A:B:C Share Ratio: Equal Interest Over Time

Dec 15, 2025 by Andrew McMorgan 44 views

Hey guys, let's dive into a cool math problem that'll test your understanding of simple interest and ratios! We've got a sum of money being split among three people, A, B, and C. The twist? The simple interest earned on their individual shares, calculated over different time periods (2, 3, and 4 years respectively) at a flat rate of 10% per annum, ends up being the same for all of them. Your mission, should you choose to accept it, is to figure out the ratio of their shares: A:B:C. This is a classic problem, and once you grasp the concept, it's super straightforward. We'll break it down step-by-step, so even if you're not a math whiz, you'll be able to follow along and impress your friends with your newfound knowledge. Ready to get your math hats on?

Understanding Simple Interest and Ratios

Before we jump into solving the problem, let's quickly refresh what simple interest and ratios are all about. Simple interest is the most basic form of interest calculation. It's calculated only on the principal amount (the initial sum of money). The formula is pretty standard: SI = (P * R * T) / 100, where P is the principal, R is the rate of interest per annum, and T is the time period in years. Now, ratios are a way to compare two or more quantities. When we express a ratio as A:B:C, it means for every 'A' units of the first quantity, we have 'B' units of the second, and 'C' units of the third. In our case, we're dealing with the ratio of shares, meaning how the total sum was divided among A, B, and C.

The core of this problem lies in the equality of the simple interest earned by each person. Let's say the total sum is S, and it's divided into shares P_A, P_B, and P_C for individuals A, B, and C, respectively. So, S = P_A + P_B + P_C. We are given that the interest rate (R) is 10% per annum for everyone. The time periods (T) are different: T_A = 2 years, T_B = 3 years, and T_C = 4 years. The crucial piece of information is that the simple interest earned is equal for all of them. Let's denote this equal interest amount as 'I'.

So, for person A, the simple interest is I = (P_A * R * T_A) / 100. For person B, it's I = (P_B * R * T_B) / 100. And for person C, it's I = (P_C * R * T_C) / 100. Since the interest (I) and the rate (R) are the same for all three, we can set up equations based on their principal shares (P_A, P_B, P_C).

This problem elegantly connects the concepts of simple interest calculation with the representation of proportions through ratios. It's not just about plugging numbers into a formula; it's about understanding the relationship between the principal amount, interest rate, time, and the resulting interest. When these interests are equal, it implies a specific proportional relationship between the principal shares, even though the time periods differ. This is where the challenge and the fun lie, guys!

Setting Up the Equations

Alright, let's get down to the nitty-gritty and set up the equations based on the information given. We know the simple interest formula: SI = (P * R * T) / 100.

Let P_A, P_B, and P_C be the shares of A, B, and C, respectively. Let R be the annual interest rate, which is 10% or 0.10. Let T_A, T_B, and T_C be the time periods for A, B, and C, which are 2 years, 3 years, and 4 years, respectively.

The problem states that the simple interest earned on their shares after these respective years is equal. Let's call this equal interest 'I'.

So, for A: I = (P_A * R * T_A) / 100 I = (P_A * 10 * 2) / 100 I = (20 * P_A) / 100 I = P_A / 5 (Equation 1)

For B: I = (P_B * R * T_B) / 100 I = (P_B * 10 * 3) / 100 I = (30 * P_B) / 100 I = (3 * P_B) / 10 (Equation 2)

For C: I = (P_C * R * T_C) / 100 I = (P_C * 10 * 4) / 100 I = (40 * P_C) / 100 I = (2 * P_C) / 5 (Equation 3)

Since the interest 'I' is the same for all three, we can equate these expressions. This is the key step to finding the ratio of their shares.

From Equation 1, we can express P_A in terms of I: P_A = 5 * I

From Equation 2, we can express P_B in terms of I: I = (3 * P_B) / 10 => P_B = (10 * I) / 3

From Equation 3, we can express P_C in terms of I: I = (2 * P_C) / 5 => P_C = (5 * I) / 2

Now we have the shares P_A, P_B, and P_C all expressed in terms of the common interest 'I'. This allows us to find the ratio P_A : P_B : P_C. We just need to substitute the expressions we found:

P_A : P_B : P_C = (5 * I) : ((10 * I) / 3) : ((5 * I) / 2)

Notice that 'I' is common to all parts of the ratio. We can divide each part by 'I' without changing the ratio:

P_A : P_B : P_C = 5 : (10 / 3) : (5 / 2)

We now have a ratio with fractions, which isn't usually the preferred format. The next step is to convert this into a ratio of whole numbers. We can do this by multiplying each part of the ratio by the least common multiple (LCM) of the denominators (3 and 2).

The LCM of 3 and 2 is 6.

Multiplying each part by 6:

For P_A: 5 * 6 = 30 For P_B: (10 / 3) * 6 = 10 * (6 / 3) = 10 * 2 = 20 For P_C: (5 / 2) * 6 = 5 * (6 / 2) = 5 * 3 = 15

So, the ratio of the shares P_A : P_B : P_C is 30 : 20 : 15.

