5.18: Solve Real World Problems Involving Compound Interest
Have you ever had a bank account? Did you ever calculate interest? Take a look at this dilemma and you will learn to understand compound interest.
“Wow! $4320 is a terrific amount!” Jeremy exclaimed when Candice filled him in on the balance in the savings account.
“It is great,” Marcus commented, “But I think we would have made out better if the interest had been compounded monthly.”
“Really, what does that mean?” Jeremy asked.
“It means that the interest is earned and then reinvested and you earn interest on the earned interest,” Marcus explained.
“Really?” Candice asked.
“Yes, let me explain,” Marcus said.
Before Marcus explains, it is time to learn about compound interest. Once you have learned the information in this Concept, you will be ready to figure out if Marcus is correct.
Guidance
Interest is important. It is not only big business for banks and investors but it is a way of securing a comfortable retirement and reaching financial goals. Understanding interest helps you to make the best decisions. The last section dealt with simple interest which illustrates the basic idea of interest.
In most cases in the real world, however, interest is calculated not with the simple interest formula \begin{align*}I = prt\end{align*} but with the compound interest formula \begin{align*}A = P(1+r)^t\end{align*}. Basically, this formula accounts for the fact that as you invest and earn interest, your balance grows. You are not only due interest, then, on your original balance, but on the new balance which includes the first installment(s) of interest. You get paid interest on the interest.
Take a look at this situation.
You invest $100 for 3 years at 10% interest. After one year, you’d have the original amount plus the 10% interest. You’d have $110. Then, in the second year, you’d get paid the same 10% interest but not on just $100 but on $110. In the second year you’d earn $11 in interest and you’d have $121. In the third year you’d earn $12.10 and your ending balance would be $133.10. You got paid interest on interest and this made a difference in your final balance.
The situation above shows compounding every year. That means that you get paid interest once a year on your balance. In the real world, interest is oftentimes compounded monthly or daily. For compound interest, we’ll use the formula \begin{align*}A = P(1+r)^t\end{align*} where \begin{align*}A\end{align*} is your final balance, \begin{align*}P\end{align*} is the principal amount, \begin{align*}r\end{align*} is the interest rate for the period (daily, monthly, semi-annually, annually, etc.) and \begin{align*}t\end{align*} is the number of time periods for which the money is invested. If compounding occurs monthly, there will be 12 periods per year.
In this situation above, \begin{align*}P\end{align*} was $100, \begin{align*}r\end{align*} was 10%, and \begin{align*}t\end{align*} was 3 periods since the interest was compounded yearly. We would substitute these values in the compound interest formula:
\begin{align*}A &= P(1+r)^t\\ A &= 100(1+.10)^3\\ A &= 100(1.10)^3\\ A &= 100(1.331)\\ A &= 133.10\end{align*}
Using the formula we arrive at the same quantity discussed.
Exactly! We use the formula to figure out the interest in a systematic way! You may be wondering what the big difference is between simple interest and compound interest. At first glance, it doesn’t seem like there is a big difference, however, if you figure out the math, the difference will become clear.
Let’s see how different it might be using the same principal for the same amount of time at the same rate. However, one account will be paid simple interest and the other will be compounded yearly.
Account 1—Simple Interest | Account 2—Compound Interest |
---|---|
Principal Amount: $20,000 | Principal Amount: $20,000 |
Interest Rate: 8% per year | Interest Rate: 8% per year |
Time Frame: 20 years | Time Frame: 20 years |
Simple Interest formula \begin{align*}I = prt\end{align*} | Compound Interest formula \begin{align*}A = P(1+r)^t\end{align*} |
\begin{align*}p = 20000, r = .08, t = 20\end{align*} | \begin{align*}P = 20000, r = .08, t = 20\end{align*} |
\begin{align*}I = prt\end{align*} | \begin{align*}A = P(1+r)^t\end{align*} |
\begin{align*}I = 20000 \cdot .08 \cdot 20\end{align*} | \begin{align*}A = 20000(1+.08)^{20}\end{align*} |
\begin{align*}I = 32000\end{align*} | \begin{align*}A = 20000(1.08)^{20}\end{align*} |
\begin{align*}20000+32000 = 52000\end{align*} | \begin{align*}A = 20000(4.661)\end{align*} |
\begin{align*}A = 93220\end{align*} |
Simple interest gives you a balance of $52,000...not bad. Compound interest gives you a final balance of $93,220. That’s a huge difference! The total is over $40,000 more by compounding the interest.
Compare the final balances on a principal of $10,000 paid simple interest of 5% for a year or compound interest of 5% per year compound monthly. Then compare after 5 years and 10 years.
Years | Simple Interest | Compound Interest |
---|---|---|
1 | $10,500 | $10,512 |
5 | $12,500 | $12,834 |
10 | $15,000 | $16,470 |
So you can see that there is big difference between the simple interest calculation and the compound interest calculation.
Calculate the following compound interest calculated yearly.
Example A
Principal = $3000, Rate = 4%
Solution: \begin{align*}$4,803\end{align*}
Example B
Principal = $5000, Rate = 3%
Solution: \begin{align*}$7,100\end{align*}
Example C
Principal = $12,000, Rate = 2%
Solution: \begin{align*}$15,120\end{align*}
Now let's go back to the dilemma from the beginning of the Concept.
First, let’s figure out the balance if the interest had been compound interest.
\begin{align*}A &= P(r + 1)^t\\ A &= 4000(.04 + 1)^{24}\\ A &= 4000 (1.04)^{24}\\ A &= 4000(2.56)\\ A &= \$10,240\end{align*}
In this case, the amount of the balance with compounded interest would have been more than double the balance with simple interest. Marcus was correct after all!
Vocabulary
- Simple Interest
- interest calculated by only considering the principal times the rate times the time.
- Compound Interest
- when interest is earned, it is reinvested and you earn interest on the interest which is then added to the balance.
Guided Practice
Here is one for you to try on your own.
A fireman invests $40,000 in a retirement account for 2 years. The interest rate is 6%. The interest is compounded monthly. What will his final balance be?
Solution
Notice that we divide by 12 because there are twelve months in a year and the interest is compounded monthly.
\begin{align*}r &= \frac{.06}{12} = .005\\ P &= 40000, t = 24\\ A &= 40000(1+.005)^{24}\\ A &= 40000(1.005)^{24}\\ A &= 40000(1.127)\\ A &= 45080\end{align*}
Final balance is $45,080.
Video Review
Khan Academy Intro to Compound Interest
Practice
Directions: Calculate the simple interest by using \begin{align*}I = PRT\end{align*}.
- Principal = $2000, Rate = 5%, Time = 3 years
- Principal = $12,000, Rate = 4%, Time = 2 years
- Principal = $10,000, Rate = 5%, Time = 5 years
- Principal = $30,000, Rate = 2.5%, Time = 10 years
- Principal = $12,500, Rate = 3%, Time = 8 years
- Principal = $34,500, Rate = 4%, Time = 10 years
- Principal = $16,000, Rate = 3%, Time = 5 years
- Principal = $120,000, Rate = 5%, Time = 4 years
Directions: Calculate the following compound interest calculated yearly.
- Principal = $3000, Rate = 4%
- Principal = $5000, Rate = 3%
- Principal = $12,000, Rate = 2%
- Principal = $34,000, Rate = 5%
- Principal = $18,000, Rate = 3%
- Principal = $7800, Rate = 4%
- Principal = $8500, Rate = 3%
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Here you'll solve real-world problems involving compound interest.