Understanding the Family Expenditure Problem

This problem involves a family's monthly income and how it is allocated among various expenditures and savings. We are given the total income, the percentage saved, and several relationships between the amounts spent on different categories: food, health, education, entertainment, and rent. Our goal is to find the specific amount spent on education.

Calculating Total Expenditure

First, let's find out the total amount the family spends in a month. The total income is Rs. 35,000, and they save 8% of this income every month.

Savings = 8% of Rs. 35,000

We can calculate the savings amount using the formula:

\text{Savings} = \left( \frac{8}{100} \right) \times 35000

\text{Savings} = 8 \times 350 = 2800 \text{ Rs.}

The total expenditure is the total income minus the savings.

\text{Total Expenditure} = \text{Total Income} - \text{Savings}

\text{Total Expenditure} = 35000 - 2800 = 32200 \text{ Rs.}

So, the family spends a total of Rs. 32,200 on all the categories combined.

Setting Up Relationships Between Expenditures

Let's represent the expenditure on each category with variables:

• Expenditure on Food = \(F\)
• Expenditure on Health = \(H\)
• Expenditure on Education = \(E\)
• Expenditure on Entertainment = \(N\)
• Expenditure on Rent = \(R\)

We are given the following relationships:

1. The expenditure on health is 50% more than that of food: \(H = F + 0.50F = 1.5F\)
2. Food is three times the expenditure on entertainment: \(F = 3N \implies N = \frac{F}{3}\)
3. The expenditure on health is half of the expenditure on education: \(H = 0.5E \implies E = 2H\)
4. The expenditure on rent is one-third of the combined expenditure on food, health, and education: \(R = \frac{F + H + E}{3}\)

The sum of all expenditures equals the total expenditure:

F + H + E + N + R = 32200

Expressing All Expenditures in Terms of One Variable

To solve for the unknown values, let's express all expenditures in terms of a single variable, say \(F\) (expenditure on Food).

• \(H = 1.5F\)
• \(E = 2H = 2 \times (1.5F) = 3F\)
• \(N = \frac{F}{3}\)
• \(R = \frac{F + H + E}{3} = \frac{F + 1.5F + 3F}{3} = \frac{5.5F}{3}\)

Solving for the Variable F

Now substitute these expressions into the total expenditure equation:

F + H + E + N + R = 32200

F + 1.5F + 3F + \frac{F}{3} + \frac{5.5F}{3} = 32200

Combine the terms with \(F\) and the terms with \(F/3\):

(F + 1.5F + 3F) + \left( \frac{F}{3} + \frac{5.5F}{3} \right) = 32200

5.5F + \frac{F + 5.5F}{3} = 32200

5.5F + \frac{6.5F}{3} = 32200

To add the terms, find a common denominator (which is 3):

\frac{3 \times 5.5F}{3} + \frac{6.5F}{3} = 32200

\frac{16.5F + 6.5F}{3} = 32200

\frac{23F}{3} = 32200

Now, solve for \(F\):

23F = 32200 \times 3

23F = 96600

F = \frac{96600}{23}

Performing the division:

F = 4200 \text{ Rs.}

So, the expenditure on Food is Rs. 4200.

Calculating the Expenditure on Education

The question asks for the expenditure on education, which is represented by \(E\). We established the relationship \(E = 3F\).

E = 3 \times F

E = 3 \times 4200

E = 12600 \text{ Rs.}

The expenditure incurred on education is Rs. 12,600.

Verification (Optional but Recommended)

Let's quickly calculate all expenditures to ensure they add up correctly and the relationships hold:

• Food \(F\) = 4200
• Health \(H\) = 1.5 * 4200 = 6300 (50% more than 4200 is 4200 + 2100 = 6300. Correct.)
• Education \(E\) = 3 * 4200 = 12600 (Health is half of Education: 6300 = 0.5 * 12600. Correct.)
• Entertainment \(N\) = 4200 / 3 = 1400 (Food is three times Entertainment: 4200 = 3 * 1400. Correct.)
• Rent \(R\) = (F + H + E) / 3 = (4200 + 6300 + 12600) / 3 = 23100 / 3 = 7700 (Rent is one-third of combined F, H, E. Correct.)

Total Expenditure = \(F + H + E + N + R = 4200 + 6300 + 12600 + 1400 + 7700 = 32200\).

Total Income = 35000, Savings = 2800. Total Expenditure = 35000 - 2800 = 32200. The total expenditure matches. All conditions are satisfied.

Revision Table: Key Information

Additional Information: Solving Word Problems

Solving word problems involving percentages and relationships requires a systematic approach:

1. Read Carefully: Understand the total amount, the parts it's divided into, and all the given relationships.
2. Identify the Unknowns: Determine what you need to find.
3. Assign Variables: Use letters to represent the unknown quantities. Choose one key variable if possible.
4. Translate to Equations: Write down the given relationships as mathematical equations. Percentages should be converted to decimals or fractions.
5. Solve the System: Use substitution or elimination methods to find the values of the variables. If you expressed everything in terms of one variable, solve that equation first.
6. Calculate the Final Answer: Use the values of the variables to find the specific quantity requested in the question.
7. Verify: Plug your calculated values back into the original relationships and the total to ensure consistency. This helps catch errors.

In this problem, setting up the relationships correctly was crucial. Expressing all variables in terms of Food (F) allowed us to form a single equation with one unknown, which we could then solve.