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Themed maths · 7 February

e Day maths activities for IGCSE and GCSE

7 February is e Day: written month first, 2/7 matches e = 2.718… This lesson shows where that number comes from, through compound interest: a 5-minute starter, a 35-minute main activity, an extension and full worked answers.

Level
IGCSE and GCSE (Higher and Extended)
Time
40 minutes, plus a 10-minute extension
Topics
Percentages and multipliers; Compound interest; Depreciation and exponential growth; Reverse percentages
Equipment
The starter is non-calculator. A calculator is needed for the main activity.

Download student sheet (PDF)Answers (PDF)

Suggested timings

PartTimeWhat
Starter5 minQuick questions on the board
Main: task A12 minCompound interest
Main: task B12 minCompound more often?
Main: task C11 minDepreciation and doubling
Extension10 minFast finishers or homework

Starter (5 minutes)

No calculator.

  1. Find 10% of £350.
  2. £200 increases by 10%, and then by another 10%. What is it worth now?
  3. Write down the multiplier for an increase of 4% and for a decrease of 15%.
  4. Work out 210.

Main activity (35 minutes)

Task A: Compound interest (12 min)

£5000 is invested at 3% a year compound interest. Give money to the nearest penny.

  1. Find the value after 1 year.
  2. Find the value after 6 years.
  3. How much interest is earned in the 6 years?
  4. How much more is this than 3% simple interest for 6 years?

Task B: Compound more often? (12 min)

£1000 is invested for one year. The bank offers 12% a year.

  1. Find the value after one year if the 12% is added once, at the end of the year.
  2. Instead, 1% is added every month. Find the value after 12 months.
  3. Instead, 12% ÷ 365 is added every day. Find the value after 365 days.
  4. However often the interest is added, the value never goes above 1000 × 2.71828…0.12. Mathematicians call 2.71828… the number e. Work out 1000 × e0.12, using the ex key.

Task C: Depreciation and doubling (11 min)

A car costs £18 000 new and loses 15% of its value each year.

  1. Find its value after 3 years.
  2. After how many whole years is the car first worth less than £5000?
  3. £1000 is invested at 6% a year compound interest. After how many whole years has it first doubled? Compare with 72 ÷ 6.

Extension (10 minutes)

For fast finishers, or as homework.

  1. After a 4% increase, the price of a bike is £520. What was the price before the increase?
  2. Work out (1 + 1/n)n for n = 1, 10 and 100, to 4 decimal places. What number are the answers getting close to?

For teachers

Teacher notes and full worked answers
  • e Day is 7 February because 2/7, written month first, matches e = 2.7…
  • Task B shows where e comes from without any algebra: let groups race to compute it with bigger and bigger n.
  • Common slip: using 1.3 instead of 1.03 as the multiplier for 3%.

Starter

  1. £35
    • 350 ÷ 10 = 35
  2. £242
    • 200 × 1.1 = 220, then 220 × 1.1 = 242
  3. 1.04 and 0.85
    • 100% + 4% = 104% = 1.04
    • 100% − 15% = 85% = 0.85
  4. 1024
    • 25 = 32 and 32 × 32 = 1024

Task A: Compound interest

  1. £5150.00
    • 5000 × 1.03 = 5150
  2. £5970.26
    • 5000 × 1.036 = 5970.261…
  3. £970.26
    • 5970.26 − 5000 = 970.26
  4. £70.26
    • Simple interest: 5000 × 0.03 × 6 = 900.
    • 970.26 − 900 = 70.26

Task B: Compound more often?

  1. £1120.00
    • 1000 × 1.12 = 1120
  2. £1126.83
    • 1000 × 1.0112 = 1126.825…
  3. £1127.47
    • 1000 × (1 + 0.12/365)365 = 1127.474…
  4. £1127.50
    • 1000 × e0.12 = 1127.496…
    • Adding interest every day gets within 3p of this limit.

Task C: Depreciation and doubling

  1. £11 054.25
    • 18 000 × 0.853 = 11 054.25
  2. 8 years
    • 18 000 × 0.857 = 5770.7…, which is still more than £5000.
    • 18 000 × 0.858 = 4905.1…, which is less than £5000.
  3. 12 years; 72 ÷ 6 = 12
    • 1.0611 = 1.898…, 1.0612 = 2.012…
    • So 12 years. The ‘rule of 72’ gives 72 ÷ 6 = 12 too.

Extension

  1. £500
    • 520 ÷ 1.04 = 500
  2. 2, 2.5937, 2.7048: getting close to e = 2.71828…
    • (1 + 1)1 = 2
    • 1.110 = 2.59374…
    • 1.01100 = 2.70481…

The same answers are in the answers PDF. Every answer was recomputed by computer before publishing.

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