This ratio can be simplified further by dividing each number by their greatest common divisor (GCD), which is 5.

30 / 5 = 6 20 / 5 = 4 15 / 5 = 3

Therefore, the ratio of their shares is 6 : 4 : 3.

This detailed setup shows how we systematically used the simple interest formula and the condition of equal interest to derive the ratio of the principal amounts. It's all about algebraic manipulation and understanding the proportionality involved. Pretty neat, huh?

Solving for the Ratio

Now that we've set up our equations and expressed the shares in terms of the common interest 'I', the final step is to actually find the ratio A:B:C. We have:

P_A = 5I P_B = (10/3)I P_C = (5/2)I

We want to find the ratio P_A : P_B : P_C. Substituting the expressions we found:

P_A : P_B : P_C = 5I : (10/3)I : (5/2)I

As 'I' is common to all terms, we can cancel it out. This leaves us with:

P_A : P_B : P_C = 5 : (10/3) : (5/2)

To get rid of the fractions and express the ratio in its simplest whole number form, we need to find the least common multiple (LCM) of the denominators, which are 3 and 2. The LCM of 3 and 2 is 6.

Now, we multiply each term in the ratio by 6:

For A: 5 * 6 = 30 For B: (10/3) * 6 = (10 * 6) / 3 = 60 / 3 = 20 For C: (5/2) * 6 = (5 * 6) / 2 = 30 / 2 = 15

So, the ratio becomes 30 : 20 : 15.

This ratio can be further simplified by dividing all terms by their greatest common divisor (GCD). The GCD of 30, 20, and 15 is 5.

30 ÷ 5 = 6 20 ÷ 5 = 4 15 ÷ 5 = 3

Therefore, the ratio of the shares A:B:C is 6 : 4 : 3.

This is our final answer, guys! It means that for every 6 units of the sum A received, B received 4 units, and C received 3 units. This specific distribution ensures that when simple interest is calculated at 10% for 2, 3, and 4 years respectively, the interest amounts are equal.

Let's quickly check this. Assume the common interest 'I' is $60 (just picking a number divisible by 5, 10/3, and 5/2 to make it easy).

If P_A = 6k, P_B = 4k, P_C = 3k, and the ratio is 6:4:3. Let's see if the interest is equal. Suppose the ratio is 6x:4x:3x.

Interest for A = (6x * 10 * 2) / 100 = 120x / 100 = 1.2x Interest for B = (4x * 10 * 3) / 100 = 120x / 100 = 1.2x Interest for C = (3x * 10 * 4) / 100 = 120x / 100 = 1.2x

As you can see, the interest is indeed equal (1.2x) for all three, confirming our ratio is correct! This kind of verification is super helpful to ensure you haven't made any calculation errors. The core idea is that the share is inversely proportional to the product of rate and time when interest is equal. Since rate is constant, share is inversely proportional to time. So, P_A : P_B : P_C = (1/T_A) : (1/T_B) : (1/T_C) = (1/2) : (1/3) : (1/4). Multiplying by LCM(2,3,4)=12 gives 6:4:3. This is a neat shortcut if you remember the inverse relationship!

Final Answer and Verification

So, after all the calculations, we've arrived at the ratio of the shares for A, B, and C. The ratio is 6 : 4 : 3. This means if the total sum distributed was, for example, $1300 (6+4+3 = 13 parts), then A would get $600, B would get $400, and C would get $300. Let's quickly verify if the simple interest earned on these amounts over their respective time periods at 10% is indeed equal.

For A: Share = $600, Time = 2 years, Rate = 10% Simple Interest (SI_A) = (600 * 10 * 2) / 100 = (12000) / 100 = $120
For B: Share = $400, Time = 3 years, Rate = 10% Simple Interest (SI_B) = (400 * 10 * 3) / 100 = (12000) / 100 = $120
For C: Share = $300, Time = 4 years, Rate = 10% Simple Interest (SI_C) = (300 * 10 * 4) / 100 = (12000) / 100 = $120

As you can see, SI_A = SI_B = SI_C = $120. The simple interests are equal, which confirms that our calculated ratio of 6 : 4 : 3 is absolutely correct! This problem beautifully illustrates the inverse relationship between the principal amount and the time period when the simple interest and the rate remain constant. Essentially, the person who has their money invested for a longer period needs to have a smaller principal share to earn the same amount of interest compared to someone whose money is invested for a shorter duration.

This kind of problem is common in competitive exams and real-life financial planning scenarios. Understanding how different factors like time and principal affect interest, and how to manipulate ratios to find unknown distributions, is a valuable skill. So, next time you encounter a similar problem, remember the steps: set up the equations based on the given conditions (equal interest here), express the unknown shares in terms of a common variable (like 'I'), form the ratio, and then simplify it to its lowest whole number terms. Don't forget to do a quick verification, as it's a great way to catch any mistakes and build confidence in your answer. Keep practicing, guys, and you'll become masters of these financial math puzzles in no time!

Therefore, the correct option is 6:4:3.