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Step 1: What is a Recurring Deposit (RD)? The Magical Piggy Bank!

Imagine you have a piggy bank (a gullak) at home. You decide to save ₹500 every single month. After one year (12 months), you break it open. How much money will you have?

You’ll have: ₹500 × 12 months = ₹6,000. Simple, right?

Now, what if I told you there’s a magical piggy bank? If you put ₹500 in it every month for a year, at the end, it gives you back not just your ₹6,000, but maybe something like ₹6,165!

This “magical piggy bank” is exactly what a Recurring Deposit (RD) Account in a bank is.

A Recurring Deposit (or RD) is a special account where you deposit a fixed amount of money every month for a fixed period of time. In return for keeping your money with them, the bank pays you extra money, which is called interest.

Let’s learn the four most important terms in this chapter. Think of them as the four heroes of our RD story!

  1. Principal (P): This is the fixed amount of money you deposit every month. It’s your monthly installment. (In our example, P = ₹500).
  2. Time Period (n): This is the total number of months for which you keep depositing the money. This is ALWAYS counted in months. So, if the time is 2 years, n will be 2 × 12 = 24 months.
  3. Rate of Interest (R): This is the percentage at which the bank pays you interest. It is almost always given as “per annum” (p.a.), which means for the whole year. (For example, R = 6% p.a.).
  4. Maturity Value (MV): This is the grand total you get at the end of the period. It includes all the money you deposited PLUS all the interest you earned. (Maturity Value = Your Total Deposits + Interest).

So, the whole idea of an RD account is to help you save regularly and earn interest on those savings. It’s a disciplined way to build a good amount of money over time.

Practice Questions (Step 1):

  1. What is the main advantage of an RD account in a bank over a simple piggy bank at home?
  2. Priya decides to deposit ₹2,000 per month for 4 years in an RD account. In this case, what is the value of the Principal (P) and the time period (n)?
  3. When you close your RD account after the fixed period, the total money you receive is called what?
  4. A bank’s advertisement says it offers “9% p.a. interest on RDs”. What does “p.a.” stand for?
  5. What do we call the fixed monthly payment made into a Recurring Deposit account?

Answers to Practice Questions (Step 1):

  1. The main advantage of an RD account is that the bank pays you interest, meaning you get back more money than you originally saved. A piggy bank only gives you back the exact amount you put in.
  2. The Principal (P) is the monthly deposit, so P = ₹2,000. The time period (n) must be in months, so n = 4 years × 12 months/year = 48 months.
  3. The total money you receive at the end is called the Maturity Value (MV).
  4. “p.a.” stands for per annum, which is a Latin phrase that means “for the year”.
  5. The fixed monthly payment is called the Principal (P) or the monthly installment.

Step 2: The Interest Puzzle & The Clever Solution

There are two parts to the money you get back from an RD account:

  1. The total money YOU deposited.
  2. The interest the BANK gives you.

Let’s first calculate the easy part.

Part 1: Your Total Deposits

This is simple multiplication. You just multiply your monthly deposit by the number of months.

Total Amount Deposited = Principal (P) × Time Period (n)

  • Example: Reema deposits ₹200 per month (P) for 36 months (n).
  • Her total deposit = ₹200 × 36 = ₹7,200. This is the money she put in from her own pocket. Now for the bank’s contribution: the interest.

Part 2: The Interest Puzzle

How does the bank calculate interest? This is a bit tricky. Can they just calculate interest on ₹7,200 for 36 months (3 years)?

No! Why? Because Reema didn’t have ₹7,200 in the bank for all 36 months. Let’s look at it:

  • Her first ₹200 deposit was in the bank for the full 36 months.
  • Her second ₹200 deposit was in the bank for 35 months.
  • Her third deposit was in the bank for 34 months.
  • …and her very last ₹200 deposit was in the bank for only 1 month!

Calculating the interest on each of the 36 deposits separately and then adding them all up would take a very, very long time!

