Percents · Grades 7

Simple Interest: The Formula and What Each Letter Means

Quick answer

Simple interest is found with I = Prt: the principal times the rate times the time. 500 dollars at 4% for 3 years earns 500 × 0.04 × 3 = 60 dollars. The rate must be a decimal and the time must be in the same units the rate is quoted in, almost always years. Simple interest is charged on the original amount only, never on interest already earned.

What you'll learn

  • Use the simple interest formula to find interest earned or owed
  • Rearrange the formula to find the principal, rate or time
  • Explain how simple interest differs from compound interest

The formula

I=PrtI = Prt
LetterMeansWatch out for
IIthe interest — the extra moneynot the total
PPthe principal — the amount you started with
rrthe rate, as a decimal4%4\% is 0.040.04, not 44
ttthe time, in yearsmonths must be converted

$500 at 4%4\% for 33 years:

I=500×0.04×3=$60I = 500 \times 0.04 \times 3 = \$60

Why it is only a percent, repeated

There is nothing new here. PrtPrt is a percent of a number worked out once and then counted tt times.

500×0.04⏟one year’s interest  =  $20×  3  =$60\underbrace{500 \times 0.04}_{\text{one year's interest} \;=\; \$20} \times \;3\; = \$60

One year earns $20. Three years earn three lots of $20. That is the whole formula, and seeing it that way is what makes the rearrangements below follow rather than needing to be memorised.

The two things that go wrong

The rate must be a decimal. 4%4\% is 0.040.04. Using 44 makes the answer a hundred times too big, and the size of the error is so large it is usually visible: an account earning more than it holds is not plausible.

The time must match the rate. Rates are quoted per year, so months have to be converted.

9 months=912=0.75 years9 \text{ months} = \frac{9}{12} = 0.75 \text{ years}

Interest is not the total

The formula gives the interest, not the balance.

total=P+I\text{total} = P + I

$500 that earns $60 is worth $560. Questions ask for one or the other, and answering with the wrong one is the most common mark lost on this topic.

Rearranging for a missing letter

Because the three quantities are multiplied, any one can be found by dividing — the same move as any other literal equation.

I=PrtP=Irtr=IPtt=IPrI = Prt \qquad P = \frac{I}{rt} \qquad r = \frac{I}{Pt} \qquad t = \frac{I}{Pr}

Nothing needs memorising here beyond “divide by the other two”.

Simple against compound

Simple interest is always taken on the original amount. Compound interest is taken on whatever is there now, which includes interest already earned.

$1000 at 10%10\%:

YearSimpleCompound
11$1100$1100
22$1200$1210
33$1300$1331
1010$2000$2594

They agree for one year and separate afterwards. Compound interest is a percent increase applied repeatedly, which is why it pulls away — each year’s rise is calculated on a bigger number than the last.

Most real savings accounts and loans compound. Simple interest is the version worth learning first, and it is what a question means unless it says otherwise.

Worked examples

Common mistakes

Practice problems

  1. Find the simple interest on $400 at 5%5\% for 22 years.

    Answer

    $40

    Full solution

    I=400×0.05×2=40I = 400 \times 0.05 \times 2 = 40.

  2. Find the simple interest on $1500 at 3%3\% for 44 years.

    Answer

    $180

    Full solution

    I=1500×0.03×4=180I = 1500 \times 0.03 \times 4 = 180.

  3. $900 is invested at 6%6\% for 33 years. What is the account worth at the end?

    Hint

    The question asks for the total, not the interest.

    Answer

    $1062

    Full solution

    I=900×0.06×3=162I = 900 \times 0.06 \times 3 = 162, and the total is 900+162=1062900 + 162 = 1062.

  4. Find the interest on $1200 at 4%4\% for 66 months.

    Hint

    Six months is half a year.

    Answer

    $24

    Full solution

    t=612=0.5t = \tfrac{6}{12} = 0.5, so I=1200×0.04×0.5=24I = 1200 \times 0.04 \times 0.5 = 24.

