Ratios
 A comparison of two numbers by division is called a Ratio.
 A ratio can be expressed in three different ways: $\frac{a}{b},a:b,\mathrm{or}atob$.
 A ratio $\frac{a}{b}$ is read as "a ratio of a to b".
 A ratio is a fraction that can be simplified.
 Two ratios are called equal ratios or equivalent ratios if are equal.
 $\frac{2}{5}$ and $\frac{4}{10}$are equal ratios as $\frac{4}{10}=\frac{2}{5}\times \frac{2}{2}$ (numerator and denominator are multiplied by the same number).
 A table of equivalent ratios is called a ratio table.

2 4 6 8 10 5 10 15 20 25  All ratios in the ratio table are equivalent.

Proportion
 Two ratios are said to be in proportion if their crossproduct are equal.
 $\frac{\mathrm{a}}{\mathrm{b}}\mathrm{and}\frac{\mathrm{c}}{\mathrm{d}}$ are in proportion if $a\times d=b\times c$.
Solved Examples
1. Find the ratio of 7 to 10.
The ratio of 7 to 10 is $\frac{7}{10}$
2. Find the ratio of 16 oranges to 21 apples.
The ratio of 16 to 21 is $\frac{16}{21}$
3. Complete the following ratio table:
Number of Lemons  5  10  ? 
Glasses of Lemonade  10  ?  30 
To keep the ratio table equal, the top and the bottom numbers should be multiplied by the same number.
Number of Lemons  5  10 ($5\times 2$)  15 ($5\times 3$) 
Glasses of Lemonade  10  20 ($10\times 2$)  30 ($10\times 3$) 
4. Do the following ratios form a proportion: $\frac{2}{5}\mathrm{and}\frac{7}{15}$?
The crossproduct should be the same for the ratios to form a proportion. Since $2\times 15\ne 7\times 5$, the ratios are not in proportion.
5. 6 pastries cost $5. What should 18 pastries cost?
Put the provided information as a proportion: $\frac{6}{5}=\frac{18}{?}$.
Since 6 is multiplied by 3 to get 18, 5 must be multiplied by 3.
The answer is $15.
Cheat Sheet
 The quantity that comes first in the ratio statement becomes the numerator in the ratio fraction.
 A proportion consists of 4 terms. It is a comparison of two ratios.
 If the numerator and denominator of a ratio are multiplied by the same number, the resulting ratio is equivalent to the first one.
 $\frac{a}{b}=\frac{3a}{3b}=\frac{5a}{5b}$
 Their crossproduct should be the same for the ratios to be in proportion.
Blunder Area
 In the case of word problems, pay attention to which value goes as a numerator and which one goes as a denominator.
 Not all ratios are in proportion.
 Rishi Jethwa
 10 Comments
 57 Likes