Ratio and Proportion are explained majorly based on fractions. When a fraction is represented in the form of a:b, then it is a ratio whereas a proportion states that two ratios are equal.
The ratio is the comparison between similar types of quantities, it is an abstract quantity and does not have any units.
1. A ratio remains the same if both antecedent and consequent are multiplied or divided by the same non-zero number,
2. Two ratios in their fraction notation can be compared just as we compare real numbers.
3. If two ratios a/b and c/d are equal
4. Key Points to Remember about Ratio:
(4 / 15) A = (2 / 5) B A = (2 / 5) * (15 / 4) B A = (3 / 2) B ⇒ A / B = 3 : 2 B's share = Rs. 1210 * (2 / 5) = Rs. 484
(4 / 15) A = (2 / 5) B
A = (2 / 5) * (15 / 4) B
A = (3 / 2) B
⇒ A / B = 3 : 2
B's share = Rs. 1210 * (2 / 5) = Rs. 484
The compounded ratio of the ratios:
(a : b), (c : d), (e : f) is (ace : bdf)
Quantity of milk = (60 * 2 / 3) litres = 40 litres Quantity of water in it = (60 -40) litres = 20 litres New ratio = 1 : 2 Let the quality of water to be added further be x litres. Then, milk : water = 40 / ( 20 + x) Now, 40 / ( 20 + x) = 1 / 2 ⇒ 20 + x = 80 ⇒ x = 60 ∴ Quantity of water to be added = 60 litres
Quantity of milk = (60 * 2 / 3) litres = 40 litres
Quantity of water in it = (60 -40) litres = 20 litres
New ratio = 1 : 2
Let the quality of water to be added further be x litres.
Then, milk : water = 40 / ( 20 + x)
Now, 40 / ( 20 + x) = 1 / 2
⇒ 20 + x = 80
⇒ x = 60
∴ Quantity of water to be added = 60 litres
The equality of two ratios is called proportion i.e. If a/b = c/d, then a, b, c, d are said to be in proportion.
Product of means = Product of extremes
Thus, a : b :: c : d ⇔ (b x c) = (a x d)
Example 1: Let us consider one more example of a number of students in a classroom. Our first ratio of the number of girls to boys is 3:5 and that of the other is 4:8, then the proportion can be written as:
3 : 5 :: 4 : 8 or 3/5 = 4/8
Here, 3 & 8 are the extremes, while 5 & 4 are the means.
Let the third number be x. Then, first number = 120% of x = (120x / 100) = 6x / 5 Second number = 150% of x = (150x / 100) = 3x / 2 ∴ Ratio of first two numbers = (6x / 5 : 3x / 2) = 12x : 15x = 4 : 5
Let the third number be x.
Then, first number = 120% of x = (120x / 100) = 6x / 5
Second number = 150% of x = (150x / 100) = 3x / 2
∴ Ratio of first two numbers = (6x / 5 : 3x / 2) = 12x : 15x = 4 : 5
If a:b = c:d is a proportion, then
To understand the concept of ratio and proportion, go through the difference between ratio and proportion given here.
|1||The ratio is used to compare the size of two things with the same unit||The proportion is used to express the relation of two ratios|
|2||It is expressed using a colon (:), slash (/)||It is expressed using the double colon (::) or equal to the symbol (=)|
|3||It is an expression||It is an equation|
|4||Keyword to identify ratio in a problem is “to every”||Keyword to identify proportion in a problem is “out of”|
The LCM process gets very cumbersome when we have to find the ratio out of multiple ratios.
We have the following simple method for that for a chain of ratios of any length.
Suppose you have the ratio train as follows
⇨ A : B = 1 : 2
⇨ B : C = 2 : 3
⇨ C : D = 5 : 6
⇨ D : E = 7 : 8
If we were to find A : B : C : D : E, then the LCM method would have taken quite a long time which is infeasible in examinations of limited hours.
The short cut is as follows:
A : B : C : D : E can be written directly as:
⇨ 1 × 2 × 5 × 7 : 2 × 2 × 5 × 7 : 2 × 3 × 5 × 7 : 2 × 3 × 6 × 7 : 2 × 3 × 6 × 8
⇨ 70 : 140 : 210 : 252 : 288
The thought algorithm for this case goes as:
To get the combined ratio of A : B : C : D : E, from A : B, B : C, C : D, and D : E
In the combined ratio of A : B : C : D : E.
Q.1. In a library, the ratio of number of story books to that of non-story books was 4:3 and total number of story books was 1248. When some more story books were bought, the ratio became 5:3. Find the number of story books bought.
Answer: Option A
Q.2. Rs. 8400 is divided among A, B, C and D in such a way that the shares of A and B, B and C, and C and D are in the ratios of 2:3, 4:5, and 6:7 respectively. The share of A is
Answer: Option A
Q.3. The ratio of the present age of father to that of son is 7:2. After 10 years their ages will be in the ratio of 9:4. The present ages of the father is
Answer: Option A
Q.4. Ajay and Raj together have Rs. 1050. On taking Rs. 150 from Ajay, Ajay will have same amount as what Raj had earlier. Find the ratio of amounts with Ajay and Raj initially.
Answer: Option D
Q.5. Price of each article of type P, Q, and R is Rs. 300, Rs. 180 and Rs. 120 respectively. Suresh buys articles of each type in the ratio 3:2:3 in Rs. 6480. How many articles of type Q did he purchase?
Answer: Option A
Q.6. Divide Rs. 60 in the ratio 1:2 between Mike and John.
Solution: Let Mike’s part be x.
Then John’s part is 2x.
Thus, x+2x = 60
3x = 60
x = (60/3)
x = 20.
Therefore, Mike’s part = x = Rs. 20
John’s part = 3x = Rs. (2*20) = Rs. 40
Q.7. Three Jars contain alcohol to water in the ratios 3:5, 1:3, and 1:1. If all the three solutions are mixed, what will be the ratio of alcohol to water in final solution?
Solution: Here we are not given the quantities of the solution in three jars. Only the ratio of alcohol to water is given. If the ratio of the quantity of solution would have been there, we could determine the ratio of alcohol to water in the final solution. Hence, the answer here will be cannot be determined.
Q.8. If there are Rs. 495 in a bag in denominations of one-rupee, 50-paisa, and 25-paisa coins which are in the ratio 1:8:16. How many 50 paisa coins are there in bag?Solution: Assume, you have x numbers of one rupee coin. Now coins are in the ratio 1:8:16. This means that if we have x number of one rupee coins, we have 8x number of 50 paisa coins and 16x number of 25 paisa coins. Here order in which ratios are mentioned in the question is very important. In this case, order is one rupee, 50 paisa and 25 paisa and ratio is 1:8:16. Thus,