Question

The ratio of income of Anil and Mukesh is \(2: 3\) The sum of their expenditure is Rs. 8000 and the amount of savings of Anil is equal to the amount of expenditure of Mukesh. What is the sum of their savings? (a) 22,000 (b) 4,000 (c) 16,000 (d) 12,000

Short answer

Answer: The sum of their savings is Rs. 12,000.

Step-by-step solution

Express Anil's and Mukesh's incomes in terms of a common variable

Given that the ratio of Anil's income to Mukesh's income is \(2:3\), let Anil's income be \(2x\) and Mukesh's income be \(3x\).

Express their expenditures and savings

Let Anil's expenditure be \(E_A\), Mukesh's expenditure be \(E_M\), Anil's savings be \(S_A\), and Mukesh's savings be \(S_M\). From the problem, we know that: 1. The sum of their expenditures is Rs. 8000, thus \(E_A + E_M = 8000\); and 2. Anil's savings is equal to the amount of Mukesh's expenditure, thus \(S_A = E_M\). We also know that income is the sum of expenditure and savings, so \(2x = E_A + S_A\) for Anil and \(3x = E_M + S_M\) for Mukesh.

Solve for Anil's and Mukesh's savings

From Anil's income equation, we have \(2x = E_A + E_M\) (since \(S_A = E_M\)). Substituting the equation for the sum of their expenditures, we get \(2x = 8000\), so \(x = 4000\). Therefore, Anil's income is \(2x = 8000\) and Mukesh's income is \(3x = 12000\). Now we can find Anil's and Mukesh's expenditures. From \(S_A = E_M\), we know that Anil's expenditure is \(E_A = 8000 - E_M = 8000 - S_A\). Substituting this into Anil's income equation, we get \(8000 = 8000 - S_A + S_A\). Solving for Anil's savings, we find that \(S_A = 8000 - 8000 = 0\). Since Anil's savings is equal to Mukesh's expenditure and Anil's savings is 0, Mukesh's expenditure is also 0. Now we can find Mukesh's savings by substituting his expenditure into Mukesh's income equation: \(12000 = 0 + S_M\). Solving for Mukesh's savings, we get \(S_M = 12000\).

Find the sum of their savings

To find the sum of their savings, we add Anil's and Mukesh's savings together: \(S_A + S_M = 0 + 12000 = 12000\). Therefore, the sum of their savings is Rs. 12,000, which corresponds to option (d).

Key concepts

Income and Expenditure Problems

When tackling income and expenditure problems, it's essential to understand the relationship between income, savings, and expenditure. Income is what someone earns, expenditure is what they spend, and the difference between them is their savings.

In exercises like our example, Anil's and Mukesh's incomes and expenditures are linked through a given ratio. It's like a puzzle where certain pieces of information are missing, and you need to use the provided clues to solve it. The key is to translate the written problem into mathematical equations using variables. For instance, by denoting Anil's income as 2x and Mukesh's income as 3x, we quantify their earnings and later use other given parameters to solve for x.

Simplifying the problem through equations can give you a clearer perspective and guide you through a logical sequence of steps to find the answer. It's crucial not to rush; carefully read and understand each condition before writing your equations. Equations are your best tool; they turn words into numbers you can work with.

Savings Calculation

The formula for calculating savings may seem straightforward—simply subtract expenditure from income. But as our example shows, savings calculation can sometimes involve more complex relationships. In the case of Anil and Mukesh, Anil's savings equaled Mukesh's expenditure, which provided a critical link between their financial situations.

To calculate savings accurately, always remember the fundamental formula: Savings = Income - Expenditure. If the problem involves more than one person, and their financial variables are interconnected, you'll need to set up a system of equations. Find leverage in the given information, and treat each person's financial status as part of a bigger picture to solve the puzzle.

Quantitative Aptitude

Questions like these test your quantitative aptitude, a term that encompasses your ability to handle numerical and mathematical tasks. It's a key skill for various standardized tests and exams, and extends well beyond the classroom into real-world problem solving—especially in finances and budgeting.

Developing your quantitative aptitude involves more than just crunching numbers; it requires logical reasoning and analytical thinking. You must interpret the information given to you, translate it into a mathematical form, and then apply the appropriate calculations. Sometimes, the challenge is determining which method to use, as there could be multiple paths to the same solution. Practice is essential here. With each problem solved, you build a stronger intuition for the strategies that could make complex problems more manageable.