What Is Interest?

Interest is money calculated as a percentage of an amount of money over time.

Interest appears in many financial situations.

If money is placed in a savings account, interest may be earned.

If money is borrowed, interest may be charged.

For example, suppose $1,000 earns $50 in interest.

The balance becomes:

$1,000 + $50 = $1,050

Interest has increased the amount of money by $50.

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Why Does Interest Exist?

Interest is connected with the use of money over time.

When money is saved or invested, some financial products may provide a return for keeping money with the provider.

When money is borrowed, a lender may charge for providing the money.

This means interest can work in two directions:

Savings → interest may increase your money

Borrowing → interest may increase what you owe

The exact terms depend on the financial product.

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Principal

The principal is the starting amount of money.

Suppose:

$2,000

is placed into a savings account.

The principal is:

P = $2,000

If $500 is borrowed, the original principal of the loan is:

P = $500

The principal is the amount on which interest calculations begin.

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Interest Rate

The interest rate is the percentage used to calculate interest.

For example:

5% per year

means an annual interest rate of:

5%

Convert to decimal form:

5% = 0.05

Other examples:

2% = 0.02

4.5% = 0.045

7% = 0.07

12% = 0.12

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Time

Interest depends on how long money is saved or borrowed.

A rate such as:

4% per year

must be considered together with the amount of time.

For simple interest:

2 years produces twice as much interest as 1 year.

5 years produces five times as much interest as 1 year.

This happens because simple interest is calculated from the same original principal each year.

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Simple Interest

Simple interest is interest calculated only on the original principal.

The equation is:

I = Prt

where:

• I = interest
• P = principal
• r = interest rate as a decimal
• t = time

When the interest rate is annual, time is normally measured in years.

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Example: Simple Interest

Suppose:

Principal:

P = $1,000

Interest rate:

r = 5% = 0.05

Time:

t = 3 years

Use:

I = Prt

Calculate:

I = 1000 × 0.05 × 3

I = $150

Therefore:

Interest earned = $150

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Finding the Final Amount

The final amount is the principal plus the interest.

A = P + I

For the previous example:

A = $1,000 + $150

A = $1,150

After 3 years:

Principal = $1,000

Interest = $150

Final amount = $1,150

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Interest Each Year

For simple interest:

$1,000 at 5% per year

produces:

$1,000 × 0.05 = $50

each year.

Therefore:

The same amount of interest is added each year because the interest is calculated from the original $1,000.

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Simple Interest Produces Linear Growth

Because the same amount is added during each equal time period, simple interest creates linear growth.

For example:

$1,000 → $1,050 → $1,100 → $1,150 → $1,200

The increase is always:

+$50

This is different from compound growth, where the amount added can increase over time.

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A Formula for the Final Amount

Since:

A = P + I

and:

I = Prt

we can write:

A = P + Prt

Factor out P:

A = P(1 + rt)

Therefore, for simple interest:

A = P(1 + rt)

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Worked Example: Savings

A student saves:

$800

at a simple interest rate of:

3% per year

for:

4 years

Calculate:

I = Prt

I = 800 × 0.03 × 4

I = $96

Final amount:

A = 800 + 96

A = $896

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Worked Example: Larger Principal

Suppose:

$5,000

earns simple interest at:

4% per year

for:

6 years

Interest:

I = 5000 × 0.04 × 6

I = $1,200

Final amount:

$5,000 + $1,200 = $6,200

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Interest for Part of a Year

Time does not always have to be a whole number of years.

Suppose:

$2,400

earns:

5% simple interest per year

for:

6 months

Convert 6 months to years:

6/12 = 0.5 years

Calculate:

I = 2400 × 0.05 × 0.5

I = $60

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Another Time Conversion

Suppose money earns annual simple interest for:

9 months

Convert:

9/12 = 0.75 years

If:

P = $1,600

and:

r = 6%

then:

I = 1600 × 0.06 × 0.75

I = $72

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Finding the Principal

The simple interest equation can be rearranged.

Starting with:

I = Prt

To find principal:

P = I / rt

Example:

$120 of simple interest is earned at 4% per year for 3 years.