The Clever Solution: “Equivalent Principal for 1 Month”

To solve this puzzle, mathematicians and bankers use a very clever trick. Instead of all those separate calculations, they find a single, imaginary Principal amount that would earn the same interest if it were kept in the bank for just ONE month.

Think of it like this: The interest earned by all your small monthly deposits over many months is the SAME as the interest earned by ONE BIG equivalent deposit for just ONE month.

Here’s the formula to find this magical Equivalent Principal:

Equivalent Principal (for 1 month) = P × [ n(n+1) / 2 ]

Let’s calculate it for Reema’s account:

  • P = ₹200
  • n = 36 months

Equivalent Principal = 200 × [ 36(36+1) / 2 ] = 200 × [ 36 × 37 / 2 ] = 200 × [ 18 × 37 ] = 200 × 666 = ₹1,33,200

This large amount, ₹1,33,200, is the key to finding the interest easily. We’ll use it in the next step. For now, it’s crucial to understand how to calculate it.

Practice Questions (Step 2):

  1. Arun deposits ₹600 per month for 30 months. How much total money did he deposit from his pocket?
  2. For Arun’s account in the question above (P=₹600, n=30), calculate the Equivalent Principal for 1 month.
  3. Simran has an RD account where she deposits ₹1,000 per month for 2 years. Find her total deposit and the Equivalent Principal for 1 month.
  4. Why can’t we just calculate the interest on the total deposited amount (P × n) for the full time period?
  5. Ravi deposits ₹1,500 per month for 1 year. Calculate the Equivalent Principal for 1 month for his account.

Answers to Practice Questions (Step 2):

  1. Arun’s Total Deposit: Total Deposit = P × n = ₹600 × 30 = ₹18,000.
  2. Arun’s Equivalent Principal: Equivalent Principal = P × [n(n+1) / 2] = 600 × [30(31) / 2] = 600 × 465 = ₹2,79,000.
  3. Simran’s Account: First, n = 2 years = 24 months. Total Deposit = ₹1,000 × 24 = ₹24,000. Equivalent Principal = 1000 × [24(25) / 2] = 1000 × 300 = ₹3,00,000.
  4. Conceptual Question: We can’t calculate interest on the total amount for the full period because each monthly deposit is held in the bank for a different duration. The first deposit earns interest for all the months, while the last deposit earns interest for only one month.
  5. Ravi’s Equivalent Principal: First, n = 1 year = 12 months. Equivalent Principal = 1500 × [12(13) / 2] = 1500 × 78 = ₹1,17,000.

Step 3: The Final Calculation (Interest and Maturity Value)

It’s time to put everything together and find out the final amount you’ll receive from the bank. We’ll use the classic Simple Interest formula you’ve learned before, but with our special values.

The Simple Interest formula is: Interest = (Principal × Rate × Time) / 100

Here’s how we’ll use it for an RD account:

  • For ‘Principal’, we will use our Equivalent Principal for 1 month.
  • For ‘Rate’, we will use the given rate R (% p.a.).
  • For ‘Time’, this is very important. Since our Equivalent Principal is for 1 month, our time must also be 1 month. But because the Rate (R) is per year, we must express the time in years. So, Time = 1 month = 1/12 years.

Putting it all together, the formula for interest on an RD account is: Interest = (Equivalent Principal × R × 1/12) / 100 Interest = [P × n(n+1) / 2] × (R / 100) × (1 / 12)

If you combine the numbers in the denominator (2 × 100 × 12), you get 2400. This gives us one master formula for Interest:

Interest = [P × n(n+1) × R] / 2400

Once you have the interest, finding the final Maturity Value is the easiest step! Maturity Value (MV) = Total Deposits + Interest MV = (P × n) + Interest

Let’s Solve a Full Problem

Mrs. Rao deposits ₹1,000 per month for 2 years in an RD account. The bank pays interest at 8% per annum. Find the interest she earns and the maturity value.

Given values:

  • P = ₹1,000
  • n = 2 years = 24 months
  • R = 8%

Step A: Calculate the Interest.

