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Comparing Quantities (same type)
Compare by Difference - by how much ?
e.g. Cost of a car is Rs 250000 and that of a motorbike is Rs 50000,Cost of car − Cost of bike = Rs (250000 − 50000) = Rs 200000
We say that, car is costlier than bike by = Rs 200000
Compare by Division - by how many times ?
If we compare the above example by division,Cost of car / Cost of bike = 250000/50000 = 5/1
We say that, car is 5 times costlier than bike.
➲ Same unit cancel each other out during division and the result we get is number .
Ratio
"When we compare two quantities in terms of 'how many times', the comparison is known as the Ratio."➣ The Ratio is the comparison by the division.
➣ We denote ratio by using symbol "∶"
➣ In earlier example, the ratio of cost of car to the cost of car = 250000/50000 = 5/1 = 5 : 1
➣ Two quantities can be compared only if they are of same type and in the same unit.
i.e.two lengths can be compared only when they are in same unit e.g., cm to cm, m to m etc.
[Rate: It is a special ratio which compare different kinds of quantities or units. The word "per" is used instead of "to". e.g., miles per hour (speed), miles per litre (mileage), price per kg (cost) etc.]
➣ The order in which quantities are taken to express their ratio is important, the ratio 3 : 2 is different from 2 : 3.
Equivalent Ratios
➣ Ratios which are same in their lowest form.e.g 30/20 = 3 : 2 ,
24/16 = 3 : 2 .
∴ 30/20 & 24/16 are equivalent ratios.
➣ We can get equivalent ratios by multiplying or diving the numerator and denominator by same number.
➣ It is same as equivalent fractions.
e.g., 9 : 6 = 9×2/6×2 = 18/12 = 18 : 12
∴ 18 : 12 is an equivalent ratio of 9 : 6
also, 9 : 6 = 9÷3/6÷3 = 3/2 = 3 : 2
So, 3 : 2 is another equivalent ratio of 9 : 6.
Try These
1. In a class, there are 20 boys and 40 girls.(a)What is the ratio of the number of boys to the number of girls?
(b)What is the ratio of the number of girls to the number of boys?
(c)What is the ratio of the number of boys to the total number of students?
2. 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.
3. Divide Rs 60 in the ratio 1 : 2 between Kriti and Kiran.
Answer
1. Number of boys = 20Number of girls = 40
Total number of students = 20+40 = 60
(a) Ratio of the number of boys to the number of girls = 20/40 = 1 : 2
(b) Ratio of the number of girls to the number of boys = 40/20 = 2 : 1
(c) Ratio of the number of boys to the total number of students = 20/60 = 1 : 3
2. Here times are in different units, to compare between them we must convert them to same unit.
Time taken by Saurabh to reach school = 15 minutes
Time taken by Sachin to reach school = 1 hour = 60 minutes∴ Ratio of the time taken by Saurabh to the time taken by Sachin = 15/60 = 1 : 4
3. Total part in which Rs 60 is divided = 1+2 =3
Kriti gets 1 part of it i.e., = 1/3 of Rs 60 = 1/3 × Rs 60 = Rs 20
Kiran gets 2 part of it i.e., = 2/3 of Rs 60 = 2/3 × Rs 60 = Rs 40
We can check our result as, Ratio of the amount Kriti gets to the amount Kiran gets = 20/40= 1/2.
Hence, our answer is correct.
Proportion
If two ratios are equal, we say that they are in proportion and use the symbol ‘::’ or ‘=’ to equate the two ratios.➣ 3, 10, 15 and 50 are in proportion which is written as 3 : 10 :: 15 : 50 and is read as 3 is to 10 as 15 is to 50 or it is written as 3 : 10 = 15 : 50
➣ Proportion has many applications, some of which are scale drawings, maps, constructions etc.
To make a drawing accurate, the dimensions in drawing must be in proportion with the dimensions in original i.e., ratio of different lengths/curves in drawing must be equal to the ratio of respective lengths/curves in original.
➣ National flags are always made in a fixed ratio of length to its breadth. These may be different for different countries but are mostly around 1.5 : 1 or 1.7 : 1.
➣ In a statement of proportion, the four quantities involved when taken in order are known as respective terms.
First and fourth terms are known as extreme terms.
Second and third terms are known as middle terms.
➣ Product of extremes = Product of means
➣ If two ratios are not equal, then we say that they are not in proportion.
➣ The order of terms in the proportion is important e.g., 2, 5, 6 and 15 are in proportion, but 2, 5, 15 and 6 are not, since 2/5 is not equal to 15/6 .
Try These
Check whether the given ratios are equal, i.e. they are in proportion. If yes, then write them in the proper form.1). 1 : 5 and 3 : 15
2). 2 : 9 and 18 : 81
3). 15 : 45 and 5 : 25
4). 4 : 12 and 9 : 27
5). Rs 10 to Rs 15 and 4 to 6.
Answer
1) 1 : 5 = 1/5 and 3 : 15 = 3/15 = 1/5∴ Ratios are equal, i.e. they are in proportion, 1 : 5 :: 3 : 15
2) 2 : 9 = 2/9 and 18 : 81 = 18/81 = 2/9
∴ Ratios are equal, i.e. they are in proportion, 2 : 9 :: 18 : 81
3) 15 : 45 = 15/45 = 1/3 and 5 : 25 = 5/25 = 1/5
∴ Ratios are not equal, i.e. they are not in proportion.
4) 4 : 12 = 4/12 = 1/3 and 9 : 27 = 9/27 = 1/3
∴ Ratios are equal, i.e. they are in proportion, 4 :12 :: 9 : 27
5) Rs 10 : Rs 15 = 10/15 = 2/3 and 4 : 6 = 4/6 = 2/3
∴ Ratios are equal, i.e. they are in proportion, Rs 10 : Rs 15 :: 4 : 6
Problem Solving
Unitary Method
"The method in which first we find the value of one unit and then the value of required number of units is known as Unitary Method."➤ In this method, we first found the value for one unit or the unit rate. This is done by the comparison of two different properties (or quantities) given.
➢We often use per to mean for each. e.g., km per hour, rupees per kg etc., denote unit rates.
➢Then we multiply the unit rate with required number of units whose value is to be found
➤ If the cost of 6 cans of juice is Rs 210, then what will be the cost of 4 cans of juice?
➣ The cost of 6 cans of juice = Rs 210
The cost of 1 cans of juice = Rs 210/6 = Rs 35
Then, The cost of 4 cans of juice = Rs 35 × 4
= Rs 140
∴ The cost of 4 cans of juice is Rs 140.
➤ Cost of 4 dozens bananas is Rs 60. How many bananas can be purchased for Rs 12.50?
➣ In Rs 60, no of bananas that can be purchased = 4 dozens = 4×12 = 48
In Re 1, no of bananas that can be purchased = 48/60 = 4/5
In Rs 12.50, no of bananas that can be purchased = 4/5 ×12.50
= 4 ×2.5
= 10
∴ 10 bananas can be purchased for Rs 12.50.
Algebraic Method
➣ First we assign a variable x to the quantity to be found .➣ Then, put it in the proportion according to the condition.
➣ Solve the equation for x. It is the required answer
➤ In a computer lab, there are 3 computers for every 6 students. How many computers will be needed for 24 students?
➣ Let, the computers needed for 24 students be x.
A/Q (According to question)
3/x = 6/24
⇒ 3×24 = x×6 (cross-multiplication)
⇒ 72 = 6x
⇒ 6x = 72 (re-arranging)
⇒ x = 72/6
⇒ x = 12
∴ The computers needed for 24 students = x = 12.
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