Question

Mr. Bajaj invested \(\frac{1}{7}\) of his total investment at \(4 \%\) and \(\frac{1}{2}\) at \(5 \%\) and rest at \(6 \%\) for the one year and received total interest of Rs. \(730 .\) What is the total sum invested? (a) Rs. 70000 (b) Rs. 14000 (c) Rs. 24000 (d) Rs. 38000

Short answer

Answer: The total sum invested by Mr. Bajaj is Rs. 14000.

Step-by-step solution

Understand the given information

We know that Mr. Bajaj invested \(\frac{1}{7}\) of his total investment at \(4 \%\), \(\frac{1}{2}\) at \(5 \%\), and the rest at \(6 \%\). We also know that he received a total interest of Rs. 730. Let's assume the total investment is x.

Calculate the remaining portion invested

Mr. Bajaj invested the rest of the amount at \(6 \%\). To find the proportion of the total investment made at \(6 \%\), we need to subtract the sum of \(\frac{1}{7}\) and \(\frac{1}{2}\) from 1. Remaining portion = 1 - \(\left(\frac{1}{7} + \frac{1}{2}\right)\) = 1 - \(\frac{9}{14}\) = \(\frac{5}{14}\). So, Mr. Bajaj invested \(\frac{5}{14}\) of his total investment at \(6 \%\).

Set up an equation to represent total interest

We can now set up an equation to represent the total interest earned by Mr. Bajaj. Total interest = (interest from \(4 \%\) investment) + (interest from \(5 \%\) investment) + (interest from \(6 \%\) investment) We know the total interest is Rs. 730, so we can write: \(730 = \left(\frac{1}{7}x \times \frac{4}{100}\right) + \left(\frac{1}{2}x \times \frac{5}{100}\right) + \left(\frac{5}{14}x \times \frac{6}{100}\right)\)

Solve the equation

To solve for x (total investment), we will now simplify the equation and isolate x: \(730 = \frac{4x}{700} + \frac{5x}{200} + \frac{30x}{1400}\) Next, we will make each term have a common denominator to simplify the equation further: \(730 = \frac{8x}{1400} + \frac{35x}{1400} + \frac{30x}{1400}\) Now, we can combine the terms: \(730 = \frac{73x}{1400}\) Now, let's isolate x by multiplying both sides by \(\frac{1400}{73}\): \(x = \frac{730 \times 1400}{73}\) After calculating the value of x, we get: \(x = 14000\) Thus, the total sum invested by Mr. Bajaj is Rs. 14000, which aligns with option (b).

Key concepts

Simple Interest

Simple interest is one of the basic concepts in financial mathematics. It allows us to calculate the interest earned or paid on a specific amount of money, known as the principal, over a period of time. Unlike compound interest, which recalculates the principal plus accumulated interest, simple interest is calculated only on the original principal.

The formula to calculate simple interest is:

• \( I = P imes R imes T \)

where:

• *\( I \) is the interest accumulated*
• *\( P \) is the principal amount*,
• *\( R \) is the rate of interest per annum (expressed as a decimal)*,
• *\( T \) represents the time period, usually in years*.

This simple formula makes it easy for individuals to quickly determine their potential earnings from an investment or the cost of a loan over time. This foundation helps build more complicated financial computations and is essential for students to grasp early on.

In the context of the given problem, Mr. Bajaj's interest from his investments was calculated using this principle by separating each investment proportion with its respective rate and duration in the problem's context.

Proportion Calculations

Proportion calculations are vital in determining how different parts of a whole are allocated. In financial contexts, proportion tells us how segments of an investment or budget are distributed among different options or categories. Calculating proportions requires understanding ratios and fractions.

For instance, in the exercise Mr. Bajaj's investment choices were spread across various interest rates. He invested different proportions of his total funds at differing rates, creating a unique setup where calculating the exact interest became crucial.

Understanding how proportions work, the calculation in the problem is as follows:

• First, \( \frac{1}{7} \) of the total investment was at one rate.
• Similarly, \( \frac{1}{2} \) was invested at another rate.
• The remaining proportion was then calculated to understand how much was invested at the last rate, as shown by the formula: \( 1 - (\frac{1}{7} + \frac{1}{2}) = \frac{5}{14} \).

By mastering proportions, students can independently resolve complex financial dilemmas by allocating assets efficiently for the desired outcomes.

Linear Equations

Linear equations are mathematical expressions that represent a straight line when graphed on a coordinate plane. They take the form of \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is a variable to be solved.

In practical terms, linear equations allow us to solve problems involving constant rates or proportions. In the problem of Mr. Bajaj's investment, a linear equation was created to represent the total interest received based on different parts of his investments.

To find the total sum invested, we set up the equation using the individual interests for each portion:

• \( 730 = \left(\frac{1}{7}x \times \frac{4}{100}\right) + \left(\frac{1}{2}x \times \frac{5}{100}\right) + \left(\frac{5}{14}x \times \frac{6}{100}\right) \)

Through simplification and solving, we find the value of \( x \), showcasing how linear equations help unravel such financial complexities. Understanding such equations helps in modeling various real-world scenarios efficiently.

Financial Mathematics

Financial mathematics involves applying mathematical techniques to solve problems in finance, such as investment planning, risk assessment, or understanding financial markets.

This branch of mathematics integrates concepts like interest rates, proportions, and linear equations to build a comprehensive understanding of managing and growing financial resources.

In the problem at hand, several aspects of financial mathematics were employed. Mr. Bajaj's investments were evaluated considering different interest rates and proportions through step-by-step calculations:

• Identifying the parts of the total investment allocated to different interest rates was key.
• Finally, using the principles of simple interest and linear equations helped solve the core problem: finding the total investment.

The tools of financial mathematics are invaluable in navigating personal finance decisions and professional financial analysis. By mastering these, students equip themselves to handle diverse financial tasks with greater confidence and accuracy.