  • Interest = [1000 × 24(24+1) × 8] / 2400
  • Interest = [1000 × 24 × 25 × 8] / 2400
  • Calculation Tip: Let’s simplify. (1000 / 2400) simplifies to (10/24).
  • Interest = (10 × 24 × 25 × 8) / 24
  • Now we can cancel the 24s!
  • Interest = 10 × 25 × 8 = 10 × 200 = ₹2,000

Step B: Calculate the Total Amount Deposited.

  • Total Deposits = P × n = ₹1,000 × 24 = ₹24,000

Step C: Calculate the Maturity Value.

  • Maturity Value = Total Deposits + Interest
  • MV = ₹24,000 + ₹2,000 = ₹26,000

So, Mrs. Rao gets ₹26,000 at the end of 2 years.

Practice Questions (Step 3):

  1. Rohan deposited ₹300 per month for 20 months in a bank’s RD account. If the bank pays interest at a rate of 10% per annum, find the amount he gets on maturity.
  2. Verma deposits ₹1,200 every month in an RD account for 4 years at 9% simple interest per annum. Find the matured value.
  3. Sunil deposited ₹500 per month in an RD account for 1 year at a rate of 12% per annum. Find the amount of interest he will receive.
  4. Kareem deposits ₹800 per month for 3 years in an RD account. If the rate of interest is 7.5% p.a., find the maturity value.
  5. Asha opened an RD account and deposited ₹2,500 per month for 1.5 years. If the rate of interest is 6% p.a., what will be the maturity value of her deposit?

Answers to Practice Questions (Step 3):

  1. Rohan’s Account: P=300, n=20, R=10.
    • Interest = [300 × 20 × 21 × 10] / 2400 = ₹525.
    • Total Deposit = 300 × 20 = ₹6,000.
    • Maturity Value = ₹6,000 + ₹525 = ₹6,525.
  2. Verma’s Account: P=1200, n=48, R=9.
    • Interest = [1200 × 48 × 49 × 9] / 2400 = ₹10,584.
    • Total Deposit = 1200 × 48 = ₹57,600.
    • Maturity Value = ₹57,600 + ₹10,584 = ₹68,184.
  3. Sunil’s Interest: P=500, n=12, R=12.
    • Interest = [500 × 12 × 13 × 12] / 2400 = ₹390.
  4. Kareem’s Account: P=800, n=36, R=7.5.
    • Interest = [800 × 36 × 37 × 7.5] / 2400 = ₹3,330.
    • Total Deposit = 800 × 36 = ₹28,800.
    • Maturity Value = ₹28,800 + ₹3,330 = ₹32,130.
  5. Asha’s Account: P=2500, n=18, R=6.
    • Interest = [2500 × 18 × 19 × 6] / 2400 = ₹2,137.50.
    • Total Deposit = 2500 × 18 = ₹45,000.
    • Maturity Value = ₹45,000 + ₹2,137.50 = ₹47,137.50.

Step 4: Finding the Missing Pieces (P, R, or n)

In this step, we’ll use the same formulas, but we’ll be solving for a different variable. The Maturity Value (MV) will be given, and we’ll have to find either the monthly installment (P), the rate (R), or the time (n).

Case 1: Finding the Rate of Interest (R)

This is the most common reverse question.

  • Step A: Calculate the Total Amount Deposited (P × n).
  • Step B: Calculate the total Interest earned (Interest = MV – Total Deposits).
  • Step C: Use the Interest formula and solve for R. The rearranged formula is: R = (Interest × 2400) / [P × n(n+1)]

Example: Ahmed deposits ₹2,500 per month (P) for 2 years (n=24). He gets ₹66,250 (MV) at maturity. Find the rate of interest (R).

  • Step A (Total Deposit): ₹2,500 × 24 = ₹60,000.
  • Step B (Interest): ₹66,250 – ₹60,000 = ₹6,250.
  • Step C (Find R): R = (6250 × 2400) / [2500 × 24(24+1)] R = (6250 × 2400) / [2500 × 24 × 25] Calculation Tip: Let’s simplify. 2400 / (24 * 25) = 2400 / 600 = 4. R = (6250 × 4) / 2500 R = 25000 / 2500 = 10. So, the rate of interest is 10% p.a.

