What is the Ratio?
A ratio is the comparison of two quantities and that can be the comparisons of parts to parts, parts to wholes, and wholes to parts are all examples of ratios. In ratios, units may or may not be included.
x : y is the ratio of x to y.
Example: In a sports room, if there are five bats and ten balls.
So, the relevant numbers are 5 and 10 which are the two numbers we are comparing. We go through the sample of 5 bats to 10 balls then the ratio can be represented as 5 : 10, which is equivalent to the ratio 1 : 2.
Value of the Ratio
The fraction \( \frac{a}{b} \) is associated with the ratio a : b. This term describes the multiplicative relationship between the quantities in a ratio.
Equivalent Ratios
If the values of the ratios are the same, then the two ratios are equivalent. The steps to find equivalent ratios are:
- First convert the ratio into fractions.
- Now, multiply or divide the fraction with the same number.
Example: Hannah wants to make 5 batches of cookies, which calls for 5 cups of flour and 3 cups of butter. Hannah can use equivalent ratios to figure out how much flour and butter she’ll need.
To begin, write a ratio for the number of cups of flour and butter.
\( \frac{5~cups~of~flour}{3~cups~of~butter} \)
Next, find an equivalent ratio by multiplying the numerator and denominator by 5.
\( \frac{5\times 5}{3\times 5}=\frac{25}{15} \)
According to the equivalent ratio, Hannah will need 25 cups of flour and 15 cups of butter to make 5 batches of cookies.
Tape Diagram
A tape diagram is a representation of ratio or fraction as equal size parts.
The portion in the tape diagram represents the parts of the ratio.
Example: There are 3 plates for every 4 bowls in a hotel. The number of plates is 240. How many bowls are there in the hotel?
Solution: The ratio of plates in hotels to bowls in hotels is 3 : 4. Use the tape diagram to represent the condition.
First part represents \( \frac{240}{3}=80 \) number of plates.
So, the second part has 4 blocks, 4 × 80 = 320 bowls.
Solved Examples
Example 1: Find the ratio of horses to elephants when we have 4 horses and 16 elephants.
Solution:
We have 4 horses to 16 elephants
1 horse for every 4 elephants
So, we can write 1 : 4 as a ratio of horse to elephant.
Example 2: On a ring, the ratio of diamonds to rubies is 1 : 6. Find and interpret the ratio’s value.
Solution:
\( \frac{1}{6} \) is the value of the ratio 1 : 6. As a result, the multiplicative relationship is equal to \( \frac{1}{6} \).
Hence, the number of diamonds is \( \frac{1}{6} \) times the number of rubies.
Example 3: On a bracelet, the ratio of diamonds to rubies is 8. Find and interpret the ratio’s value.
Solution:
There are 8 diamonds per ruby because the number of diamonds is eight times the number of rubies.
So, the ratio of diamonds to rubies is 8 : 1.
Example 4: Determine if the given ratios are equivalent or not.
(a) 9 : 2 and 36 : 8
(b) 4 : 1 and 16 : 8
Solution:
Part (a)
We have 9 : 2 and 36 : 8
To get the numbers in the second ratio, multiply each number in the first ratio by four.
\( \frac{9}{2}\times \frac{4}{4}=\frac{36}{8} \)
As values of the ratio are equivalent the given ratio is equivalent.
Part (b)
We have 4 : 1 and 16 : 8
To get the numbers in the second ratio, multiply each number in the first ratio by different amounts.
\( \frac{4}{1}\times \frac{4}{8}=\frac{16}{8} \)
As values of the ratio are not equivalent. So, the given ratio is not equivalent.
Example 5: John is walking at a pace of seven meters per two seconds. Steve runs fourteen meters every five seconds. Is it true that they’re both speed walking at the same time? If not, who is the quickest?
Solution:
Write the ratio to represent the pace of both:
John’s pace = 7 : 2
Steve’s pace = 14 : 5
We have to find the equivalent ratios for the both to check whether they are speed walking at the same pace.
So, \( \frac{7}{2}\times \frac{2}{2.5}=\frac{14}{5} \)
As the multiplicand is different the ratio is not equivalent.
They are not walking at the same speed. So, take John’s pace and multiply it by the equivalent ratio and compare both ratios to decide who is faster.
Let the equivalent ratio be 2.
So, \( \frac{7}{2}\times \frac{2}{2}=\frac{14}{4} \)
Every four seconds, John walks 14 metres and walking the same distance takes 5 seconds for Steve.
Hence, John is walking faster than Steve.
Example 6: In a game, the tape diagram represents the ratio of blueberries to green apples caught. You managed to get ten green apples. How many blueberries did you manage to capture?
Solution:
The one part for green corresponds to ten apples. As a result, the three parts for blue represent, 3 × 10 = 30 blueberries.
Hence, you can get 30 blueberries.
Simplify a ratio by dividing all of the numbers in the ratio by the same number to get smaller numbers. All of the numbers in the ratio must be whole numbers. If no number larger than one can divide all of the numbers in the ratio, the ratio is fully simplified.
If a bowl of fruit contains three oranges and nine lemons, the ratio of oranges to lemons is three to nine or 3 : 9, which is equivalent to the ratio 1 : 3.
In a sports room, if there are two hockey sticks and eight baseball bats, the ratio of hockey sticks to baseball bats is two to eight or 2 : 8, which is equivalent to the ratio 1 : 4.
If the two ratios are equal to each other then they are called equivalent ratios.