Dividing a Sum of Money According to Given Ratios

In this article, we will explore the problem of dividing a sum of money among multiple parties based on given ratios. This involves basic arithmetic and ratio calculations, which are essential skills in various real-world scenarios, including finance, economics, and everyday budgeting.

Let's start with the first problem: A, B, and C are given a sum of money, 12240, to be divided among them such that A gets 2/3 of what B gets, and B gets 3/4 of what C gets. How much is the share of C?

Solution

We can denote the shares of A, B, and C as A, B, and C respectively. The problem statements can be translated into mathematical equations as follows:

A (frac{2}{3}B) B (frac{3}{4}C)

Our goal is to find the share of C (C). We begin by expressing everything in terms of C.

Step 1: Express B in terms of C

From the equation B (frac{3}{4}C), we can directly express B. Let us denote this as:

B (frac{3}{4}C)

Step 2: Express A in terms of C

Next, we use the equation A (frac{2}{3}B). Substituting the value of B from Step 1:

A (frac{2}{3} times frac{3}{4}C)

A (frac{2 times 3}{3 times 4}C)

A (frac{1}{2}C)

Now we have the shares in terms of C:

A (frac{1}{2}C) B (frac{3}{4}C) C C

Step 3: Find the total sum in terms of C

The total sum is given as 12240. We can now express the total sum as:

A B C (frac{1}{2}C frac{3}{4}C C)

To combine these, we need a common denominator. The least common multiple of 2, 4, and 1 is 4. Rewriting each term:

A (frac{1}{2}C frac{2}{4}C)

B (frac{3}{4}C)

C (frac{4}{4}C)

Adding them together:

A B C (frac{2}{4}C frac{3}{4}C frac{4}{4}C frac{9}{4}C)

Step 4: Solve for C

Given that the total sum is 12240, we can set up the equation:

(frac{9}{4}C 12240)

Multiplying both sides by (frac{4}{9}) gives us:

C 12240 times (frac{4}{9})

Calculating the above expression:

C 1360 times 4 5440

Therefore, the share of C is 5440.

Additional Examples

Problem 2

Given A:B 6:5 and B:C 10:9, find the share of C if the total amount is 1280 rupees.

We can rewrite the ratios as follows:

A:B 6:5 Rightarrow A:B 12:10 Therefore, A:B:C 12:10:9

Let the shares of A, B, and C be 12x, 1, and 9x respectively. The total sum is 1280, which gives us the equation:

12x 1 9x 1280

Solving for x:

31x 1280

x (frac{1280}{31}) 40

Therefore, the share of C is 9 times 40 360 rupees.

Problem 3

Given A:B 6:5 and B:C 10:9, solve for C.

We can rewrite the ratios as follows:

A:B 6:5 Rightarrow A:B 12:10 Therefore, A:B:C 12:10:9

Let the shares of A, B, and C be 12x, 1, and 9x respectively. The total sum ABC is 1240, which gives us the equation:

12x 1 9x 1240

Simplifying, we get:

31x 1240

x (frac{1240}{31}) 40

Therefore, the share of C is 9 times 40 360 rupees.

Problem 4

Given a total amount of Rs. 1900, and the share of A is 1 1/2 times the share of B, and the share of B is 3/2 times the share of C, find the share of C.

The ratios can be set up as follows:

A:B 3:2 B:C 3:2

The combined ratio is A:B:C 3 times 3 : 2 times 3 : 2 times 2 9:6:4

Let the shares of A, B, and C be 9x, 6x, and 4x respectively. The total sum is Rs. 1900, which gives us the equation:

9x 6x 4x 1900