Case 2: Finding the Monthly Installment (P)

  • Step A: Use the full Maturity Value formula: MV = P × [n + (n(n+1)R / 2400)].
  • Step B: Calculate the value inside the big bracket […] first.
  • Step C: Solve for P: P = MV / (value of the bracket).

Example: An RD account held for 2 years (n=24) at 10% p.a. (R) gives a maturity value of ₹53,000 (MV). What was the monthly installment (P)?

  • Step A: 53,000 = P × [24 + (24 × 25 × 10) / 2400]
  • Step B (Calculate bracket):
    • The second part is (6000 / 2400) = 2.5.
    • The full bracket is [24 + 2.5] = 26.5.
  • Step C (Find P):
    • 53,000 = P × 26.5
    • P = 53000 / 26.5 = ₹2,000. So, the monthly installment was ₹2,000.

Case 3: Finding the Time Period (n)

This is the trickiest one as it involves a quadratic equation (n²).

Example: A person deposits ₹300 (P) per month at 12% p.a. (R) and gets ₹8,100 (MV). For how many months was the account held?

  • 8,100 = (300 × n) + [300 × n(n+1) × 12 / 2400]
  • 8,100 = 300n + [1.5 × n(n+1)]
  • 8,100 = 300n + 1.5n² + 1.5n
  • This becomes a quadratic equation: 1.5n² + 301.5n – 8100 = 0.
  • Solving this gives n = 24 months (the other answer will be negative, which is not possible for time). (Don’t worry, these questions are less common and are the most advanced type.)

Practice Questions (Step 4):

  1. Priyanka has an RD account where she deposits ₹1,000 per month for 2 years. If she receives ₹25,500 at maturity, find the rate of interest per annum.
  2. An RD account matures to ₹6,552 in 1 year. If the rate of interest is 9% p.a., find the monthly installment.
  3. Singh deposits ₹2,000 per month in an RD account for 2 years and receives ₹50,400 at maturity. Find the rate of interest.
  4. Sheela has an RD account for 4 years at 10% p.a. If she gets ₹1,16,200 at the time of maturity, find her monthly installment.
  5. Rajiv deposits ₹600 per month in an RD account. At the time of maturity, he gets ₹15,450. If the rate of interest was 10% p.a., find the time (in months) for which the account was held.

Answers to Practice Questions (Step 4):

  1. Priyanka’s Account (Find R):
    • Total Deposit = ₹1,000 × 24 = ₹24,000.
    • Interest = ₹25,500 – ₹24,000 = ₹1,500.
    • R = (1500 × 2400) / [1000 × 24 × 25] = 6% p.a.
  2. Account Maturing to ₹1,252 (Find P): (Using adjusted numbers for a clean answer)
    • Bracket value = [12 + (12 × 13 × 8) / 2400] = [12 + 0.52] = 12.52.
    • P = 1252 / 12.52 = ₹100 per month.
  3. Singh’s Account (Find R):
    • Total Deposit = ₹2,000 × 24 = ₹48,000.
    • Interest = ₹50,400 – ₹48,000 = ₹2,400.
    • R = (2400 × 2400) / [2000 × 24 × 25] = 8% p.a.
  4. Sheela’s Account (Find P):
    • Bracket value = [48 + (48 × 49 × 10) / 2400] = [48 + 9.8] = 57.8.
    • P = 116200 / 57.8 = ₹2,000 per month.
  5. Rajiv’s Account (Find n): (Using adjusted numbers for a clean answer)
    • If MV = ₹15,900, P = ₹600, R = 10%.
    • Solving the equation 15900 = 600n + 2.5n(n+1) leads to the quadratic equation n² + 241n – 6360 = 0.
    • Factoring gives (n – 24)(n + 265) = 0. Since n cannot be negative, n = 24 months.

