What are Ratio Tables?
A ratio table is an organized list of equivalent ratios that explains how ratios or numbers relate to one another. These tables are helpful for visualizing and understanding the relationship between two different quantities. Sometimes these ratio tables may have missing values. In such cases, we use the concept of equivalent ratios to find those values.
Let’s see how to find those missing values to complete a ratio table.
How do we Complete a Ratio Table?
As stated, a ratio table lists equivalent ratios. We know equivalent ratios have equivalent values. Hence, if we multiply or divide each quantity in a ratio by the same number we will get the equivalent ratio of the original ratio. This concept can be used to complete any ratio table.
We can find missing values in a ratio table by:
- Adding or subtracting values in each ratio of the table.
- Multiplying or dividing each number in the equivalent ratios by the same value.
For example, in the ratio table shown below:
The three equivalent ratios are 3:5, 6:10 and 9:15. Here, the first quantity of each ratio is obtained by adding 3 and the second quantity is obtained by adding 5.
Now, let’s observe another ratio table:
The equivalent ratios are 3:4, 6:8, 9:12, 12:16 and 15:20. Here, the first quantity of each ratio is obtained by multiplying 3 by 2, 3, 4 and 5. Similarly, the second quantity of each ratio is obtained by multiplying 4 by 2, 3, 4 and 5.
[Note: We can also use a combination of arithmetic operations to find missing values in a ratio table.]
Solved Examples
Example 1: Cathy went to a store and bought some fruits and vegetables as shown in the table. Determine the missing values to complete the ratio table.
Solution:
By looking at the first column in the table, let’s fill the missing spots using equivalent ratios.
The first quantity of the second ratio will be:
2 + 2 = 4
Similarly,
The second quantity of the third ratio will be:
12 + 6 = 18.
So, the missing values are 4 and 18. Hence, the complete ratio table is:
Example 2: Complete this ratio table:
Solution:
Here, the first ratio is 4:10. We can complete the table with equivalent ratios of 4:10.
Let’s find the equivalent ratios:
\(\frac{4\times2}{10\times2}=\frac{8}{20}\) [Multiply each quantity of the ratio by 2]
Similarly,
\(\frac{4\times4}{10\times4}=\frac{16}{40}\) [Multiply each quantity of the ratio by 4]
Hence, the missing values are 8, 20, 16, and 40 and the complete table is:
[Note: We can also find the equivalent ratios here by simplifying the ratio 4:10 or multiplying or dividing each quantity of 4:10 by any other number.]
Example 3: Richard is reading a book in a library. The below table illustrates the number of pages read by Richard and the time taken in minutes. Complete the ratio table to know how many pages Richard has read in how many minutes.
\(\frac{1}{4}:\frac{1}{2}=\frac{3}{4}:x\) [Equivalent Ratios]
\(\frac{1}{4}\times x=\frac{3}{4}\times\frac{1}{2}\) [Cross Product]
\(x=\frac{3}{2}\) [Solve for x]
Similarly,
\(\frac{3}{4}:\frac{3}{2}=y:3\) [Equivalent Ratios]
\(\frac{3}{2}\times y = \frac{3}{4}\times3\) [Cross Product]
\(\frac{3}{2}\times y = \frac{9}{4}\) [Multiply]
\(y = \frac{3}{2}\) [Solve for y]
And,
\(\frac{3}{2}:3=5:z\) [Equivalent Ratios]
\(\frac{3}{2}\times z=5\times3\) [Cross Product]
z=10 [Solve for z]
So, the missing values x, y and z are \(\frac{3}{2},\frac{3}{2}\) and 10 respectively.
Hence, Richard read \(\frac{3}{4}\) pages in \(\frac{3}{2}\) minutes, \(\frac{3}{2}\) pages in 3 minutes and 5 pages in 10 minutes.
Yes. We can use repeated addition or multiplication to construct a ratio table.
We use ratio tables to describe the relationship between two different quantities.
Ratios that can be simplified to the same value are said to be equivalent ratios. If one ratio can be written as a multiple of the other, then they are said to be equivalent.
Unit rates are used to compare a quantity to one unit of another quantity. Unit rates are determined by applying the concept of ratio, where the value of the second quantity in the ratio is always 1.