Question

The simple interest on certain sum at \(5 \%\) for 9 month is Rs. I0 greater than the simple interest on the same sum @ \(3 \%\) for 14 months. What is the sum of interest in both the cases (ie., total sum of interest)? (b) Rs. 290 (a) Rs. 130 (d) Rs. 330 (c) Rs. 120

Short answer

Answer: (b) Rs. 290

Step-by-step solution

Recall the Simple Interest Formula

Recall that the formula for simple interest is: Simple Interest (S.I) = Principal (P) × Rate (R) × Time (T) / 100 We will use this formula to calculate the simple interest in both cases.

Analyze the given information

Let P be the principal. The first case has a rate of 5% and the interest is for 9 months (3/4 of a year). So, the interest is: S.I1 = P × 5 × 3/4 / 100 The second case has a rate of 3% and the interest is for 14 months (7/6 of a year). So, the interest is: S.I2 = P × 3 × 7/6 / 100 We are given that, S.I1 = S.I2 + I0

Solve for principal (P)

We need to find the principal (P) first. Substitute the values of S.I1 and S.I2 in the given equation and solve for P: P × 5 × 3/4 / 100 = P × 3 × 7/6 / 100 + I0 Solving for P, we get: P = 60I0 / 13

Find the sum of interest

Now we can find the sum of interest by adding the interests in both cases: Total sum of interest = S.I1 + S.I2 Substituting the values of P, we get: Total sum of interest = 60I0 / 13 × 5 × 3/4 / 100 + 60I0 / 13 × 3 × 7/6 / 100 Now, let's use the given options to find the correct value of the total sum of interest.

Check the given options

Let's check each option by substituting the possible values of I0: (a) Total sum of interest = Rs. 130 I0 = 1/2 × (13 × 130 / 60) = 13 (b) Total sum of interest = Rs. 290 I0 = 1/2 × (13 × 290 / 60) = 30 (c) Total sum of interest = Rs. 120 I0 = 1/2 × (13 × 120 / 60) = 12 (d) Total sum of interest = Rs. 330 I0 = 1/2 × (13 × 330 / 60) = 33 Comparing these values, we see that I0 = 30 corresponds to the given condition in the problem since it results in an integer value. Thus, the total sum of interest is Rs. 290. Therefore, the correct answer is (b) Rs. 290.

Key concepts

Understanding the Simple Interest Formula

Simple interest is a quick and straightforward way to calculate the interest earned or paid on a certain amount of money over time. It's used in various financial situations from bank accounts and loans to investments. To master this concept, we must become familiar with the simple interest formula which is represented as:
\[ \text{Simple Interest (S.I)} = \frac{\text{Principal (P)} \times \text{Rate (R)} \times \text{Time (T)}}{100} \]
This formula encapsulates the relationship between the principal—or the initial sum of money—the annual interest rate (usually as a percentage), and the time the money is invested or borrowed for. When we apply this formula, it's vital to remember that the time is often expressed in years. Thus, for periods shorter or longer than a year, we need to convert the time into an annualized format, just as we saw in the exercise where months were converted into a fraction of the year.

Converting Time Periods

For instance, 9 months out of a 12-month year is represented as a fraction, \[ \frac{9}{12} \] which simplifies to \[ \frac{3}{4} \]. Similarly, 14 months would convert to \[ \frac{14}{12} \] or \[ \frac{7}{6} \]. Ensuring correct time conversion is crucial to the accurate computation of simple interest. Moreover, the resulted interest from this calculation gives us insight into the cost or yield of financial operations over time.

Building Quantitative Aptitude

Quantitative aptitude refers to one's capacity to handle numbers and solve quantitative problems in a logical and detail-oriented manner. This skill is not only fundamental for excelling in competitive exams but also for making informed financial decisions in real life. In our textbook exercise, quantitative aptitude is demonstrated through understanding the problem statement, deducing the required formula, and manipulating mathematical expressions to reach a logical conclusion.

Steps to Sharpen Quantitative Skills

• Read and understand the problem entirely before attempting a solution.
• Identify what numerical information you have and what you need to find out.
• Use the appropriate formulas, in this case, the simple interest formula.
• Perform precise calculations, keeping units and conversions in mind.

As we improve our quantitative aptitude, we grow our proficiency in reasoning and problem-solving, which are both crucial for achieving the correct answer in mathematical problems.

Problem-Solving in Mathematics

Being adept at problem-solving in mathematics means being able to understand, conceptualize, and execute a series of steps to move from a question to a credible solution. It's a skill that hooks on logic, creativity, and critical thinking. The process generally involves several stages, such as comprehension, strategizing, performing operations, and verifying results.

In the context of the textbook solution provided, we can identify these stages in effect. Initially, understanding the question and recognizing that it relates to simple interest sets the baseline. Then, deciding to use the simple interest formula is a strategic choice. The actual computation of the interest values, along with the manipulation of variables, represents the operational phase. Lastly, checking the options for consistency with the problem statements serves as verification of the solution.

Effective Strategies to Tackle Math Problems

• Break down complex problems into smaller, manageable parts.
• Choose the right formula and understand its application.
• Cross-check your computation to avoid miscalculations.
• Finally, use logical reasoning to assess whether your solution makes practical sense.

Employing a structured approach to mathematical problem-solving not only aids in arriving at the correct answer but also strengthens one’s overall logical and analytical skills.