Calculate:

P = 120 / (0.04 × 3)

P = 120 / 0.12

P = $1,000

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Finding the Interest Rate

Rearrange:

I = Prt

to:

r = I / Pt

Suppose:

Principal = $2,000

Interest = $240

Time = 3 years

Calculate:

r = 240 / (2000 × 3)

r = 0.04

Convert to a percentage:

0.04 = 4%

Therefore:

interest rate = 4% per year

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Finding the Time

Rearrange:

I = Prt

to:

t = I / Pr

Suppose:

Principal = $1,500

Rate = 5%

Interest = $225

Calculate:

t = 225 / (1500 × 0.05)

t = 225 / 75

t = 3 years

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Percentage Growth

Interest is one example of percentage growth.

Percentage growth means that a quantity increases by a percentage of some reference amount.

Suppose a quantity grows from:

$500 to $550

Increase:

$550 − $500 = $50

Percentage growth:

50 / 500 × 100%

= 10%

Therefore:

the amount increased by 10%

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Percentage Increase Formula

The percentage increase is:

Percentage Increase = Increase / Original Amount × 100%

where:

Increase = New Amount − Original Amount

For example:

Original amount:

$800

New amount:

$920

Increase:

$920 − $800 = $120

Percentage increase:

120 / 800 × 100%

= 15%

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Growth Multipliers

A percentage increase can also be calculated using a multiplier.

For example:

A 10% increase means:

100% + 10% = 110%

Convert:

110% = 1.10

Therefore:

New Amount = Original Amount × 1.10

Example:

$600 × 1.10 = $660

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More Growth Multipliers

5% growth → ×1.05

10% growth → ×1.10

15% growth → ×1.15

20% growth → ×1.20

25% growth → ×1.25

For example:

$400 increases by 25%.

$400 × 1.25 = $500

Increase:

$100

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Simple Interest vs Percentage Growth

Suppose $1,000 earns 10% simple interest each year.

Each year's interest is:

10% of the original $1,000

Therefore:

$100 per year

Balances:

Year 0: $1,000

Year 1: $1,100

Year 2: $1,200

Year 3: $1,300

The percentage rate is applied to the original principal each year.

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Compound Interest

Many real savings products use compound interest rather than simple interest.

With compound interest, interest can be calculated on:

the principal plus previously added interest

This means interest can itself begin earning interest.

Compound interest will be explored more fully in more advanced financial mathematics, but it is useful to understand the basic difference.

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Simple vs Compound Growth

Consider $1,000 at 10% annually.

With simple interest:

Year 1:

$1,100

Year 2:

$1,200

Year 3:

$1,300

If 10% compound growth is applied annually:

Year 1:

$1,000 × 1.10 = $1,100

Year 2:

$1,100 × 1.10 = $1,210

Year 3:

$1,210 × 1.10 = $1,331

The difference grows over time because compound growth repeatedly applies the percentage to a changing balance.

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Why Time Matters

Consider simple interest of:

5% per year on $2,000

Interest each year:

$2,000 × 0.05 = $100

After 1 year:

$100 interest

After 5 years:

$500 interest

After 10 years:

$1,000 interest

With simple interest, doubling the time doubles the total interest when the principal and rate remain unchanged.

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Why the Interest Rate Matters

Compare $2,000 invested for 5 years.

Option A:

3% simple interest

Interest:

2000 × 0.03 × 5 = $300

Option B:

5% simple interest

Interest:

2000 × 0.05 × 5 = $500

The higher rate produces more interest when the principal and time are the same.

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Why the Principal Matters

Compare two amounts at:

4% simple interest for 3 years

$1,000:

I = 1000 × 0.04 × 3 = $120

$5,000:

I = 5000 × 0.04 × 3 = $600

A larger principal produces more interest when the rate and time are unchanged.

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Comparing Savings Options

Suppose $3,000 could be placed in either of two hypothetical simple-interest savings options.

Option A:

3% per year for 4 years

Option B:

4% per year for 3 years

Option A:

I = 3000 × 0.03 × 4

I = $360

Final amount:

$3,360

Option B:

I = 3000 × 0.04 × 3

I = $360

Final amount:

$3,360

Despite different rates and times, the interest is the same in this example.

