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Simple interest

Ratio, rates and money · level 5, lesson 6 of 7

A question to start

A bank advertises a savings account that pays 3% interest. You put in 1000 dollars and take it out again 4 months later. Do you get 3% of 1000, which is 30 dollars, on top?

No. You get about 10 dollars. The 3% is a rate per year, and you only kept the money in the bank for a third of a year. Interest is a rate, just like a speed: "3% a year" is like "80 km per hour", and how far you get depends on how long you keep going.

The idea

Renting money

Interest is the price of using someone else's money for a while. It works in both directions.

  • When you save money in a bank, the bank uses your money (it lends it to other people), so the bank pays you interest.
  • When you borrow money, you use the lender's money, so you pay the lender interest.

Think of it as rent: you pay rent to use an apartment, and you pay interest to use money.

The words

Word Meaning Letter
Principal The amount saved or borrowed at the start \(P\)
Interest rate The percent charged or paid per year (written as a decimal in formulas) \(r\)
Time How long, in years \(t\)
Interest The extra money paid or earned \(I\)
Amount Principal plus interest, the total at the end \(A\)

"Per year" is sometimes written per annum (Latin for "per year") or "p.a.". Unless a rate clearly says otherwise, an interest rate is a yearly rate.

Simple interest: I = Prt

With simple interest, the interest is worked out on the original principal only, every year the same:

\[I = P \times r \times t\]

Why this formula? In one year, the interest is the rate times the principal: \(r \times P\). For example, 3% of 1000 dollars is \(0.03 \times 1000 = 30\) dollars. Simple interest pays that same amount every year. After 2 years it is \(2 \times 30\); after \(t\) years it is \(t \times 30\), which is \(t \times r \times P\). Nothing more to it than "the yearly interest, times the number of years".

The formula also shows that interest is directly proportional to each of the three: double the principal, the rate or the time (with the other two fixed), and the interest doubles. That is lesson 2 again.

The total at the end is the principal plus the interest:

\[A = P + I = P + Prt = P(1 + rt)\]

A picture: the same step every year

Here is 1000 dollars at 10% simple interest (a big rate, so the steps are easy to see). Blue is the principal, orange the interest so far.

year 0 1 2 3 4 5 1000 1500

The total goes 1000, 1100, 1200, 1300, 1400, 1500: the same amount added each year. That makes simple interest linear growth: on a graph the totals lie on a straight line (but not through the origin, because it starts at 1000). In level 6 you will meet compound interest, where the orange part itself earns interest and the steps get bigger every year.

Time that is not whole years

Since \(r\) is a yearly rate, \(t\) must be in years.

  • Months: divide by 12. 4 months is \(t = \dfrac{4}{12} = \dfrac13\) year.
  • Days: divide by 365. 90 days is \(t = \dfrac{90}{365}\) year. (Some banks use 360 days instead, which gives slightly more interest; the fine print says which.)

That answers the question: \(I = 1000 \times 0.03 \times \dfrac{4}{12} = 10\) dollars.

Using the formula backwards

\(I = Prt\) has four letters. If you know any three, you can find the fourth by dividing:

\[P = \frac{I}{rt}, \qquad r = \frac{I}{Pt}, \qquad t = \frac{I}{Pr}.\]

Where you meet simple interest

Real savings accounts and most loans use compound interest, but simple interest still appears: some short loans, some bonds that pay the same interest every year, car loans in some places, and as a first quick estimate. Real interest rates also change over time, so the rates in this lesson are examples: check the current rate before you save or borrow.

A warning about short loans

A rate per year makes very different loans comparable. Suppose a short-term lender charges a fee of 15 dollars for every 100 dollars borrowed for 2 weeks. That sounds small. But 15% for 14 days, as a simple yearly rate, is \(0.15 \times \dfrac{365}{14} \approx 3.91\), which is about 391% a year. Compare that with a credit card at about 20% a year. Always turn a fee into a yearly rate before you decide.

