A man who recently died left a sum of Rs. 3,90,000 to be divided among his wife, five sons and four daughters. He directed that each son should receive 3 times as much as each daughter receives and that each daughter should receive twice as much as their mother receives. What was the wife’s share?
Rs. 10,000
Understanding the Problem: Dividing Money Among Family
The question asks us to find the share of the wife from a total sum of Rs. 3,90,000 that is to be divided among his wife, five sons, and four daughters based on specific conditions regarding their respective shares. We are given relationships between the shares of sons, daughters, and the wife.
Defining the Relationships for Shares
Let's represent the share of each family member using variables based on the given directions:
- The man directed that each son should receive 3 times as much as each daughter receives.
- He also directed that each daughter should receive twice as much as their mother (wife) receives.
To make calculations easier, we can express all shares in terms of a single variable. Let's choose the wife's share as the base.
Let the wife's share be \(W\).
According to the second condition:
Each daughter's share = 2 \(\times\) Wife's share
Each daughter's share = \(2W\)
According to the first condition:
Each son's share = 3 \(\times\) Each daughter's share
Each son's share = 3 \(\times\) \((2W)\)
Each son's share = \(6W\)
Calculating the Total Amount Based on Shares
We have 1 wife, 5 sons, and 4 daughters. The total sum is Rs. 3,90,000. The total sum is the sum of the total amount received by the wife, the total amount received by all sons, and the total amount received by all daughters.
Total amount = (Number of wives \(\times\) Wife's share) + (Number of sons \(\times\) Each son's share) + (Number of daughters \(\times\) Each daughter's share)
Total amount = \((1 \times W) + (5 \times 6W) + (4 \times 2W)\)
Total amount = \(W + 30W + 8W\)
Total amount = \(39W\)
Solving for the Wife's Share
We know the total amount is Rs. 3,90,000. So, we can set up the equation:
\[39W = 3,90,000\]Now, we solve for \(W\), which is the wife's share:
\[W = \frac{3,90,000}{39}\] \[W = 10,000\]So, the wife's share is Rs. 10,000.
Verifying the Shares
Let's check if the total share matches the given amount using \(W = 10,000\).
- Wife's share: \(W = 10,000\)
- Each daughter's share: \(2W = 2 \times 10,000 = 20,000\)
- Each son's share: \(6W = 6 \times 10,000 = 60,000\)
Total amount distributed = (1 \(\times\) Wife's share) + (5 \(\times\) Each son's share) + (4 \(\times\) Each daughter's share)
Total amount distributed = \((1 \times 10,000) + (5 \times 60,000) + (4 \times 20,000)\)
Total amount distributed = \(10,000 + 300,000 + 80,000\)
Total amount distributed = \(390,000\)
This matches the total amount given in the question, Rs. 3,90,000. Therefore, the calculated wife's share is correct.
Revision Table: Money Division
| Family Member | Number | Share Relation (to Wife) | Share Amount (Rs.) | Total for Group (Rs.) |
|---|---|---|---|---|
| Wife | 1 | \(W\) | 10,000 | \(1 \times 10,000 = 10,000\) |
| Daughter | 4 | \(2W\) | 20,000 | \(4 \times 20,000 = 80,000\) |
| Son | 5 | \(6W\) | 60,000 | \(5 \times 60,000 = 300,000\) |
| Total Amount | \(10,000 + 80,000 + 300,000 = 390,000\) | |||
Additional Information: Ratio and Proportion Problems
This problem is a typical example of a ratio and proportion problem, often encountered in aptitude tests and general math.
- Ratio: A ratio compares two quantities. In this case, we established ratios between the shares (e.g., Son's share : Daughter's share = 3:1, Daughter's share : Wife's share = 2:1).
- Proportion: A proportion states that two ratios are equal. While not directly used as two equal ratios here, the relationships establish a proportional distribution of the total amount.
- Algebraic Approach: Using variables like \(W\) to represent the unknown base share (wife's share) is a common and effective method to solve such problems. By expressing all other shares in terms of this base variable, we can set up an equation representing the total amount and solve for the variable.
- Unit Method: Alternatively, one could think in terms of 'units'. If the wife gets 1 unit, a daughter gets 2 units, and a son gets 3 \(\times\) 2 = 6 units. The total units would be \((1 \times 1) + (5 \times 6) + (4 \times 2) = 1 + 30 + 8 = 39\) units. Since 39 units equal Rs. 3,90,000, 1 unit equals Rs. 3,90,000 / 39 = Rs. 10,000. The wife's share is 1 unit, which is Rs. 10,000. This unit method is essentially the same algebraic approach but visualised differently.
Understanding how to translate word problems into mathematical equations is key to solving these types of questions.
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