Ratio
Consider the following: Isha’s weight is 25 kg and her father’s weight is 75 kg. How many times Father’s weight is of Isha’s weight?
It is
Cost of a pen is 10 rupees and cost of a pencil is 2 rupees. How many times the cost of a pen that of a pencil? Obviously it is
In the above examples, we compared the two quantities in terms of ‘how many times’. This comparison is known as the Ratio. We denote ratio using symbol ‘:’
Consider the earlier examples again. We can say,
The ratio of father’s weight to Isha’s weight =
The ratio of the cost of a pen to the cost of a pencil =
In a class, there are 20 boys and 40 girls.What is the ratio of number of girls to the total number of students ?
In a class, there are 20 boys and 40 girls. What is the ratio of the number of boys to the number of girls?
2. Ravi walks 6 km in an hour while Roshan walks 4 km in an hour. What is the ratio of the distance covered by Ravi to the distance covered by Roshan?
=
Now consider the following example.
Length of a house lizard is 20 cm and the length of a crocodile is 4 m.
“I am 5 times bigger than you”, says the lizard. As we can see this is really absurd. A lizard’s length cannot be 5 times of the length of a crocodile. So, what is wrong? Observe that the length of the lizard is in centimetres and length of the crocodile is in metres. So, we have to convert their lengths into the same unit.
Convert bigger unit to smaller unit.
Length of the crocodile = 4 m = 4 ×
Therefore, ratio of the length of the crocodile to the length of the lizard
=
Two quantities can be compared only if they are in the same unit.
Now what is the ratio of the length of the lizard to the length of the crocodile?
It is
Observe that the two ratios 1 : 20 and 20 : 1 are different from each other. The ratio 1 : 20 is the ratio of the length of the lizard to the length of the crocodile whereas, 20 : 1 is the ratio of the length of the crocodile to the length of the lizard.
Now consider another example.
Length of a pencil is 18 cm and its diameter is 8 mm. What is the ratio of the diameter of the pencil to that of its length?
Since the length and the diameter of the pencil are given in different units, we first need to convert them into same unit.
Thus, length of the pencil = 18 cm = 18 ×
The ratio of the diameter of the pencil to that of the length of the pencil
=
Try these
1. Saurabh takes 15 minutes to reach school from his house and Sachin takes one hour to reach school from his house. Find the ratio of the time taken by Saurabh to the time taken by Sachin.
(Note: 1 hour = 60 minutes)
Solution :
Time taken by Saurabh to reach school from his house =
Time taken by Sachin to reach school from his house =
= 1 x
Ratio of time taken by Saurabh to time taken by Sachin =
=
Hence, required ratio = 1:4
2. Cost of a toffee is 50 paise and cost of a chocolate is Rs. 10. Find the ratio of the cost of a toffee to the cost of a chocolate.
Solution:
Cost of a toffee =
Cost of a chocolate = ₹
Convert the given unit into same unit. We know that, ₹ 1 =
Therefore, ₹ 10 = 10 x
= 1000 paise
Ratio of the cost of a toffee to the cost of a chocolate =
(Enter fraction form)
Hence, the ratio of the cost of a toffee to the cost of a chocolate is
3. In a school, there were 73 holidays in one year. What is the ratio of the number of holidays to the number of days in one year?
Solution :
Number of holidays in one year =
Number of days in a year = 365
Ratio of the number of holidays to the number of days in a year =
=
Hence, required ratio = 1:5
Think of some more situations where you compare two quantities of same type in different units. We use the concept of ratio in many situations of our daily life without realising that we do so.
Compare the drawings A and B. B looks more natural than A. Why?
The legs in the picture A are
This is because we normally expect a certain ratio of the length of legs to the length of whole body.
Compare the two pictures of a pencil. Is the first one looking like a full pencil?
Why not?