  5. $5000 earns $750 in 55 years. What is the rate?

    Answer

    3%3\%

    Full solution

    r=7505000×5=75025000=0.03r = \tfrac{750}{5000 \times 5} = \tfrac{750}{25000} = 0.03, which is 3%3\%.

  6. How long does $800 take to earn $96 at 4%4\%?

    Answer

    33 years

    Full solution

    t=96800×0.04=9632=3t = \tfrac{96}{800 \times 0.04} = \tfrac{96}{32} = 3.

  7. How much must be invested at 5%5\% to earn $250 in 22 years?

    Answer

    $2500

    Full solution

    P=2500.05×2=2500.1=2500P = \tfrac{250}{0.05 \times 2} = \tfrac{250}{0.1} = 2500.

  8. A $2000 loan is taken at 9%9\% for 1818 months. How much interest is owed?

    Hint

    Convert the months to years first.

    Answer

    $270

    Full solution

    t=1812=1.5t = \tfrac{18}{12} = 1.5 years.

    I=2000×0.09×1.5=270I = 2000 \times 0.09 \times 1.5 = 270.

  9. $1000 is invested at 10%10\% for 22 years. Find the total under simple interest, then under compound interest, and say which is larger and by how much.

    Hint

    For the compound figure, apply a 10%10\% increase twice.

    Answer

    Simple $1200, compound $1210. Compound is $10 more.

    Full solution

    Simple: I=1000×0.1×2=200I = 1000 \times 0.1 \times 2 = 200, so the total is $1200.

    Compound: 1000×1.1×1.1=12101000 \times 1.1 \times 1.1 = 1210.

    The extra $10 is 10%10\% of the first year’s $100 of interest — interest earned on interest, which simple interest never charges.

  10. Leo works out the interest on $600 at 5%5\% for 33 years and gets $9000. Find his error.

    Hint

    Compare the answer with the amount invested.

    Answer

    He used r=5r = 5 instead of 0.050.05. The interest is $90.

    Full solution

    An answer of $9000 on a $600 investment is fifteen times the money invested, which no rate of 5%5\% could produce. The size alone rules it out.

    Correctly: I=600×0.05×3=90I = 600 \times 0.05 \times 3 = 90.

    Leo’s figure is exactly 100100 times too big, which is the signature of a percent used without dividing by 100100.

Frequently asked questions

What is the simple interest formula?

I = Prt, where I is the interest, P is the principal you started with, r is the rate as a decimal, and t is the time in years. The three are multiplied together.

Do I use 4 or 0.04 for a 4% rate?

Use 0.04. The formula multiplies by the rate as a decimal, so a percent has to be divided by 100 first. Using 4 makes every answer a hundred times too big.

What if the time is in months?

Convert it to years, because the rate is quoted per year. Nine months is 9/12, which is 0.75 years. Leaving it as 9 would charge nine years of interest.

What is the difference between simple and compound interest?

Simple interest is always calculated on the original amount. Compound interest is calculated on the amount currently there, so it includes interest already earned, and it grows faster over time.

How do I find the total rather than the interest?

Add the interest to the principal. The formula gives only the interest earned or owed, so a question asking what the account is worth needs one more step.

What to learn next

Formulas on this page

Key terms in this lesson

Interest
Interest is the extra money paid for the use of money — earned on savings or owed on a loan. It is worked out as a percent of an amount, over a period of time.
Principal
The principal is the starting amount of money in an interest calculation — what was invested or borrowed, before any interest is added.

Standards alignment

This lesson covers the following Common Core State Standards for Mathematics.

  • CCSS.MATH.CONTENT.7.RP.A.3Ratios and Proportional RelationshipsUse proportional relationships to solve multistep ratio and percent problems.
For teachersTick to print a blank worksheet; leave it clear to print the lesson with the full answer key.