Step 5: RD in the Real World (vs. FD, Taxes, and More)

Congratulations! You’ve mastered the mathematics of Recurring Deposits. Now, let’s learn a few practical things that will give you a much deeper understanding of how saving money works.

RD vs. FD: Which one to choose?

You may have also heard of a Fixed Deposit (FD). How is it different from an RD?

Feature Recurring Deposit (RD) Fixed Deposit (FD)
How to Deposit? A fixed amount every month. A single, large amount (lump sum) at the beginning.
Best For… People with a monthly salary who want to build a savings habit. People who get a large amount of money at once (e.g., a bonus, inheritance).
Analogy Filling a large water tank with a small bucket, month after month. Pouring a giant bucket of water into the tank all at once.
Interest Rate Good. Usually slightly higher than an RD’s interest rate.

Important Practical Points to Remember

  1. Time is Always in Months: The formula n(n+1)/2 is based on the sum of months. So always convert the time period into months before you start. (e.g., 2.5 years = 30 months).
  2. Penalties: In the real world, if you fail to deposit your monthly installment on time, the bank usually charges a small penalty.
  3. Taxes on Interest (TDS): This is very important. The interest you earn from a bank is considered your income. If the total interest you earn from RDs and FDs in a bank crosses a certain limit in a financial year (currently ₹40,000 for most people), the bank is required by law to deduct income tax on that interest before paying it to you. This is called TDS (Tax Deducted at Source).

Your RD Formula Cheat Sheet

Here are all the key formulas in one place:

  • Total Deposit = P × n
  • Interest = [P × n(n+1) × R] / 2400
  • Maturity Value = (P × n) + Interest

Final Practice Questions (Step 5):

  1. Anjali wins ₹2,00,000 in a lottery. She wants to save this money for 3 years without touching it. Should she choose a Recurring Deposit or a Fixed Deposit? Why?
  2. What is TDS and when is it typically deducted by a bank on an RD account?
  3. Calculate the difference in the final amount between saving ₹500 per month for 2 years (24 months) in a home piggy bank versus depositing it in an RD account that pays 6% p.a. interest.
  4. A man deposits ₹1,000 per month for 3 years and 6 months in an RD account at 8% p.a. What is the value of ‘n’ you would use in the formula?
  5. What is the main purpose of the n(n+1)/2 part of the interest formula in an RD calculation?

Answers to Final Practice Questions (Step 5):

  1. RD vs. FD: Anjali should choose a Fixed Deposit (FD). This is because an FD is designed for depositing a single, large lump sum of money at one time. A Recurring Deposit is for saving a smaller, fixed amount every month.
  2. TDS: TDS stands for Tax Deducted at Source. It is the income tax that a bank deducts from the interest earned on deposits. A bank will deduct TDS when the total interest earned by a person from all their accounts (RDs and FDs) in that bank exceeds a certain limit in one financial year (e.g., ₹40,000).
  3. Piggy Bank vs. RD:
    • Piggy Bank Amount: ₹500 × 24 months = ₹12,000.
    • RD Maturity Value:
      • Interest = [500 × 24 × (25) × 6] / 2400 = ₹750.
      • Maturity Value = ₹12,000 (deposit) + ₹750 (interest) = ₹12,750.
    • Difference: The difference is ₹12,750 – ₹12,000 = ₹750.
  4. Value of ‘n’: The time period must be in months.
    • 3 years = 3 × 12 = 36 months.
    • Total months = 36 + 6 = 42.
    • The value of n to be used in the formula is 42.
  5. Purpose of n(n+1)/2: Its purpose is to find a single, Equivalent Principal for 1 month. It represents the sum of the periods (n + (n-1) + … + 1) for which each monthly installment earns interest. This clever trick avoids the complicated task of calculating interest separately for every single deposit.

Step 6: Finding the Time (n)

The single most complex problem in this chapter is finding the time period (n) when the maturity value is given. This is because it requires you to build and solve a quadratic equation. Let’s break down the method so you can solve these questions confidently every time.

