Question

Adam's savings are equal to Ben's expenditure, which in turn is equal to Mary's savings. If Mary's savings are Rupees 50,000 and the incomes of Adam, Ben, and Mary are in the ratio 3:1:4, Mary's expenditure is less than thrice of Adam's expenditure and twice of Adam's expenditure is less than two times Ben's income, then which of the following could be true about Ben's income (\(I_B\))?

• A. \(18000<I_B<22000\)
• B. \(20000<I_B<25000\)
• C. \(23000<I_B<28000\)
• D. Data Inconsistent

Hint:

In complex word problems with multiple variables, the key is to define a single variable (like a common ratio 'x') and express all other quantities in terms of it. This simplifies the problem into solving inequalities for that single variable.

Answer 1

The Correct Option is B

Step 1: Understanding the Question & Defining Variables: 
The problem links the incomes, savings, and expenditures of three individuals through a series of equalities, ratios, and inequalities. We need to find the possible range for Ben's income. 
Let the incomes of Adam, Ben, and Mary be \(I_A, I_B, I_M\). 
Let their savings be \(S_A, S_B, S_M\). 
Let their expenditures be \(E_A, E_B, E_M\). 
The incomes are in the ratio 3:1:4. Let 'x' be the common ratio factor. 
\[ I_A = 3x, \quad I_B = x, \quad I_M = 4x \] 

Step 2: Key Formula or Approach: 
The fundamental relationship for personal finance is: 
\[ \text{Income} = \text{Expenditure} + \text{Savings} \implies \text{Expenditure} = \text{Income} - \text{Savings} \] Our strategy is to express all unknown quantities in terms of the variable 'x' and then use the given inequalities to establish a valid range for 'x'. This range will directly correspond to the range for Ben's income. 

Step 3: Detailed Explanation: 
First, use the given equalities and values: 
We are given \(S_M = 50,000\). 
We are also given \(S_A = E_B = S_M\). 
Therefore, \(S_A = 50,000\) and \(E_B = 50,000\). 
Next, express the expenditures of Adam and Mary in terms of 'x': 
\[ E_A = I_A - S_A = 3x - 50,000 \] \[ E_M = I_M - S_M = 4x - 50,000 \] Now, we apply the two inequalities given in the problem: 
Inequality 1: Mary's expenditure is less than thrice of Adam's expenditure. 
\[ E_M<3 \times E_A \] \[ 4x - 50,000<3(3x - 50,000) \] \[ 4x - 50,000<9x - 150,000 \] \[ 150,000 - 50,000<9x - 4x \] \[ 100,000<5x \] \[ 20,000<x \] Inequality 2: Twice of Adam's expenditure is less than two times Ben's income. 
\[ 2 \times E_A<2 \times I_B \] Dividing by 2, we get: 
\[ E_A<I_B \] \[ 3x - 50,000<x \] \[ 2x<50,000 \] \[ x<25,000 \] Combining the results from both inequalities, we get the range for 'x': 
\[ 20,000<x<25,000 \] Since Ben's income is \(I_B = x\), the range for Ben's income is: 
\[ 20,000<I_B<25,000 \] 

Step 4: Final Answer: 
The possible range for Ben's income is between Rupees 20,000 and Rupees 25,000. This corresponds to option (B).