Worked examples

1. Saving for whole years. 2000 dollars is saved at 3% a year simple interest for 4 years.

  • \(I = 2000 \times 0.03 \times 4 = 240\) dollars.
  • \(A = 2000 + 240 =\) 2240 dollars.

2. Part of a year. 1000 dollars at 3% a year for 4 months.

  • \(t = \dfrac{4}{12}\), so \(I = 1000 \times 0.03 \times \dfrac{4}{12} =\) 10 dollars.

3. A short loan. You borrow 500 dollars at 12% a year simple interest and pay it all back after 3 months.

  • \(I = 500 \times 0.12 \times \dfrac{3}{12} = 15\) dollars.
  • You pay back \(500 + 15 =\) 515 dollars.

4. Finding the rate. 800 dollars earns 48 dollars of simple interest in 2 years. What is the rate?

  • \(r = \dfrac{I}{Pt} = \dfrac{48}{800 \times 2} = \dfrac{48}{1600} = 0.03\), which is 3% a year.

5. Finding the time. How long does 1500 dollars take to earn 300 dollars at 4% a year simple interest?

  • One year's interest is \(0.04 \times 1500 = 60\) dollars.
  • \(t = 300 \div 60 =\) 5 years. (This is \(t = \dfrac{I}{Pr}\), done in two steps.)

6. Days. 3650 dollars at 5% a year for 90 days.

  • \(I = 3650 \times 0.05 \times \dfrac{90}{365} =\) 45 dollars. (One day's interest is \(3650 \times 0.05 \div 365 = 0.50\) dollars, and \(90 \times 0.50 = 45\).)

Try it

  1. Find the simple interest and the total amount on 600 dollars at 5% a year for 3 years.
  2. Find the simple interest on 2400 dollars at 2.5% a year for 6 months.
  3. You borrow 1200 dollars at 8% a year simple interest for 9 months. How much do you pay back?
  4. 5000 dollars earns 400 dollars simple interest in 2 years. What is the yearly rate?
  5. How long does 2000 dollars take to earn 150 dollars at 3% a year simple interest?
  6. What principal earns 90 dollars in 3 years at 2% a year simple interest?
  7. At 5% a year simple interest, how long does it take for money to double? Does the answer depend on the principal?
  8. Find the simple interest on 730 dollars at 6% a year for 100 days.
Answers
  1. \(I = 600 \times 0.05 \times 3 = 90\) dollars. \(A = 600 + 90 = 690\) dollars.
  2. \(I = 2400 \times 0.025 \times \dfrac{6}{12} = 30\) dollars.
  3. \(I = 1200 \times 0.08 \times \dfrac{9}{12} = 72\) dollars, so you pay back \(1200 + 72 = 1272\) dollars.
  4. \(r = \dfrac{400}{5000 \times 2} = \dfrac{400}{10\,000} = 0.04\), which is 4% a year.
  5. One year's interest is \(0.03 \times 2000 = 60\) dollars, so \(t = 150 \div 60 = 2.5\) years.
  6. \(P = \dfrac{I}{rt} = \dfrac{90}{0.02 \times 3} = \dfrac{90}{0.06} = 1500\) dollars.
  7. Doubling means the interest equals the principal: \(Prt = P\). Divide both sides by \(P\): \(rt = 1\), so \(t = 1 \div 0.05 = 20\) years. The \(P\) cancelled, so it does not depend on the principal: 10 dollars and 10 000 dollars both take 20 years.
  8. \(I = 730 \times 0.06 \times \dfrac{100}{365} = 12\) dollars. (One day: \(730 \times 0.06 \div 365 = 0.12\), and \(100 \times 0.12 = 12\).)

Common mistake

Using the yearly rate for a part of a year. "3% a year" for 4 months is not 3% of the principal. Change the time into years first: \(t = \tfrac{4}{12}\).

Writing the percent as a whole number. In \(I = Prt\), the rate must be a decimal: 3% is \(r = 0.03\), not 3. Using 3 makes the interest 100 times too big. A quick sense check catches it: 3% of 1000 dollars a year cannot be 3000 dollars.

Where it leads