**High School Number Sense Lesson 80: Estimating Interest Earned on Money**

Today's lesson, which wraps up our "once-through" of the most common concepts on a high school test, is probably the most practical lesson to understand. My education is in personal finance, and it baffles me how many people do not grasp the concept of compounding interest. One of my favorite quotes (by an unknown author) goes something like this: "He who understands the idea of compound interest

**earns it**, while he who doesn't understand it

**pays it**.

This concept appeared

**10 times**on last year's tests, always at

**question # 80**.

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The biggest key to getting these questions right is to know the difference between simple interest and compound interest:

**Simple interest**is interest earned only on the principal (the original amount invested)**Compound interest**is interest earned on the principal and interest.

Since simple interest should be simple, let's start there.

**How to Calculate Simple Interest:**

Use the formula

**I = Prt**, where:

**P**= the principal (original amount) invested**r**= the rate of interest the investment earns**t**= the time invested

*One way to reduce errors is to remove the % sign while removing 2 zeros from the principal amount.*

**Example 1: * The interest on $5000 for 2.5 years at a simple interest rate of 1.5% is ___ dollars (integer)**

- I = Prt = ($5000)(1.5%)(2.5)
- I = (50)(1.5)(2.5) = 50 x 3.75 =
**187.50** - Since the answer has to be an integer, write
**187**.

**Example 2: * The simple interest on $4,500.00 at 4.5% for 4.5 years is ___ dollars (integer)**

- I = Prt = ($4500)(4.5%)(4.5)
- I = (45)(4.5)(4.5) = 45 x 20.25
- Round 20.25 down to 20. 45 x 20 =
**900** - The exact answer is
**$911.25**, and the acceptable (+/- 5%) range is**from 866 to 956**.

**Example 3: * The simple interest on $1250 at 2.5% for 3.75 years is ___ dollars (integer)**

- I = Prt = ($1250)(2.5%)(3.75)
- I = (12.5)(2.5)(3.75)
- Use
**Mult125var**to multiply 3.75 x 12.5: 3.75 x 1/8 x 100 = .46875 x 100 =**46.875** - We are left with 2.5 x 46.875, so change it to 2.5 x 47. Use
**Mult25**: 47 x 1/4 x 10 = 11.75 x 10 =**117.5** - Since the answer has to be an integer, write
**117**. - The exact answer is
**$117.1875**, and the acceptable (+/- 5%) range is**from 112 to 123**.

**How to Calculate Compound Interest:**

**I = P(1 + r/n)**, where:

^{nt}*

**P**= the principal (original amount) invested

*

**r**= the rate of interest the investment earns

*

**n**= the number of periods compounded per year

*

**t**= the number of years invested

*This gives you the total amount after the investment period, which includes the principal. If you are asked for only the interest, subtract the principal back out.*

**Example 4: * The compound interest on $3000 for 2 years at 6% compounded annually is ___ dollars (integer)**

**I = P(1 + r/n)**.

^{nt}2. (3000)(1 + .06/1)

^{1 x 2}= 3000(1.06)

^{2}

3. Use

**Mult101-109**to determine that 1.06

^{2}is

**1.1236**

4. 3000 x 1.1236 = 3 x 1123.6 =

**3370.8**

5. Subtract the 3000 out. 3370.8 - 3000 =

**370.8**. Write

**371**.

6. The acceptable (+/- 5%) range is

**from 353 to 389**.

**Example 5: * The compound interest on $2000 for 4 years at 8% compounded annually is ___ dollars (integer)**

**I = P(1 + r/n)**

^{nt}2. (2000)(1 + .08/1)

^{1 x 4}= 2000(1.08)

^{4}

3. Use

**Mult101-109**to determine that 1.08

^{2}is

**1.1664**. Round this to 1.17.

4. Use

**Mult101-109**to determine that 1.17

^{2}is

**1.3689**. Round this to 1.37.

5. 2000 x 1.37 = 20 x 137 =

**2740**.

6. Subtract the 2000 to get

**740**.

7. The acceptable (+/- 5%) range is

**from 685 - 757**.

**Example 6: * The interest on $2000 for 4 years at 6% compounded semiannually is ___ dollars (integer)**

**I = P(1 + r/n)**

^{nt}2. (2000)(1 + .06/2)

^{2 x 4}= 2000(1.03)

^{8}

3. 1.03

^{8}is [(1.03

^{2})

^{2}]

^{2}

4. Use

**Mult101-109**to determine that 1.03

^{2}is

**1.0609**. Round this to 1.06.

5. Use

**Mult101-109**to determine that 1.06

^{2}is

**1.1236**. Round this to 1.12.

6. Use

**Mult101-109**to determine that 1.12

^{2}is

**1.2544**.

7. 2000 x 1.2544 = 2 x 1254.4 =

**2508.8**.

8. Subtract the 2000 to get

**508.8**. Write

**509**.

9. The actual answer is

**533.54**, and the acceptable (+/- 5%) range is

**from 507 to 560**.

**

*Notice how close we are to the bottom of the range. Rounding and then compounding causes a greater error.*

**Up Next for High School: DivDeciDeci**