A father gives 8% of his monthly income to both his sons as pocket money. The elder son gets 85% of the total amount given to both sons. He spends 90% of the amount and saves Rs. 17. What is the monthly income of his father?
Let's break down this percentage problem step by step to find the father's monthly income. We know how much the elder son saves, and we can work backwards through the percentages to find the original amount.
Understanding the Problem Percentages
The problem gives us information about percentages at different levels:
- Percentage of father's income given as pocket money to both sons.
- Percentage of total pocket money given to the elder son.
- Percentage of the elder son's share that is spent (which implies the percentage saved).
Step-by-Step Calculation of Father's Income
We start with the amount the elder son saves and use the given percentages to find the larger amounts.
Step 1: Calculate the Elder Son's Total Share
The elder son spends 90% of his share. This means he saves the remaining percentage. Savings percentage = 100% - Spending percentage
Savings percentage = $100\% - 90\% = 10\%$.
We are given that the elder son saves Rs. 17. This Rs. 17 is 10% of his total share of pocket money.
Let the elder son's total share be $E$.
$10\%$ of $E = 17$
In decimal form, $0.10 \times E = 17$.
To find $E$, we divide 17 by 0.10:
$E = \frac{17}{0.10} = 170$.
So, the elder son's total share of the pocket money is Rs. 170.
Step 2: Calculate the Total Pocket Money for Both Sons
The elder son gets 85% of the total amount given to both sons. His share is Rs. 170.
Let the total pocket money given to both sons be $T$.
The elder son's share ($E$) is 85% of the total pocket money ($T$).
$85\%$ of $T = E$
$85\%$ of $T = 170$
In decimal form, $0.85 \times T = 170$.
To find $T$, we divide 170 by 0.85:
$T = \frac{170}{0.85} = 200$.
The total pocket money given to both sons is Rs. 200.
Step 3: Calculate the Father's Monthly Income
The father gives 8% of his monthly income as pocket money to both sons. This total pocket money is Rs. 200.
Let the father's monthly income be $M$.
The total pocket money ($T$) is 8% of the father's monthly income ($M$).
$8\%$ of $M = T$
$8\%$ of $M = 200$
In decimal form, $0.08 \times M = 200$.
To find $M$, we divide 200 by 0.08:
$M = \frac{200}{0.08} = 2500$.
The father's monthly income is Rs. 2,500.
Summary of Calculations
We can summarize the steps and results in a table:
| Step | Description | Calculation | Result |
|---|---|---|---|
| 1 | Elder son's savings percentage | $100\% - 90\% = 10\%$ | 10% |
| 2 | Elder son's total share | Savings / 10% = $17 / 0.10$ | Rs. 170 |
| 3 | Total pocket money for both sons | Elder son's share / 85% = $170 / 0.85$ | Rs. 200 |
| 4 | Father's monthly income | Total pocket money / 8% = $200 / 0.08$ | Rs. 2,500 |
Based on the calculations, the father's monthly income is Rs. 2,500.
Revision Table - Key Percentages and Amounts
| Item | Amount/Percentage | Related to |
|---|---|---|
| Father's Monthly Income | Rs. 2,500 | Base value |
| Total Pocket Money (8% of Income) | Rs. 200 | 8% of Rs. 2,500 |
| Elder Son's Share (85% of Total Pocket Money) | Rs. 170 | 85% of Rs. 200 |
| Elder Son's Savings (10% of his Share) | Rs. 17 | 10% of Rs. 170 |
Additional Information - Percentage Calculations
Understanding how to work with percentages is crucial for solving problems like this. Here are some key points:
- A percentage is a fraction out of 100. For example, 8% means $\frac{8}{100}$ or 0.08.
- To find a percentage of a number, multiply the number by the percentage in decimal form. For example, 8% of 2500 is $0.08 \times 2500$.
- To find the original number when you know a percentage of it, divide the known value by the percentage in decimal form. For example, if 10% of a number is 17, the number is $\frac{17}{0.10}$.
- Working backwards from a known value (like savings) through the percentages is a common strategy for these types of problems.
This question involves successive percentages applied to decreasing amounts. It's important to calculate each step based on the correct intermediate total.
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