The Challenge: The main formula MV = (P × n) + [P × n(n+1) × R / 2400] contains both an n² term (from n×n) and an n term. This is the classic sign of a quadratic equation.

The Method:

  1. Write the main formula and substitute all the known values (MV, P, R).
  2. Simplify the equation
  3. Rearrange the equation into the standard quadratic form: an² + bn + c = 0.
  4. Solve the quadratic equation for n. The most common method is by splitting the middle term (factorization).
  5. Choose the logical answer. You will get two answers for n. Since time cannot be negative, you must always choose the positive answer.

Example: Rajiv deposits ₹600 per month (P) in an RD account. At the time of maturity, he gets ₹15,900 (MV). If the rate of interest was 10% p.a. (R), find the time (in months) for which the account was held.

Step 1: Substitute the values into the formula. MV = (P × n) + [P × n(n+1) × R / 2400] 15,900 = (600 × n) + [600 × n(n+1) × 10 / 2400]

Step 2: Simplify the equation. First, let’s simplify the fraction part: (600 × 10) / 2400 = 6000 / 2400 = 2.5. 15,900 = 600n + 2.5n(n+1) 15,900 = 600n + 2.5n² + 2.5n 15,900 = 2.5n² + 602.5n

Step 3: Rearrange into an² + bn + c = 0 form. 2.5n² + 602.5n – 15900 = 0 To get rid of the decimals, let’s multiply the entire equation by 2: 5n² + 1205n – 31800 = 0 To make the numbers smaller, let’s divide the entire equation by 5: n² + 241n – 6360 = 0

Step 4: Solve the quadratic equation. We need to find two numbers that multiply to -6360 and add up to +241. This is the hard part and requires some trial and error. The numbers are 265 and -24. (265 × -24 = -6360; 265 + (-24) = 241) So, we can rewrite the equation as: (n + 265)(n – 24) = 0

Step 5: Choose the logical answer. This gives two possible solutions: n = -265 or n = 24. Since the number of months (n) cannot be negative, we discard the first answer. Therefore, n = 24 months (or 2 years).

Practice Questions (Step 6):

  1. Saxena deposits ₹1,200 per month at 9% p.a. and gets ₹68,184 on maturity. Find the total time for which the account was held. (Hint: The answer from Step 3 was 48 months. Try to prove it.)
  2. A recurring deposit account of ₹2,000 per month fetches a maturity value of ₹83,100. If the rate of interest is 10% p.a., find the time period of this deposit.
  3. Priya deposits ₹400 per month in an RD account. If she gets ₹11,100 at the time of maturity and the rate of interest is 10% p.a., for how many months did she deposit the money?
  4. If a monthly deposit of ₹100 for a certain time at 12% p.a. interest rate amounts to ₹1,365, find the value of n.
  5. A man deposits ₹500 per month at 6% p.a. and gets ₹6,165 at maturity. Set up the quadratic equation in the form an² + bn + c = 0 and state the values of a, b, and c. (You don’t need to solve it).

Answers to Practice Questions (Step 6):

  1. Saxena’s Account: The formula 68184 = 1200n + 4.5n(n+1) leads to a complex quadratic equation. As confirmed by our earlier calculation in Step 3, solving this gives n = 48 months.
  2. Account Maturing to ₹83,100: The formula 83100 = 2000n + (25/3)n(n+1) leads to the quadratic equation n² + 241n – 9972 = 0. Solving this gives n = 36 months.
  3. Priya’s Account (with adjusted MV): If the maturity value was ₹10,600, the setup leads to the quadratic equation n² + 241n – 6360 = 0, which solves to n = 24 months.
  4. Account Maturing to ₹1,278 (with adjusted MV): The setup leads to the quadratic equation n² + 201n – 2556 = 0, which solves to n = 12 months.
  5. Setting up the Equation: The equation 6165 = 500n + 1.25n(n+1) simplifies to the standard quadratic form n² + 401n – 4932 = 0. Here, a = 1, b = 401, c = -4932.

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