Understanding the Share Division Problem

The question asks us to divide a total sum of Rs. 46,800 among four people: A, B, C, and D. We are given specific ratios regarding their combined shares and the ratio between B and C's shares. We are also given the exact share of A. Our goal is to find the difference between A and B's shares (let's call this x), the difference between C and D's shares (let's call this y), and finally calculate the value of (x - y).

Let's break down the given information:

• Total sum = Rs. 46,800
• Participants: A, B, C, D
• Ratio of combined share of A and D to combined share of B and C is 8 : 5. This can be written as $(A + D) : (B + C) = 8 : 5$.
• Ratio of the share of B to that of C is 5 : 4. This can be written as $B : C = 5 : 4$.
• A's share = Rs. 18,400
• x is the difference between shares of A and B, so $x = |A - B|$.
• y is the difference between shares of C and D, so $y = |C - D|$.
• We need to find the value of $(x - y)$.

Calculating the Combined Shares (A+D) and (B+C)

The total sum is Rs. 46,800, and it is divided into two groups: (A + D) and (B + C) in the ratio 8 : 5. The total ratio parts are $8 + 5 = 13$.

The combined share of A and D is:

$\text{Share}(A + D) = \frac{8}{13} \times \text{Total sum}$

$\text{Share}(A + D) = \frac{8}{13} \times 46800$

$\text{Share}(A + D) = 8 \times \frac{46800}{13}$

$\text{Share}(A + D) = 8 \times 3600 = 28800$ Rs.

The combined share of B and C is:

$\text{Share}(B + C) = \frac{5}{13} \times \text{Total sum}$

$\text{Share}(B + C) = \frac{5}{13} \times 46800$

$\text{Share}(B + C) = 5 \times 3600 = 18000$ Rs.

Let's quickly check if these add up to the total sum: $28800 + 18000 = 46800$. Yes, they do.

Determining Individual Shares of A, B, C, and D

We already know A's share is Rs. 18,400.

We know that $\text{Share}(A + D) = 28800$ and $A = 18400$.

So, $18400 + D = 28800$.

D's share = $28800 - 18400 = 10400$ Rs.

Now let's find the shares of B and C. We know that $\text{Share}(B + C) = 18000$ and the ratio $B : C = 5 : 4$. The total ratio parts for B and C are $5 + 4 = 9$.

B's share is:

$\text{Share}(B) = \frac{5}{9} \times \text{Share}(B + C)$

$\text{Share}(B) = \frac{5}{9} \times 18000$

$\text{Share}(B) = 5 \times \frac{18000}{9}$

$\text{Share}(B) = 5 \times 2000 = 10000$ Rs.

C's share is:

$\text{Share}(C) = \frac{4}{9} \times \text{Share}(B + C)$

$\text{Share}(C) = \frac{4}{9} \times 18000$

$\text{Share}(C) = 4 \times 2000 = 8000$ Rs.

Let's summarize the individual shares:

Let's check if the sum of individual shares equals the total sum: $18400 + 10000 + 8000 + 10400 = 46800$. Yes, it matches the total sum.

Calculating x and y

x is the difference between the shares of A and B. $x = |A - B|$.

$x = |18400 - 10000|$

$x = |8400| = 8400$ Rs.

y is the difference between the shares of C and D. $y = |C - D|$.

$y = |8000 - 10400|$

$y = |-2400| = 2400$ Rs.

Finding the Value of (x - y)

Now we need to calculate $(x - y)$.

$x - y = 8400 - 2400$

$x - y = 6000$ Rs.

The value of $(x - y)$ is 6000.

Revision Table: Key Calculations

Additional Information on Ratio and Proportion

Ratio and proportion are fundamental concepts used to compare quantities and solve problems involving division in specific proportions. A ratio expresses the relationship between two or more quantities, often written as a:b or a/b. A proportion is an equation stating that two ratios are equal.

• Understanding Ratios: If a sum of money is divided in the ratio m:n, the total number of parts is m+n. The first share is $\frac{m}{m+n}$ of the total sum, and the second share is $\frac{n}{m+n}$ of the total sum.
• Combined Ratios: In this problem, we dealt with ratios of combined shares ($(A+D):(B+C)$) and ratios of individual shares ($B:C$). It's crucial to handle each ratio separately based on the total sum or the relevant combined sum it applies to.
• Difference: The difference between two quantities is found by subtracting the smaller quantity from the larger one, or by taking the absolute value of their subtraction, as done for x and y here ($|A-B|$ and $|C-D|$).

Solving problems like these requires careful breakdown of the given ratios and step-by-step calculation of each person's share before finding the required differences and their final calculation.