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Daily Maths

Simple interest

Simple Interest Type 1 Core + Extended 8 min read

On this page
  1. In this note
  2. Formula bank
  3. What simple interest is
  4. Working backwards: rate, time, principal
  5. Ways they test it
  6. Mistakes to avoid
  7. Summary
  8. Practice
  9. Can I do this?
  10. Quiz

In this note

Syllabus 1.13: simple interest
  • 1.13Find the simple interest and the total value (Type 1).
  • 1.13Find the rate (Type 2), the time or the amount invested (Type 3).
  • 1.13Handle months, "complete years" and two accounts compared.

Before you start: percentages. Next: compound interest.

Formula bank

Simple interestLearn it\(I=\dfrac{P\times R\times T}{100}\)\(P\) = amount invested or borrowed, \(R\) = % rate per year, \(T\) = years.Use it when "simple interest".
Total valueLearn it\(A=P+I\)the money at the end.Use it when "value of the investment", "total owed".
Find the rateLearn it\(R=\dfrac{100I}{PT}\)find I first (total − P).Use it when Type 2.
Find the timeLearn it\(T=\dfrac{100I}{PR}\)Use it when Type 3.
Find the principalLearn it\(P=\dfrac{100I}{RT}\)Use it when Type 3.

What simple interest is

Put money in a bank and it pays you interest each year; borrow and you pay it. With simple interest the interest is always a percentage of the ORIGINAL amount, so you get the SAME amount every year.

Simple interest grows by the same amount each yearYear 1$480Year 2$480Year 5$4805 × $9.60
$480 at 2%: $9.60 every year, so $48 after 5 years.

General methodHow to answer any simple interest question

  1. Write down P, R and T, with T in YEARS (18 months = 1.5).
  2. If you know the TOTAL, find the interest first: I = total − P.
  3. Put the three you know into \(I=\dfrac{PRT}{100}\) and solve for the fourth.
  4. Check whether the question wants the interest or the total value.

Example $2500 is invested at 3.5% per year simple interest for 4 years. Find the total value.

Interest from the formula, then add P.

  1. \(I=\dfrac{2500\times3.5\times4}{100}=350\)
  2. \(A=2500+350\)

Answer $2850

Try it yourself

1
$600 at 5% for 3 years. Interest?
Show answer

Answer $90

2
$1200 at 4% for 18 months. Interest?
Hint
T = 1.5.
Show answer

Answer $72

3
$2000 at 3% for 6 months. Interest?
Show answer

Answer $30

Working backwards: rate, time, principal

Example $5000 grows to $5700 in 8 years with simple interest. Find the rate.

Interest first, then rearrange for R.

  1. \(I=5700-5000=700\)
  2. \(R=\dfrac{700\times100}{5000\times8}=1.75\)

Answer 1.75% per year

Example $1500 at 2.4% earns $180. For how many years?

Rearrange for T.

  1. \(T=\dfrac{180\times100}{1500\times2.4}=5\)

Answer 5 years

Doubling: money doubles when \(I=P\), so \(100=R\times T\). At 4%, \(T=25\) years for ANY amount.

Try it yourself

1
$400 grows to $460 in 3 years. Rate (%)?
Show answer

Answer 5%

2
$500 at 4% earns $40. How many years?
Show answer

Answer 2

3
3.5% simple interest earns $126 in 3 years. How much was invested?
Show answer

Answer $1200

Ways they test it

Type 1 Type 1: interest or value

"$2400 at 3.25% for 4 years. (a) Interest. (b) Total value." (3 marks)

Spot it P, R and T all given.

  1. Formula.
  2. Add P if they want the value.

For example $2400, 3.25%, 4 years

  1. \(I=312\)
  2. \(2400+312\)

Answer $312; $2712

Type 2 Type 2: the rate

"$6500 is worth $7553 after 6 years. Find r." (3 marks)

Spot it a final value and an unknown rate.

  1. I = final − P.
  2. Rearrange for r.

For example $6500 → $7553 in 6 years

  1. \(I=1053\)
  2. \(r=\dfrac{1053\times100}{6500\times6}\)

Answer 2.7

Type 3 Type 3: time or principal

"$1500 at 2.4% earns $180. How many years?"

Spot it the time or amount unknown.

  1. Rearrange the formula.

Type 4 Complete years

"Least number of complete years until it is worth more than $1200?"

