|Table of contents|
|Basics of Ratio|
|Ratio & Proportion|
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When two ratios are equal, the four terms involved, taken in order are called proportional, and they are said to be in proportion.
If a, b, c are in continued proportion then second quantity ‘b’ is called the mean proportional between ‘a’ and ‘c’ and a, b and c are known as 1st, 2nd and 3rd proportion respectively.
Example 1. What is the mean proportional to 5/36, 3.2/25?
Let the mean proportion of 5/36 and 3.2/25 be x
Example 2. What should be added to each of the numbers 19, 26, 37, 50 so that the resulting number should be in proportion?
(c) - 2
(d) - 5
Let the required number be x.
According to question,
19 + x, 26 + x, 37 + x and 50 + x are in proportion
19 + x : 26 + x = 37 + x : 50 + x
(19 + x) (50 + x) = (37 + x) (26 + x)
⇒ x2 + 69x + 950 = x2 + 63x + 962
x2 - x2 + 69x - 63x = 962 - 950
Hence the resulting number = 2.
Basics of Proportion
(i) Let us take four quantities a, b, c & d such that they are in proportion i.e. a : b : : c : d then,
This operation is called componendo and Dividendo.
(ii) If a/b = c/d = e/f = then
Partnership & Share
If there is profit in the business run by two partners A and B then,
Example 3. Saman begins business with a capital of Rs. 50,000 and after 3 months takes Manu into partnership with a capital of Rs. 75000. Three months later Amandeep joined the firm with a capital of Rs. 1, 25,000. At the end of the year the firm makes a profit of Rs. 99,495. How much of this sum should Amandeep receive?
Solution. Money invested by Saman for 12 months = Rs. 50,000
Money invested by Manu for 9 months = Rs. 75000
Money invested by Amandeep for 6 months = Rs. 1,25,000
Share of Saman: Manu: Amandeep
= 50,000 × 12 : 75,000 × 9 ; 1,25,000 × 6
= 6,00,000 : 6,75,000 : 7,50,000 = 600 : 675 : 750 = 8 : 9 : 10
Total profit = Rs. 99,495.
Profit of Amandeep
If x is directly proportional to y, x1/y1 = x2/y2
If x is in direct relation to y then, x = K1 + yK2
Example 4. If 6 men can lay 8 bricks in one day, then how many men are required to lay 60 bricks in the same time?
(a) 45 men
(b) 40 men
(c) 60 men
(d) 50 men
Since the time is same so to do more work we need more persons. Hence this is the problem of direct
M1/B1 = M2/B2 ⇒ M2 = 6 × 60/8 = 45 men
Example 5. The cost of New Year party organized in TCY is directly related to the number of persons attending that party. If 10 persons attend the party the cost per head is Rs 250 and if 15 people attend, the cost per head is Rs. 200. What will be the total cost of the party if 20 persons attend it?
Solution. This is the problem of direct relation
Let the total cost of party is
Cost = K1 + K2 N (where K1 & K2 are fixed and variable costs and N is number of persons)
250 × 10 = K1 + 10K2 .......... (1)
200 × 15 = K1 + 15K2 .......... (2)
Solving them we get K1 = 1500 and K2 = 100
So total cost for 20 persons = 1500 + 20 × 100 = Rs. 3500
If x is inversely proportional to y, then xy = K
⇒ x1y1 = x2y2
Example 6. If 6 men can build a wall in 9 days then 60 men can build a similar wall in ______ days?
Solution. Work = Men x Days
Example 7. A can do a piece of work in 12 days, B is 60% more efficient than A. Find the number of days required for B to do the same piece of work.
Solution. Ratio of the efficiencies is A : B = 100 : 160 = 5 : 8. Since efficiency is inversely proportional to the
number of days, the ratio of days taken to complete the job is 8 : 5
So, the number of days taken by
Example 8. Ten years ago, the ratio of ages of A and B is 3 : 4, now, it is 4 : 5. What is the present age of A?
Solution. 10 years ago, let their ages be 3k and 4k.
So, present ages are 3k + 10 and 4k + 10.
The ratio is given as 4 : 5.
k = 10
Percentage of A = 3k + 10 = 40 years.
Example 9. Three friends A, B, C earn Rs. 2000 together. If they want to distribute this money such that ‘A’ should get Rs. 300 more than B and Rs. 100 more than C, in what ratio, they have to distribute the money?
Solution. A = 300 + B = 100 + C
So, A = 300 + B and C = 200 + B
A + B + C = 2000
∴ B = 500 ⇒ A = 800 and C = 700 So, the required ratio = 8 : 5 : 7
Example 10. Four friends A, B, C, D share Rs. 10,500 in the ratio 1/3 : 1/4 : 1/6 : 1/8 how much more money A and C together get than B and D together?
Solution. The given ratio is
Multiply with 24 = 8 : 6 : 4 : 3.
Take A = 8k, B = 6k, C = 4k, D = 3k.
Total = 21k = 10,500
⇒ k = 500
Required answer = (A + C) - (B + D)
= (8k + 4k) - (6k + 3k) = 3k = Rs. 1500