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ARITHMETIC

As we can multiply or divide both terms of a fraction by the same number without changing its value, so we can multiply or divide both terms of a ratio by the same number without changing its value. Thus, $3:6={\frac {1}{2}}$ , or $6:12={\frac {1}{2}}$ , or $1:2={\frac {1}{2}}$ .

The term antecedent is another name for numerator or dividend; the term consequent is another name for denominator or divisor.

Find the value of the following ratios:

1.	$3:27$	$12:24$	${\frac {3}{4}}:{\frac {1}{2}}$
2.	$8:72$	$24:12$	$5{\text{ft.}}:20{\text{in.}}$
3.	$\$2{\frac {1}{2}}:\$15$	$44:10$	${\frac {3}{4}}{\text{yd.}}:{\frac {3}{4}}{\text{ft.}}$

4.⁠Divide 30 cents between two boys in the ratio of 2 to 3.

Each time a division is made, 5 cents are taken, of which the first boy gets 2 of the 5, or ${\frac {2}{5}}$ and the other boy gets 3 of the 5, or ${\frac {3}{5}}$ .

${\frac {2}{5}}{\text{ of }}30=12$ ⁠ ${\frac {3}{5}}{\text{ of }}30=18$ ⁠12 cents and 18 cents.

The ratio $8:4=2$ .⁠The ratio $12:6=2$ .

Thus the two ratios being equal to the same number, are equal to each other, and may be written $8:4=12:6$ .

A proportion is an equality of ratios.

The first and last terms of a proportion are called the extremes.

The second and third terms of a proportion are called the means.

The test of a proportion is to have the product of the means equal the product of the extremes. Thus the product of the means divided by either extreme will give the other extreme; and the product of the extremes divided by either means will give the other mean.

Find the missing term, which is represented by $\times$ in the following:

$15:5=12:\times$
${\frac {5\times 12}{15}}=4$	Multiply the means and divide by the given extreme.
$15:5=12:4$
$15:\times =12:4$
${\frac {15\times 4}{12}}=5$	Multiply the extremes and divide by the given mean.
$15:5=12:4$