Anubhav spent 14% of his income on electricity bills, 28% on rent and 18% on shopping. If 4/5 of the remaining amount is ₹ 5120, how much did he spend on electricity bills?
₹ 2240
Solving Anubhav's Monthly Spending Problem
Let's break down Anubhav's income and expenses step by step to find out how much he spent on electricity bills. We are given the percentages of income spent on various items and the value of a fraction of the remaining amount.
Calculating Total Percentage of Income Spent
Anubhav spends his income on three main categories:
- Electricity bills: 14%
- Rent: 28%
- Shopping: 18%
To find the total percentage of income spent, we add these percentages together:
\(\text{Total percentage spent} = 14\% + 28\% + 18\%\)
\(\text{Total percentage spent} = 60\%\)
So, Anubhav spent a total of 60% of his income on these items.
Finding the Remaining Percentage of Income
After spending 60% of his income, the percentage of income remaining is calculated by subtracting the spent percentage from the total income percentage (which is 100%).
\(\text{Percentage remaining} = 100\% - \text{Total percentage spent}\)
\(\text{Percentage remaining} = 100\% - 60\%\)
\(\text{Percentage remaining} = 40\%\)
Anubhav has 40% of his income remaining.
Determining the Actual Remaining Amount
We are told that 4/5 of the remaining amount is ₹ 5120. Let the remaining amount be \(R\).
According to the question:
\(\frac{4}{5} \times R = \text{₹ } 5120\)
To find the full remaining amount \(R\), we can multiply both sides of the equation by \(\frac{5}{4}\):
\(R = \text{₹ } 5120 \times \frac{5}{4}\)
\(R = \text{₹ } \frac{5120 \times 5}{4}\)
\(R = \text{₹ } 1280 \times 5\)
\(R = \text{₹ } 6400\)
The remaining amount is ₹ 6400.
Calculating Anubhav's Total Monthly Income
We know that the remaining amount (₹ 6400) represents 40% of Anubhav's total income. Let Anubhav's total income be \(I\).
So, we can write this as:
\(40\% \text{ of } I = \text{₹ } 6400\)
To express 40% as a decimal or fraction:
\(\frac{40}{100} \times I = \text{₹ } 6400\)
\(0.4 \times I = \text{₹ } 6400\)
Now, to find \(I\), divide ₹ 6400 by 0.4:
\(I = \frac{\text{₹ } 6400}{0.4}\)
\(I = \frac{\text{₹ } 64000}{4}\)
\(I = \text{₹ } 16000\)
Anubhav's total monthly income is ₹ 16000.
Calculating the Electricity Bill Expense
The question asks for the amount Anubhav spent on electricity bills. We know he spent 14% of his total income on electricity.
\(\text{Electricity expense} = 14\% \text{ of Total Income}\)
\(\text{Electricity expense} = 14\% \text{ of } \text{₹ } 16000\)
Convert the percentage to a decimal or fraction:
\(\text{Electricity expense} = \frac{14}{100} \times \text{₹ } 16000\)
\(\text{Electricity expense} = 0.14 \times \text{₹ } 16000\)
\(\text{Electricity expense} = \text{₹ } 14 \times 160\)
\(\text{Electricity expense} = \text{₹ } 2240\)
Anubhav spent ₹ 2240 on electricity bills.
Revision Table: Summary of Anubhav's Financials
| Category | Percentage of Income | Amount Spent |
|---|---|---|
| Electricity Bills | 14% | ₹ 2240 |
| Rent | 28% | \(28\% \text{ of } 16000 = \frac{28}{100} \times 16000 = \text{₹ } 4480\) |
| Shopping | 18% | \(18\% \text{ of } 16000 = \frac{18}{100} \times 16000 = \text{₹ } 2880\) |
| Total Spent | \(14\% + 28\% + 18\% = 60\%\) | \(2240 + 4480 + 2880 = \text{₹ } 9600\) |
| Remaining Amount | \(100\% - 60\% = 40\%\) | \(40\% \text{ of } 16000 = \text{₹ } 6400\) |
| Total Income | 100% | ₹ 16000 |
Additional Information: Working with Percentages and Fractions
This problem involves understanding how percentages and fractions relate to a whole amount (the income). Here are some key points:
- A percentage is a part per hundred. For example, 14% means 14 out of 100, or \(\frac{14}{100}\).
- Fractions like 4/5 represent a part of a whole. If 4/5 of an amount is known, you can find the whole amount by multiplying the known part by the reciprocal of the fraction (in this case, \(\frac{5}{4}\)).
- When percentages of the same whole are added, the sum represents the total part of that whole.
- If you know the percentage that a certain amount represents, you can find the total whole by setting up a proportion or using division. For example, if 40% of income is ₹ 6400, then 1% of income is ₹ \(\frac{6400}{40}\), and 100% (the total income) is ₹ \(\frac{6400}{40} \times 100\). This is equivalent to ₹ \(\frac{6400}{0.4}\).
These concepts are fundamental for solving problems involving income, expenditure, profit, loss, and discounts.
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