Determining Ratios
A ratio is a comparison between two (2) quantities, and it is often expressed as a fraction, eg. \(\frac{3}{4}\), or a decimal, 0.75. We can also express them in percentages as fractions can be converted to percentages by multiplying them by 100%.
For instance, a ratio of \(3 : 4\) or \(\frac{3}{4}\) means that for every three (3) units of one quantity, there are four (4) units of another quantity.
Ratios are often used when it comes to sharing.
Assuming your mother brought home 15 toffees to be shared between you and your brother. Your brother, who is 2 times older than you, could say that once he is older than you, he needs to get more share of the toffees than you.
Your mother could decide to share the toffees according to your ages. Since your brother is twice (2 times) your age, every time your mother gives you one of the toffees, she will give your brother 2.
This method of sharing will continue until the toffees are exhausted.
Mathematically, we can say that, your mother shared the toffees in the ratio 1 is to 2 (that is, 1:2).
Let's practice some questions to get more understanding of ratios.
Example 1
What is the ratio of boys to girls, if there are 60 boys and 120 girls in a school?
Solution
Number of boys \(= 60 \)
Number of girls \(= 120\)
\(\therefore\) Ratio of boys to girls \(\Rightarrow 60 : 120 \ or \ \frac{60}{120} \)
\(\Rightarrow\) boys : girls \(= 1:2 \ or \ \frac{1}{2}\)
\(\therefore\) The ratio of boys to girls is \(1:2\)
Example 2
Yaw is 6 years old and Kofi is 10 years old. What is the ratio of Yaw's age to Kofi's age?
Solution
Yaw's age \(= 6\)
Kofi's age \(= 10\)
\(\therefore\) Ratio of Yaw's age to Kofi's age
\(\Rightarrow\) Yaw : Kofi \(= 6:10\)
\(\hspace{3.2cm}\) \(= 3 : 5 \ or \ \frac{3}{5} \)
\(\therefore\) The ratio of Yaw's age to Kofi's age is \(3:5\)
We can also use ratios to determine the value of an unknown quantity when one quantity is given.
Consider the examples below:
Example 3
Alhassan is twice as old as his son Musah. If Musah is 25 years of age, how old is his father, Alhassan?
Solution
Musah's ratio \(= 1\)
\(\therefore\) Alhassan's ratio \(= 2\)
let Alhassan's age \(= x\)
\(\Rightarrow 1:2 = 25 : x\)
\(\hspace{1.4cm} \frac{1}{2} = \frac{25}{x}\)
\(\hspace{0.5cm}1 \times x = 2 \times 25\) (Cross multiplication)
\(\hspace{1.5cm}x = 50\)
\(\therefore\) Alhassan is 50 years of age.
Sometimes, we need to convert two (2) quantities who have different units to a common unit in order to find their ratios.
Consider the examples below:
Example 4
In a competition, Kwame run 4km and Joe run 600m. What is the ratio of Kwame's distance to Joe's distance?
Solution
We need to first convert one of the distances, so they are both expressed in the same unit. Let's convert the bigger unit (kilometer) to the smaller unit (meter).
\(1 km = 1000 m\)
\(\therefore 4 km = \frac{4 km}{1 km} \times 1000 m\) (If more, less divide)
\(\hspace{2.2cm} = 4 \times 1000 m\)
\(\hspace{2.2cm}= 4000 m\)
\(\therefore\) Kwame's distance is \(4000 m\).
Ratio of their distances:
\(\Rightarrow\)Kwame's distance : Joe's distance
\(\hspace{3.2cm} 4000 m : 600 m\)
\(\hspace{4.4cm} 40 : 6 \) (Breaking it down)
\(\hspace{4.4cm} 20 : 3 \)
\(\therefore\) Ratio of Kwame's distance to Joe's distance is \(20:3\).
Example 5
Express 80cm to 40m as a ratio in its lowest form.
Solution
Like we did in the previous example, let's convert the bigger unit to the smaller unit.
\(1 m = 100 cm\)
\(\therefore 40m = \frac{40m}{1m} \times 100cm\) (If more, less divide)
\(\hspace{2.2cm} = 40 \times 100cm\)
\(\hspace{2.2cm} = 4000cm\)
This implies;
\(80cm : 40m = 80cm : 4000cm\)
\(\hspace{3.7cm} = 80:4000\)
\(\hspace{3.7cm} = 8:400\) (breaking it down)
\(\hspace{3.7cm} = \frac{8}{400}\) (Use 8 to divide through)
\(\hspace{3.7cm} = \frac{1}{50}\)
\(\hspace{3.7cm} = 1:50\) (Change it back to ratios)
\(\therefore\) the ratio of 80cm to 40m is \(1:50\)
Also, we may sometimes wish to express one quantity in terms of another quantity before we use ratios to find their values. Example 3 above can be solved differently using this idea.
Solution
let Musah's age \(= x\)
\(\Rightarrow\) Alhassan's age \(= 2x\)
\(\therefore\) If \(x = 25\)
It implies \(2x = 2 \times x\)
\(\hspace{3.8cm} = 2 \times (25)\)
\(\hspace{3.8cm} = 50\)
\(\therefore\) Alhassan is 50 years of age.