A sum of Rs. 6342 is divided amongst A, B, C and D in the ratio 3 : 4 : 8 : 6. What is the difference between the shares of B and D?
Rs. 604
Understanding Ratio and Proportion Problems
This question involves dividing a total sum of money among four individuals (A, B, C, and D) according to a given ratio. To find the share of each individual and the difference between any two shares, we first need to determine the value of one part of the ratio.
Calculating the Total Ratio Parts
The given ratio for the division among A, B, C, and D is 3 : 4 : 8 : 6.
To find the total number of equal parts into which the sum is divided, we add the individual parts of the ratio:
Total ratio parts = Ratio of A + Ratio of B + Ratio of C + Ratio of D
Total ratio parts = $3 + 4 + 8 + 6$
Total ratio parts = $21$
Determining the Value of One Ratio Part
The total sum to be divided is Rs. 6342. This total sum corresponds to the total ratio parts, which is 21.
To find the value of one ratio part, we divide the total sum by the total ratio parts:
Value of one ratio part = $\text{Total sum} \div \text{Total ratio parts}$
Value of one ratio part = $\text{Rs. } 6342 \div 21$
Let's perform the division:
| Calculation | Result |
|---|---|
| $6342 \div 21$ | $302$ |
So, the value of one ratio part is Rs. 302.
Calculating Individual Shares
Now that we know the value of one ratio part, we can find the share of each individual by multiplying their respective ratio part by the value of one part:
- A's share = Ratio of A $\times$ Value of one part = $3 \times 302$ = Rs. 906
- B's share = Ratio of B $\times$ Value of one part = $4 \times 302$ = Rs. 1208
- C's share = Ratio of C $\times$ Value of one part = $8 \times 302$ = Rs. 2416
- D's share = Ratio of D $\times$ Value of one part = $6 \times 302$ = Rs. 1812
Let's quickly verify the total sum: $906 + 1208 + 2416 + 1812 = 6342$. This matches the original sum, confirming our calculations for individual shares are correct.
Finding the Difference Between Shares of B and D
The question asks for the difference between the shares of B and D.
B's share is Rs. 1208.
D's share is Rs. 1812.
Difference = D's share - B's share
Difference = Rs. $1812 - \text{Rs. } 1208$
| Calculation | Result |
|---|---|
| $1812 - 1208$ | $604$ |
The difference between the shares of B and D is Rs. 604.
Alternatively, we could find the difference in their ratio parts first. The ratio of B is 4, and the ratio of D is 6. The difference in ratio parts is $6 - 4 = 2$. Since the value of one ratio part is Rs. 302, the difference in their shares is $2 \times 302 = 604$. This method is quicker when only the difference is required.
Conclusion on Share Difference
The difference between the shares of B and D is Rs. 604.
Revision Table: Ratio and Share Calculations
| Individual | Ratio Part | Share (Ratio Part $\times$ 302) |
|---|---|---|
| A | 3 | $3 \times 302 = 906$ |
| B | 4 | $4 \times 302 = 1208$ |
| C | 8 | $8 \times 302 = 2416$ |
| D | 6 | $6 \times 302 = 1812$ |
| Total | 21 | 6342 |
| Difference (D - B) | $6 - 4 = 2$ | $2 \times 302 = 604$ |
Additional Information on Ratio and Proportion
Ratio and proportion problems are common in quantitative aptitude sections of many exams. A ratio expresses the relative size of two or more values. For example, a ratio of 3:4:8:6 means that for every 3 units A receives, B receives 4 units, C receives 8 units, and D receives 6 units.
A proportion is a statement that two ratios are equal. While this specific problem focuses on dividing a quantity in a given ratio rather than equating two ratios, the underlying principle involves understanding how parts relate to the whole and how individual parts relate to each other.
When a sum is divided in a ratio $a : b : c : d$, the total number of parts is $a+b+c+d$. If the total sum is $S$, then the value of one part is $S / (a+b+c+d)$. The share of any person with a ratio part $k$ is $k \times (S / (a+b+c+d))$. The difference between the shares of two individuals with ratio parts $k_1$ and $k_2$ is $|k_1 - k_2| \times (S / (a+b+c+d))$.
Mastering these concepts is crucial for solving problems involving distribution, mixtures, and other applications of ratios.
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