The ratio of monthly income and expenditure of a person is 57 : 43. If he saves Rs. 42,000 per annum, What is his monthly income?
Rs. 14,250
Understanding the Income, Expenditure, and Savings Relationship
The question provides the ratio of a person's monthly income to their monthly expenditure and their annual savings. We need to find their monthly income using this information.
The basic relationship between income, expenditure, and savings is:
\(\text{Savings} = \text{Income} - \text{Expenditure}\)
Setting up the Ratio
The ratio of monthly income to expenditure is given as 57 : 43.
Let the monthly income be \(57x\).
Let the monthly expenditure be \(43x\).
Where \(x\) is a common multiplier.
Calculating Monthly Savings from the Ratio
Using the formula above, the monthly savings in terms of \(x\) will be:
\(\text{Monthly Savings} = \text{Monthly Income} - \text{Monthly Expenditure}\)
\(\text{Monthly Savings} = 57x - 43x\)
\(\text{Monthly Savings} = (57 - 43)x\)
\(\text{Monthly Savings} = 14x\)
Converting Annual Savings to Monthly Savings
The person saves Rs. 42,000 per annum (yearly). To find the monthly savings, we divide the annual savings by the number of months in a year (12).
\(\text{Monthly Savings} = \frac{\text{Annual Savings}}{12}\)
\(\text{Monthly Savings} = \frac{42000}{12}\)
\(\text{Monthly Savings} = 3500\)
So, the monthly savings are Rs. 3,500.
Finding the Value of the Multiplier (x)
We have two expressions for monthly savings: \(14x\) and Rs. 3,500. We can equate these to find the value of \(x\).
\(14x = 3500\)
To find \(x\), divide both sides by 14:
\(x = \frac{3500}{14}\)
\(x = 250\)
Calculating the Monthly Income
The monthly income was represented as \(57x\). Now that we know \(x = 250\), we can calculate the monthly income.
\(\text{Monthly Income} = 57x\)
\(\text{Monthly Income} = 57 \times 250\)
Let's perform the multiplication:
\(57 \times 250 = 57 \times (200 + 50)\)
\(= 57 \times 200 + 57 \times 50\)
\(= 11400 + 2850\)
\(= 14250\)
So, the monthly income is Rs. 14,250.
Summary of Steps
- Identify the ratio of monthly income to expenditure (57:43).
- Represent income and expenditure as \(57x\) and \(43x\).
- Calculate monthly savings in terms of \(x\): \(14x\).
- Convert annual savings (Rs. 42,000) to monthly savings: Rs. 3,500.
- Equate the two expressions for monthly savings: \(14x = 3500\).
- Solve for \(x\): \(x = 250\).
- Calculate the monthly income using \(57x\): \(57 \times 250 = 14250\).
| Item | Ratio Part | Value (in terms of \(x\)) | Calculated Value |
|---|---|---|---|
| Monthly Income | 57 | \(57x\) | \(57 \times 250 = 14250\) |
| Monthly Expenditure | 43 | \(43x\) | \(43 \times 250 = 10750\) |
| Monthly Savings | \(57 - 43 = 14\) | \(14x\) | \(14 \times 250 = 3500\) |
| Annual Savings | - | \(12 \times 14x\) | \(12 \times 3500 = 42000\) |
The monthly income of the person is Rs. 14,250.
Revision Table: Income Expenditure Savings
| Concept | Formula/Relation | Notes |
|---|---|---|
| Basic Relation | Savings = Income - Expenditure | Applicable for same time period (e.g., monthly, annual) |
| Income from Savings & Expenditure | Income = Savings + Expenditure | Useful for checking calculations |
| Expenditure from Income & Savings | Expenditure = Income - Savings | Also useful for checking calculations |
| Ratios | Part : Part | Used to represent proportional relationships; a multiplier (like \(x\)) is often used. |
| Annual vs. Monthly | Annual = Monthly \(\times\) 12 | Need to use consistent time periods for calculations. |
Additional Information: Percentage of Savings
We can also express savings as a percentage of income or expenditure, although it wasn't required in this problem.
- Savings Percentage of Income: \(\left(\frac{\text{Savings}}{\text{Income}}\right) \times 100\%\)
- In this case: \(\left(\frac{14x}{57x}\right) \times 100\% = \left(\frac{14}{57}\right) \times 100\% \approx 24.56\%\)
Understanding these relationships is key to solving problems involving personal finance ratios.
Important Questions from Simple Ratios
If x : y = 5 : 2, then the value of (8x + 9y) : (8x + 2y) is:
If the value of \(\frac{{p - q}}{q} = \frac{4}{3}\) .find p : q
Two numbers are in the ratio of 9 : 7. If the larger number is 56 more than one-seventh of the smaller, then what is the sum of the two numbers?
The ratio of two numbers A and B is 5: 8. If 5 is added to each of A and B, then the ratio becomes 2 : 3. The difference between A and B is:
In population of a city the ratio of men and women is 12 : 11. If total population of that city is 4,60,000, then the population of women is:
