a greatest common factor of 2, so you can obtain the simplest form
of $10\over 2$ as follows.
| $\displaystyle{{10} \over 2} = {{10 \div \color{red}2\color{black}} \over {2 \div\color{red}2\color{black}}} = {5 \over 1}$ |
Divide numerator and denominator by 2. |
d.
You can convert the mixed
numbers to
fractions as follows.
| $\displaystyle 3{1 \over 2} = {{3\left( 2 \right) + 1} \over 2} = {7 \over 2}$ |
Multiply 3 by 2, add the result to 1, and then divide by 2. |
| $\displaystyle 5{1 \over 4} = {{5\left( 4 \right) + 1} \over 4} = {{21} \over 4}$ |
Multiply 5 by 4, add the result to 1, and then divide by 4. |
After you invert the divisor and multiply, you can simplify
the ratio as follows.
| $\displaystyle{7 \over 2} \bullet {4 \over {21}} = {\cancel{7} \over \cancel{2}} \bullet {{2 \bullet\cancel{2}} \over {3 \bullet\cancel{7}}} = {2 \over 3}$ |
Factor 4 and 21. Divide out the common factors. |
Greatest common factor and simplest form are defined in the Math
Help for page 30.