6th Grade - Ratios and Proportions

Ratios

• A comparison of two numbers by division is called a Ratio.
• A ratio can be expressed in three different ways: $\frac{a}{b}, a:b, \mathrm{or} a to b$.
• 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 cross-product 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 cross-product 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 cross-product 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.