Spot it "complete years", "more than".

  1. Solve for T.
  2. Round UP to a whole year; check both years.

For example $900 at 4.5%, target $1200

  1. interest needed 300; per year 40.5
  2. \(300\div40.5=7.4\ldots\)

Answer 8 years

Type 5 Two accounts

"Amina invests P at 3% for 5 years; Bilal P + 500 at 2.5% for 4 years; same interest. Find P." (4 marks)

Spot it two people, one unknown.

  1. Write each interest in P.
  2. Set equal, solve.

For example \(0.15P=0.1(P+500)\)

  1. \(0.05P=50\)

Answer \(P=1000\)

Mistakes to avoid

  • ✗ T = 18 for 18 months✓ T = 1.5T is in years.
  • ✗ Using the total as I✓ I = total − PThe formula gives the interest only.
  • ✗ Gave the interest when asked for the value✓ add PRead the question's last line.
  • ✗ 7.4 years → 7 complete years✓ 8 yearsInterest is added only at the end of a year.

Summary

  • \(I=\dfrac{PRT}{100}\); value \(A=P+I\).
  • The interest is the same every year.
  • T in years.
  • Rate, time or principal: find I first, then rearrange.
  • "Complete years": round up.

If you see …

  • a final valuesubtract P to get I.
  • months÷ 12 for years.
  • "complete years"round up and check.

Practice

Practice: easy, medium, hard

Easy

1
$480 at 2% for 5 years. Interest?
Show working
  1. \(\tfrac{480\times2\times5}{100}=48\)

Answer $48

2
$750 at 1.8% for 6 years. Interest?
Show working
  1. \(\tfrac{750\times1.8\times6}{100}=81\)

Answer $81

3
At 5% simple interest, how many years for money to double?
Show working
  1. \(100\div5=20\)

Answer 20 years

Medium

1
Kim invests $750. After 5 years it is worth $840. Find the rate r%.
Guide me

What is it asking? the rate

What do we know? P = 750, value 840, T = 5

Plan I = 90, then rearrange

Show working
  1. \(I=90\)
  2. \(r=\tfrac{90\times100}{750\times5}=2.4\)

Answer 2.4

2
An investment at 3.25% earns $234 in 4 years. How much was invested?
Guide me

What is it asking? the principal

What do we know? I = 234, R = 3.25, T = 4

Plan P = 100I ÷ (RT)

Show working
  1. \(\tfrac{234\times100}{3.25\times4}=1800\)

Answer $1800

3
$800 becomes $1016 after 9 years. Rate (%)?
Guide me

What is it asking? the rate

What do we know? I = 216

Plan rearrange

Show working
  1. \(\tfrac{216\times100}{800\times9}=3\)

Answer 3%

Hard

1
Tariq invests $900 at 4.5% simple interest, added at the end of each year. Least number of complete years until it is worth more than $1200?
Guide me

What is it asking? whole years

What do we know? needs more than $300 interest

Plan ÷ yearly interest, round up

Show working
  1. per year \(=40.5\)
  2. \(300\div40.5=7.4\ldots\)
  3. 7 years: $1183.50; 8 years: $1224

Answer 8

PartAnswer / workingMarksGuidance
40.5 per yearM1
\(300\div40.5\)M1
8A1
Total: 3 marks
2
Sam owes $1380 after 2 years 6 months at 6% simple interest. How much did he borrow?
Hint
Interest is 15% of P, so $1380 is 115% of P.
Guide me

What is it asking? the amount borrowed

What do we know? T = 2.5, R = 6

Plan total = P × 1.15

Show working
  1. \(6\times2.5=15\%\)
  2. \(1380\div1.15=1200\)

Answer $1200

PartAnswer / workingMarksGuidance
T = 2.5B1
\(1380\div1.15\)M1
1200A1
Total: 3 marks
3
Amina invests $P at 3% for 5 years. Bilal invests $500 more at 2.5% for 4 years. Same interest. Find P.
Guide me

What is it asking? P

What do we know? two interests equal

Plan \(0.15P=0.1(P+500)\)

Show working
  1. \(0.15P=0.1P+50\)
  2. \(P=1000\)

Answer $1000

PartAnswer / workingMarksGuidance
\(0.15P\) and \(0.1(P+500)\)M1M1
1000A1
Total: 3 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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