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Comparing Options Fairly

When comparing financial options, make sure the conditions are comparable.

Check:

• starting amount
• interest rate
• length of time
• whether the rate is annual or monthly
• simple or compound interest
• how often interest is calculated
• fees
• restrictions or penalties
• whether the rate can change

A higher advertised rate alone does not necessarily provide enough information for a complete comparison.

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Example: Rate vs Fees

Consider two hypothetical one-year options for $1,000.

Option A:

5% simple interest with a $30 fee

Interest:

$1,000 × 0.05 = $50

Net increase after the stated fee:

$50 − $30 = $20

Option B:

4% simple interest with no fee

Interest:

$1,000 × 0.04 = $40

The higher advertised interest rate does not necessarily produce the larger net gain once the stated fee is included.

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Interest and Borrowing

Interest also affects borrowing.

Suppose someone borrows:

$2,000

at:

6% simple interest per year

for:

2 years

Interest:

I = 2000 × 0.06 × 2

I = $240

Total amount:

$2,000 + $240 = $2,240

In this simplified example, the borrower pays $240 in interest.

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Cost of Borrowing

The original amount borrowed is not always the total amount eventually paid.

If:

Principal = $5,000

Simple interest = 8%

Time = 3 years

Then:

I = 5000 × 0.08 × 3

I = $1,200

Total:

$5,000 + $1,200 = $6,200

This simplified calculation shows why interest is an important part of evaluating borrowing costs.

Real loans may use more complex calculations and include additional fees.

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Interest Rates and Financial Decisions

Suppose two simplified loans both provide:

$4,000

Loan A:

5% simple interest for 2 years

Interest:

4000 × 0.05 × 2 = $400

Total:

$4,400

Loan B:

4% simple interest for 4 years

Interest:

4000 × 0.04 × 4 = $640

Total:

$4,640

A lower annual rate does not automatically mean a lower total interest cost when the time periods differ.

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Interpreting "Per Year"

Suppose an interest rate is:

6% per year

The words per year are important.

For simple interest:

$1,000 at 6% per year produces:

$60 per year

If the money remains for:

3 years

total simple interest is:

3 × $60 = $180

Always identify the time unit associated with the rate.

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Annual Rates and Months

Suppose:

P = $4,000

r = 3% per year

t = 9 months

Convert time:

9/12 = 0.75 years

Then:

I = 4000 × 0.03 × 0.75

I = $90

The rate and time must use compatible units.

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Growth in Other Situations

Percentage growth is not limited to money.

It can describe:

• population growth
• business sales
• prices
• measurements
• production
• website users
• quantities in scientific models

For example:

A population increases from:

5,000 to 5,400

Increase:

400

Percentage growth:

400 / 5000 × 100%

= 8%

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Percentage Points vs Percentage Growth

These are different ideas.

Suppose a rate changes from:

4% to 6%

The increase is:

2 percentage points

But relative to the original 4%:

(6 − 4) / 4 × 100% = 50%

So the rate increased by:

2 percentage points

or:

50% relative to its original value

These statements describe different comparisons.

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Interpreting Financial Tables

Suppose three hypothetical simple-interest options are shown:

Calculate interest.

Option A:

2000 × 0.03 × 4 = $240

Option B:

2000 × 0.04 × 3 = $240

Option C:

2000 × 0.05 × 2 = $200

The table alone provides the inputs. Mathematical calculations allow the outcomes to be compared.

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Interpreting Growth Graphs

Simple interest creates a straight-line pattern when the rate and principal remain constant.

When reading a growth graph, check:

• starting value
• time scale
• units
• rate of increase
• whether growth is linear or curved
• whether equal time intervals produce equal increases
• whether the graph represents simple or compound growth

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Worked Example 1: Simple Interest

Principal:

$1,200

Rate:

4% per year

Time:

5 years

Calculate:

I = 1200 × 0.04 × 5

I = $240

Final amount:

$1,440

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Worked Example 2: Short-Term Interest

Principal:

$3,000

Rate:

6% per year

Time:

6 months = 0.5 years

Calculate:

I = 3000 × 0.06 × 0.5

I = $90

Final amount:

$3,090

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Worked Example 3: Find the Rate

Principal:

