A ratio is a comparison of quantities having the same unit. A ratio helps us indicate how big or how small a quantity is when compared to another quantity. Two known quantities can be compared by dividing them. We can use division to compare the quantities \(a\) and \(b\): \(\frac{a}{b}\). Here, \(a\) is the dividend and \(b\) is the divisor. Ratio is an efficient method of representing this division operation in a different manner.
A ratio can be expressed in three ways:
- A ratio is denoted using the ‘:’ symbol. A ratio is expressed by writing the ‘:’ symbol in the middle of the two quantities that are being compared. In the ratio \(a : b\), \(a\) and \(b\) are together known as the terms of the ratio. Also, \(a\) is known as the antecedent or the first term, and \(b\) is known as the consequent or the second term.
- But ratios needn’t always be represented using the ‘:’ symbol. As ratios are basically fractions, a fraction like \(\frac{a}{b}\) can also be interpreted as a ratio.
- A ratio can also be expressed in words as \(a\) to \(b\).
∴ \(a : b\) = \(\frac{a}{b}\) = \(a\) to \(b\)
As discussed earlier, ratios can be used to compare quantities in our day-to-day lives. You can easily compare the number of boys and the number of girls in your classroom by taking the ratio.
Ratios can help make decision-making easier while performing activities like grocery shopping and cooking easier.
Suppose you need 3.5 ounces of ice cream and 10 ounces of milk while making a milkshake for one person. Ratios make it easy to scale up this recipe if we need to make this milkshake for multiple people. Let’s see how ratios can be scaled up by performing operations on them.
Two ratios that describe the same relationship are known as equivalent ratios. Equivalent ratios can be represented using a ratio table by adding or subtracting quantities in equivalent ratios, or by multiplying and dividing with the same values.
In the previous example of the recipe of milkshakes, a ratio table can be used to scale up the number of servings.
Ingredients for 1 milkshake: 3.5 ounces of ice cream for 10 ounces of milk.
With this information in hand, it is possible to scale up the recipe to the required amount.
Number of milkshakes | Ice cream (oz.) | Milk (oz.) |
1 | 3.5 | 10 |
5 | \(3.5\times 7 = 17.5\) | \(10\times 5 = 50\) |
12 | \(3.5\times 12 = 42\) | \(10\times 12 = 120\) |
Missing values in a ratio table can be calculated by performing addition, subtraction, multiplication, or division.
Example 1: If there are 25 students in a class and 15 of them are girls, find the simplest ratio of the number of boys to the number of girls.
Solution:
Total number of students = 25
Number of girls = 15
Number of boys = 25 – 15 = 10
Ratio of number of boys to the number of girls: 15 : 10
Divide both sides by 5.
\(\frac{15 \div 5}{10 \div 5} = \frac{3}{2}\)
The simplest form of the ratio 15 : 10 is 3 : 2.
Example 2: You have to mix \(\frac{1}{3}\) cup of yellow paint for every \(\frac{1}{2}\) cup of red paint to make 10 cups of orange paint. Find the required number of cups of yellow paint.
Solution:
We can use a ratio table to find the number of cups of yellow paint required to get 10 cups of orange paint,
By looking at the ratio table, we can conclude that we need 4 cups of yellow paint to get a cup of orange paint.
Example 3: Find the missing values from the ratio table.
Solution: In this case, we can perform math operations to find the missing values of the ratio table.
Note that during multiplication or division, the same operation was done on both sides. But during addition or subtraction, the values used for the operation are different on both sides: they are equivalent ratios.
The quantities or terms in a ratio should be of the same unit. If we want to compare two quantities of different units, we use rate instead of ratios.
Ratios and fractions are mathematically the same: 1 : 2 is the same as \(\frac{1}{2}\). But in most cases, they carry a different meaning. For example, \(\frac{1}{2}\) of a sandwich means half of a sandwich. But a ratio is a comparison of two different quantities, i.e. 1 : 2 represents the ratio of non-vegetarian sandwiches to vegetarian sandwiches, which means there are two vegetarian sandwiches for every non-vegetarian sandwich.
The values in the ratio table can be multiplied or divided by any values. We just need to make sure that we use the same value while multiplying or dividing both sides.
While multiplication and division operations can be done with any values, this is not possible with addition and subtraction. In the case of addition and subtraction, the operation is done with equivalent ratios