When a sum of money was equally distributed among 50 children, each child received Rs. 80. If the same amount is equally distributed among another group of children such that each child get Rs. 50, then what is the number of children in the second group?
80
Solving Money Distribution Problems
This problem involves understanding how a fixed sum of money is distributed among different numbers of children, resulting in different amounts received by each child. We are given information about the first distribution and asked to find the number of children in a second distribution given the amount each child receives.
Calculate the Total Sum of Money
In the first scenario, we know the number of children and the amount each child received. This allows us to calculate the total sum of money that was distributed.
- Number of children in the first group = 50 children
- Amount received by each child in the first group = Rs. 80
The total sum of money distributed is the product of the number of children and the amount each child received.
Total Sum $= \text{Number of children} \times \text{Amount per child}$
Total Sum $= 50 \times 80$
Total Sum $= 4000$
So, the total sum of money distributed was Rs. 4000.
Determine the Number of Children in the Second Group
The problem states that the same amount (the total sum we just calculated) is equally distributed among another group of children. We know the amount each child in this second group receives, and we need to find the number of children in this group.
- Total Sum of money = Rs. 4000
- Amount received by each child in the second group = Rs. 50
To find the number of children in the second group, we divide the total sum by the amount each child receives.
Number of children in the second group $= \frac{\text{Total Sum}}{\text{Amount per child}}$
Number of children in the second group $= \frac{4000}{50}$
Number of children in the second group $= \frac{400}{5}$
Number of children in the second group $= 80$
Thus, the number of children in the second group is 80.
Summary of Money Distribution Scenarios
| Scenario | Number of Children | Amount per Child | Total Sum |
|---|---|---|---|
| First Group | 50 | Rs. 80 | $50 \times 80 = 4000$ |
| Second Group | ? | Rs. 50 | Rs. 4000 (Same as first group) |
Using the relationship: Total Sum = Number of Children × Amount per Child, we found that for the second group, Number of Children = Total Sum / Amount per Child = $4000 / 50 = 80$.
Revision Table: Key Concepts
| Concept | Description | Formula/Relationship |
|---|---|---|
| Total Sum | The entire amount of money being distributed. | Total Sum = (Number of Recipients) × (Amount per Recipient) |
| Distribution | Dividing a total amount equally among a number of recipients. | Amount per Recipient = Total Sum / Number of Recipients |
| Finding Number of Recipients | Calculating how many individuals receive a share. | Number of Recipients = Total Sum / Amount per Recipient |
Additional Information: Inverse Proportion
This problem illustrates a concept called inverse proportion. When the total sum of money is fixed, the number of children and the amount each child receives are inversely proportional. This means if the amount per child decreases, the number of children who can receive a share increases proportionally, and vice versa.
- In our case, the amount per child decreased from Rs. 80 to Rs. 50.
- The ratio of the amounts is $80 : 50$, which simplifies to $8 : 5$.
- Since the relationship is inversely proportional, the ratio of the number of children in the first group to the second group will be the inverse ratio of the amounts, which is $5 : 8$.
- We know the number of children in the first group is 50. Let the number of children in the second group be $N_2$.
- The ratio is $50 : N_2 :: 5 : 8$.
- Using the property of ratios, $\frac{50}{N_2} = \frac{5}{8}$.
- Cross-multiplying gives $50 \times 8 = 5 \times N_2$.
- $400 = 5 \times N_2$.
- $N_2 = \frac{400}{5}$.
- $N_2 = 80$.
This confirms our previous calculation and demonstrates the inverse relationship between the amount per child and the number of children when the total sum is constant.
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