$2,500

Interest:

$300

Time:

4 years

Use:

r = I / Pt

r = 300 / (2500 × 4)

r = 0.03

Therefore:

r = 3% per year

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Worked Example 4: Find the Time

Principal:

$800

Rate:

5%

Interest:

$160

Use:

t = I / Pr

t = 160 / (800 × 0.05)

t = 4 years

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Worked Example 5: Percentage Growth

A savings balance increases from:

$1,500 to $1,620

Increase:

$120

Percentage growth:

120 / 1500 × 100%

= 8%

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Worked Example 6: Growth Multiplier

An amount of:

$750

increases by:

12%

Multiplier:

1.12

Calculate:

$750 × 1.12 = $840

Increase:

$90

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Worked Example 7: Compare Savings Options

Suppose $4,000 is placed for 3 years.

Option A:

3.5% simple interest

Interest:

4000 × 0.035 × 3 = $420

Final amount:

$4,420

Option B:

4% simple interest with a stated $75 total fee

Interest:

4000 × 0.04 × 3 = $480

Amount before fee:

$4,480

After the stated fee:

$4,405

Comparing the final amounts provides more information than comparing the advertised rates alone.

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Worked Example 8: Borrowing

Borrow:

$6,000

Simple interest rate:

7%

Time:

2 years

Interest:

6000 × 0.07 × 2 = $840

Total amount under this simplified model:

$6,840

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Worked Example 9: Compare Growth

Quantity A:

$500 → $575

Growth:

$75

Percentage growth:

75 / 500 × 100% = 15%

Quantity B:

$1,000 → $1,120

Growth:

$120

Percentage growth:

120 / 1000 × 100% = 12%

Quantity B increased by more dollars, while Quantity A had the larger percentage growth.

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Worked Example 10: Multi-Step Financial Comparison

Suppose $5,000 is saved for 4 years.

Option A:

4% simple interest

Interest:

5000 × 0.04 × 4 = $800

Final amount:

$5,800

Option B:

3.5% simple interest plus a stated $150 bonus at the end

Interest:

5000 × 0.035 × 4 = $700

Add bonus:

$5,000 + $700 + $150 = $5,850

The final amounts are:

$5,800 and $5,850

This illustrates why all relevant numerical terms should be included when comparing financial options.

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Estimating Interest

Estimation helps check answers.

Suppose:

$1,980 at 6% simple interest for 3 years

Estimate the principal as:

$2,000

6% of $2,000:

≈ $120 per year

For 3 years:

≈ $360

Exact calculation:

1980 × 0.06 × 3 = $356.40

The exact answer is close to the estimate.

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Checking Whether an Answer Is Reasonable

Suppose:

$1,000

earns:

5% simple interest

for:

2 years

5% of $1,000 is:

$50

So approximately:

$100

should be earned over two years.

If a calculation gives:

$1,000 interest

the result should immediately be questioned.

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Comparing Dollar Growth and Percentage Growth

Suppose:

Account A grows:

$500 → $600

Increase:

$100

Percentage growth:

20%

Account B grows:

$2,000 → $2,200

Increase:

$200

Percentage growth:

10%

Account B gained more dollars.

Account A had greater percentage growth.

Neither statement contradicts the other because they measure different things.

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Interest and Inflation

When evaluating money over long periods, another idea that can matter is inflation, which refers to a general increase in prices over time.

For example, if savings grow by 3% while prices also change, simply knowing the account balance does not tell us everything about how purchasing power has changed.

A more advanced financial analysis may therefore consider both:

• nominal growth in money
• changes in purchasing power

This is one reason financial decisions often require more information than a single interest rate.

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A Reliable Simple Interest Strategy

Step 1: Identify the principal, P.

Step 2: Identify the interest rate, r.

Step 3: Convert the percentage rate to a decimal.

Step 4: Identify the time, t.

Step 5: Make sure the time unit matches the rate.

Step 6: Use:

I = Prt

Step 7: Calculate the interest.

Step 8: If required, calculate:

A = P + I

Step 9: Include the correct units and currency.

Step 10: Estimate and check whether the result is reasonable.

