PART 1

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In this lesson, we will learn about the compound interest formula.

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Consider a loan of $1000, with the compound interest rate of 10%, and the duration of the loan is 3 years.

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Calculate the total amount of money that we need to pay back, after 3 years.

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Before we learn about the compound interest formula, let's see how compound interest works.

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Let's represent this green bar with the loan amount, $1000.

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Now, after the first year, the interest will be 10% of the loan amount, $1000.

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To calculate the interest, I, we multiply $1000, with the interest rate, 10%.

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Now, before we can calculate, we need to change 10% to a decimal.

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We can do so, by dividing 10% with 100. This gives 0.1.

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$1000 multiply with 0.1, gives $100.

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This is the amount of interest, after the first year.

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Hence, we can find the amount of money that we owe after the first year, by adding $1000, with $100.

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This gives $1100. Let's write this down here.

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After the 2nd year, since the interest is compounding, the interest will be 10% of the current amount, $1100.

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To calculate the interest, I, we multiply $1100 with 10%.

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As calculated earlier, 10% converted to decimal is 0.1. $1100 multiply with 0.1, gives $110.

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This is the loan's interest after the second year.

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With this, we can find the amount of money that we owe after the 2nd year, by adding $1100, with $110. This gives $1210. Let's write this down here.

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After the 3rd year, since the interest is compounding, the interest will be 10% of the current amount, $1210.

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To calculate the interest, I, we multiply $1210, with 10%.

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10% converted to a decimal is, 0.1. $1210 multiply with 0.1, gives $121.

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Therefore, the interest is $121, after the 3rd year.

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With this, we can find the total amount of money to pay back after 3 years, by adding $1210, with $121.

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This gives, $1331. Let’s write it down here.

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Hence, for a loan with compound interest, the total amount of money that we need to pay back after 3 years is, $1331.

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In the second part, we will see how we can find the compound interest formula.

PART 2


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After understanding how compound interest works, let's find the compound interest formula.

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Consider a loan of P dollars, with the compound interest rate of r%, and the loan duration is 3 years.


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Find the total amount of money that we need to pay back, after 3 years.

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Let's start by representing this green bar, with the loan amount, P.

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Now, after the first year, the loan's interest will be r%, of P.

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Hence, we can find the interest, I, by multiplying P, with r.

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With this, we can find the amount of money that we owe, by adding P, with Pr.

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This gives, P + Pr.

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Notice that, these two terms have 'P' as a common factor.

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Hence, we can factorize these 2 terms, by taking out 'P'.

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By doing so, we have P(1+r).

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So, this is the amount of money that we owe, after the first year.

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Next, let's find out the interest at the end of 2nd year.

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Before doing so, we let Q = P(1+r). Hence, this expression becomes Q.

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Since this is a compound interest, the interest will be r%, of Q.

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Hence, we can find the interest, I, by multiplying Q, with r.

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Now, we can find the amount of money that we owe, by adding Q, with Qr.

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By doing so, we have Q + Qr.

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Again, notice that these 2 terms have 'Q' as a common factor.

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Therefore, we can factorize these terms by taking out Q. This gives Q(1+r).

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Let's change Q back to P(1+r). By doing so, we have P(1+r)(1+r).

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Here, P(1+r)(1+r), gives (1+r)^2.

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So, after the second year, the amount of money that we owe is, P(1+r)^2.

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Here, we can see something interesting.

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We know that, P(1+r) is the same as, P(1+r)^1.

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Observe that, after the first year, we have P(1+r)^1. After the second year, we have P(1+r)^2.

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From this, we can guess that, after the 3rd year, the amount money that we need to pay back is, P(1+r)^3.

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Please verify this, to convince yourself.

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Hence, if the loan duration is n year, we will have, P(1+r)^n.

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Therefore, the total amount of money that we need to pay back after n years, A = P(1+r)^n.

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This is the compound interest formula.

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That is all for this lesson. Try out the practice question to further your understanding.