RATIO AND PROPORTIONS
In Maths, ratio and proportion are one of the most essential concepts. The concept of ratio defines us to compare two quantities while the proportion is an equation which shows that two ratios are equivalent. Hence, it is easy to calculate the ratio of the given proportion.
Suppose a and b are two different numbers or integers, then the ratio of these two integers can be represented as a/b or a:b. Whereas, the proportion, on the other hand, is the relation between two ratios such as a:b::c:d or a/b = c/d, where a,b,c and d are integers.
Now, let us assume that, in proportion, the two ratios are a:b & c:d. The two terms ‘b’ and ‘c’ are called ‘means or mean term,’ whereas the terms ‘a’ and ‘d’ are known as ‘extremes or extreme terms.’
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a/b = c/d or a : b :: c : d |
Ratio and Proportion Tricks:
- If u/v = x/y, then uy = vx
- If u/v = x/y, then u/x = v/y
- If u/v = x/y, then v/u = y/x
- If u/v = x/y, then (u+v)/v = (x+y)/y
- If u/v = x/y, then (u-v)/v = (x-y)/y
- If u/v = x/y, then (u+v)/ (u-v) = (x+y)/(x-y), which is known as componendo -Dividendo Rule
- If u/v = v/x, then u/x = u2/v2
- If u/v = x/y, then u = x and v =y
- If a/(b+c) = b/(c+a) = c/(a+b) and a+b+ c ≠0, then a =b = c
Examples:
1. Are the ratios 4:5 and 8:10 said to be in Proportion?
Solution:
4:5= 4/5 = 0.8 and 8: 10= 8/10= 0.8
Since both the ratios are equal, they are said to be in proportion.
2. Are the two ratios 8:10 and 7:10 in proportion?
Solution:
8:10= 8/10= 0.8 and 7:10= 7/10= 0.7
Since both the ratios are not equal, they are not in proportion.
3. Given ratio are-
a:b = 2:3
b:c = 5:2
c:d = 1:4
Find a: b: c.
Solution:
Multiplying the first ratio by 5, second by 3 and third by 6, we have
a:b = 10: 15
b:c = 15 : 6
c:d = 6 : 24
In the ratio’s above, all the mean terms are equal, thus
a:b:c:d = 10:15:6:24
4. Out of the total students in a class, if the number of boys is 5 and the number of girls being 3, then find the ratio between girls and boys.
Solution: The ratio between girls and boys can be written as 3:5( Girls: Boys). The ratio can also be written in the form of factor like 3/5.
5. Two numbers are in the ratio 2 : 3. If the sum of numbers is 60, find the numbers.
Solution: Given, 2/3 is the ratio of any two numbers.
Let the two numbers be 2x and 3x.
As per the given question, the sum of these two numbers = 60
So, 2x + 3x = 60
5x = 60
x = 12
Hence, the two numbers are;
2x = 2 x 12 = 24
and
3x = 3 x 12 = 36
24 and 36 are the required numbers.
6.
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Salaries of Ravi and Sumit are in the ratio 2:3. If the salary of each is increased by Rs. 4000, the new ratio becomes 40:57. What is Sumit's salary? |
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A. |
Rs. 17,000 |
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B. |
Rs. 20,000 |
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C. |
Rs. 34,000 |
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D. |
Rs. 38,000 |
Solution:
Option(D) is correct
Let the original salaries of Ravi and Sumit be Rs. 2x and Rs. 3x respectively.
then,
2x+4000/3x+4000=40/57
⇒57×(2x+4000) =40×(3x+4000)
6x=68,000
3x=34,000
Sumit's present salary =(3x+4000)=Rs.(34000+4000)= Rs. 38,000
7.
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If 0.75:x::5:8, then xx is equal to: |
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A. |
1.12 |
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B. |
1.20 |
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C. |
1.25 |
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D. |
1.30 |
Solution:
Option(B) is correct
x×5x=0.75×8=1.2
8.
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The sum of three numbers is 98. If the ratio of the first to second is 2:3 and that of the second to the third is 5:8, then the second number is: |
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A. |
20 |
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B. |
30 |
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C. |
48 |
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D. |
58 |
Solution:
Option(B) is correct
Let the three parts be AA, BB, CC. Then,
A:B=2:3 and B:C=5:8=5×35:8×35
⇒A:B:C=2:3:245=10:15:24
⇒B=98×1549=30=3:245
9.
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If Rs. 782 be divided into three parts, proportional to (dfrac{1}{2}:dfrac{2}{3}:dfrac{3}{4})then the first part is: |
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A. |
Rs. 182 |
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B. |
Rs. 190 |
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C. |
Rs. 196 |
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D. |
Rs. 204 |
Solution:
Option(D) is correct
Given ratio = 12:23:34=6:8:9
The first part is Rs 782×623=Rs 204
10.
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A sum of money is to be distributed among A, B, C, D in the proportion of 5:2:4:3. If C gets Rs. 1000 more than D, what is B's share? |
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A. |
Rs. 500 |
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B. |
Rs. 1500 |
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C. |
Rs. 2000 |
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D. |
None |
Solution:
Option(C) is correct
Let the shares of A, B, C and D be Rs. 5x, Rs. 2x, Rs. 4x and Rs. 3x respectively.
Then, 4x−3x=1000
⇒x=1000.
B's share =Rs.2x=Rs.(2×1000)= Rs. 2000
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