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Comparing Financial Options

When comparing savings or borrowing options:

1. Compare the same starting amount.

2. Compare over the same time period when possible.

3. Identify whether the interest is simple or compound.

4. Check the interest rate and its time period.

5. Calculate the actual interest or growth.

6. Calculate the final amount.

7. Include stated fees, bonuses, or other numerical conditions.

8. Consider restrictions or conditions that may affect the comparison.

9. Compare both dollar changes and percentage changes when useful.

10. Use the numerical evidence to understand the trade-offs.

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Common Mistakes

Mistake 1: Using the percentage as a whole number

Incorrect:

1000 × 5 × 3

Correct:

1000 × 0.05 × 3

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Mistake 2: Forgetting to convert months to years

If the rate is annual:

6 months = 0.5 years

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Mistake 3: Confusing interest with the final amount

If:

I = $200

and:

P = $1,000

then:

A = $1,200

not $200.

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Mistake 4: Calculating simple interest from the changing balance

Simple interest is calculated from the original principal.

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Mistake 5: Assuming simple and compound interest are identical

They can produce different results because they calculate growth differently.

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Mistake 6: Comparing rates without considering time

A lower annual rate over a much longer period can produce more total interest.

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Mistake 7: Ignoring fees or other conditions

The advertised interest rate may not provide enough information to compare financial products completely.

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Mistake 8: Confusing dollar growth with percentage growth

An increase of $200 is not necessarily a larger percentage increase than an increase of $100.

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Did You Know?

Interest connects financial mathematics with several major mathematical ideas.

Interest problems can involve:

• percentages
• decimals
• ratios
• proportions
• algebra
• linear relationships
• exponential relationships
• tables
• graphs
• estimation
• financial decision-making

Simple interest provides an especially useful example of linear growth, while compound interest introduces exponential growth.

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Key Terms

• Interest: Money earned or charged based on an amount of money and an interest rate.
• Principal: Original amount saved, invested, or borrowed.
• Interest rate: Percentage used to calculate interest.
• Simple interest: Interest calculated only on the original principal.
• Final amount: Principal plus accumulated interest.
• Annual rate: Interest rate stated per year.
• Growth: Increase in a quantity over time.
• Percentage growth: Increase expressed as a percentage of the original amount.
• Growth multiplier: Decimal multiplier used to calculate an increased value.
• Compound interest: Interest calculated using a balance that can include previously added interest.
• Linear growth: Growth involving equal additions over equal intervals.
• Compound growth: Growth in which repeated percentage changes can produce increasing absolute changes.
• Fee: Additional financial charge.
• Purchasing power: Amount of goods or services that money can buy.
• Inflation: General increase in prices over time.

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Key Equations

Simple interest:

I = Prt

where:

I = interest

P = principal

r = interest rate as a decimal

t = time

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Final amount:

A = P + I

or:

A = P(1 + rt)

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Percentage growth:

Percentage Growth = Increase / Original Amount × 100%

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Increase:

Increase = New Amount − Original Amount

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Growth multiplier:

New Amount = Original Amount × (1 + Growth Rate)

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Key Takeaways

• Interest is money calculated from an amount of money over time.
• Interest may be earned on savings or charged on borrowing.
• The principal is the original amount.
• Interest rates are commonly expressed as percentages.
• Simple interest is calculated only from the original principal.
• The simple interest equation is I = Prt.
• The final amount is A = P + I.
• The interest rate should be converted from a percentage to a decimal before using the formula.
• The units of time must match the time period of the interest rate.
• Simple interest produces linear growth because equal amounts of interest are added over equal time intervals.
• Percentage growth compares an increase with the original amount.
• Growth multipliers provide a convenient way to calculate percentage increases.
• Compound interest differs from simple interest because previous interest can contribute to later interest calculations.
• A higher interest rate can increase savings growth but can also increase borrowing costs.
• Time has a major effect on the total amount of interest.
• A lower annual rate does not automatically mean a lower total borrowing cost if the time periods differ.
• Savings options should be compared using consistent starting amounts, time periods, interest methods, fees, and other relevant conditions.
• Dollar growth and percentage growth provide different information.
• Estimation helps identify unreasonable financial calculations.
• Interest calculations provide numerical evidence that can help people understand the consequences of different financial choices.