Same ratio in different situations :
Consider the following :
● Length of a room is 30 m and its breadth is 20 m. So, the ratio of length of the room to the breadth of the room =
● There are 24 girls and 16 boys going for a picnic. Ratio of the number of girls to the number of boys =
The ratio in both the examples is 3:2.
Note: The ratios 30 : 20 and 24 : 16 in lowest form are same as 3 : 2. These are equivalent ratios.
Ravi and Rani started a business and invested money in the ratio 2 : 3.
After one year the total profit was Rs 4,00,000.
Ravi said “we would divide it equally”, Rani said “I should get more as I have invested more”.It was then decided that profit will be divided in the ratio of their investment.
Here, the two terms of the ratio 2 : 3 are 2 and 3.
Sum of these terms = 2 + 3 =
What does this mean?
This means if the profit is 5 rupees then Ravi should get 2 rupees and Rani should get 3 rupees. Or, we can say that Ravi gets 2 parts and Rani gets 3 parts out of the 5 parts.
i.e., Ravi should get
If the total profit were 500 rupees,
Ravi would get
and Rani would get
Now, if the profit were 4,00,000 rupees could you find the share of each?
Ravi’s share =
and Rani’s share =
Let us look at the kind of problems we have solved so far.
Example 1 :
Length and breadth of a rectangular field are 50 m and 15 m respectively. Find the ratio of the length to the breadth of the field
Solution :
Length of the rectangular field =
Breadth of the rectangular field =
The ratio of the length to the breadth is 50 : 15
We simplify this ratio by dividing the numerator and denominator by the greatest common factor for the numbers 15 and 50 i.e. 5. Thus,
the ratio can be written as =
Thus, the required ratio is 10:3.
Example 2 :
Find the ratio of 90 cm to 1.5 m.
Solution :
The two quantities are not in the same units. Therefore, we have to convert them into same units.
1.5 m = 1.5 ×
Therefore, the required ratio is 90 : 150.
Similar to the previous example, we can further simplify this ratio by dividing the numerator and denominator by the greatest common factor (for 90 and 150) which will be
Required ratio is 3:5.
Example 3 :
There are 45 persons working in an office. If the number of
females is 25 and the remaining are males, find the ratio of:
(a) The number of females to number of males.
(b) The number of males to number of females.
Solution :
Number of females =
Total number of workers =
Number of males = 45 – 25 =
Therefore, the ratio of number of females to the number of males = 25 : 20 =
And the ratio of number of males to the number of females = 20 : 25 =
Example 4 :
Give two equivalent ratios of 6 : 4.
Solution :
Ratio 6 : 4 =
Therefore,
Similarly, the ratio 6 : 4 =
So, 3:2 is another equivalent ratio of 6 : 4.
Therefore, we can get equivalent ratios by multiplying or dividing the numerator and denominator by the same number.
Example 5 :
Fill in the missing numbers :
Solution :
In order to get the first missing number, we consider the fact that
21 = 3 ×
.i.e. when we divide 21 by 7 we get 3. This indicates that to get the
missing number of second ratio, 14 must also be
When we divide, we have,
Hence, the second ratio is
Similarly, to get third ratio we multiply both numerator and denominator of second ratio by
as 2 x 3 will give us the desired given numerator in the third ratio i.e. 6
Hence, the third ratio is
Therefore,
Example 6 :
Ratio of distance of the school from Mary’s home to the distance of the school from John’s home is 2 : 1.
(a) Who lives nearer to the school?
(b) Complete the following table which shows some possible distances that Mary and John could live from the school.
| Distance from Mary’s home to school(in km.) | Distance from John’s home to school(in km.) |
|---|---|
Example 7 :
Divide Rs 60 in the ratio 1 : 2 between Kriti and Kiran.
Solution :
The two parts are 1 and 2.
Therefore, sum of the parts = 1 + 2 =
This means if there are Rs 3 , Kriti will get Rs 1 and Kiran will get Rs 2. Or, we can say that Kriti gets
Therefore, Kriti’s share =
And Kiran’s share =