Hard GCSE Simple and compound interest Questions

Plan the year-by-year table

Interest first, then the £500 payment, five times over. The exact balance carries forward every year. Exam convention: keep the full unrounded balance in the calculator all the way through and round to the nearest penny only at the very end.

Write the yearly percentage as a multiplier

A 6% increase keeps the whole 100% and adds 6% on top, so multiply by 1.06 each year.

Year 1: add the interest

6% interest on £500 is £30.00, giving £530.

Year 1: add the £500 payment

The payment goes in after the interest, so the balance is £1030.

Year 2: add the interest

6% interest on £1030 is £61.80, giving £1091.80.

Year 2: add the £500 payment

The payment goes in after the interest, so the balance is £1591.80.

Year 3: add the interest

6% interest on £1591.80 is £95.51, giving £1687.31.

Year 3: add the £500 payment

The payment goes in after the interest, so the balance is £2187.31.

Year 4: add the interest

6% interest on £2187.31 is £131.24, giving £2318.55.

Year 4: add the £500 payment

The payment goes in after the interest, so the balance is £2818.55.

Year 5: add the interest

6% interest on £2818.55 is £169.11, giving £2987.66.

Year 5: add the £500 payment

The payment goes in after the interest, so the balance is £3487.66.

Round to the nearest penny

The full-precision balance is rounded to the penny only at the end.

Work out the total she paid in

She paid in £3000 of her own money altogether.

State the answer

After 5 years the account holds £3487.66, of which £487.66 is interest.

Plan the year-by-year table

Each year: interest first, then the £100 payment. Keep going until the balance passes £3200. Exam convention: keep the full unrounded balance in the calculator all the way through and round to the nearest penny only at the very end.

Write the yearly percentage as a multiplier

A 3.5% increase keeps the whole 100% and adds 3.5% on top, so multiply by 1.035 each year.

Year 1: interest, then the payment

Interest of £87.50 is added to £2500, then the £100 payment goes in: £2687.50. The exact balance carries forward.

Year 2: interest, then the payment

Interest of £94.06 is added to £2687.50, then the £100 payment goes in: £2881.56. The exact balance carries forward.

Year 3: interest, then the payment

Interest of £100.85 is added to £2881.56, then the £100 payment goes in: £3082.42. The exact balance carries forward.

Year 4: interest, then the payment

Interest of £107.88 is added to £3082.42, then the £100 payment goes in: £3290.30. The exact balance carries forward.

Year 5: interest, then the payment

Interest of £115.16 is added to £3290.30, then the £100 payment goes in: £3505.46. The exact balance carries forward.

Check year 3 against the target

After 3 years the balance is £3082.42, which is still £117.58 SHORT of £3200. So 3 years is not enough.

Check year 4 against the target

After 4 years the balance is £3290.30, which is £90.30 OVER £3200. Year 4 is the first year the target is passed.

Check the total she has paid in

She has paid in £2900 of her own money, so £390.30 of the final balance is interest.

Say why the payments alone are not enough

Without any interest the account would hold only £2900 after 4 years, which is under the target. The interest is doing real work here.

Say why the interest alone is not enough either

And with no payments at all, £2500 would grow to only £2868.81 in 4 years. It takes BOTH the payments and the interest to pass £3200 in year 4.

Work out the interest earned in year 4 alone

Year 4 earns £107.88 of interest -- more than year 1 earned (£87.50), because the balance it is worked out on is bigger.

Confirm year 5 stays above

The balance keeps growing (£3505.46 after 5 years), so once past £3200 it stays past it.

State the answer

After 4 complete years the account is worth more than £3200 for the first time.

Plan the comparison

Account B starts with more money but grows more slowly. Work out both after 4 years. Exam convention: keep the full unrounded balance in the calculator all the way through and round to the nearest penny only at the very end.

Write the yearly percentage as a multiplier

A 5% increase keeps the whole 100% and adds 5% on top, so multiply by 1.05 each year.

Balance after year 1

Year 1: $£3000 \times 1.05 = £3150$. This is worked out from the exact starting amount as $P \times \text{multiplier}^{1}$, so no rounding error can build up.

Balance after year 2

Year 2: $£3150 \times 1.05 = £3307.50$. This is worked out from the exact starting amount as $P \times \text{multiplier}^{2}$, so no rounding error can build up.

Balance after year 3

Year 3: $£3307.50 \times 1.05 = £3472.88$. This is worked out from the exact starting amount as $P \times \text{multiplier}^{3}$, so no rounding error can build up.

Balance after year 4

Year 4: $£3472.88 \times 1.05 = £3646.52$. This is worked out from the exact starting amount as $P \times \text{multiplier}^{4}$, so no rounding error can build up.

Round Account A to the nearest penny

Account A finishes on £3646.52.

Write the yearly percentage as a multiplier

A 3% increase keeps the whole 100% and adds 3% on top, so multiply by 1.03 each year.

Balance after year 1

Year 1: $£3200 \times 1.03 = £3296$. This is worked out from the exact starting amount as $P \times \text{multiplier}^{1}$, so no rounding error can build up.

Balance after year 2

Year 2: $£3296 \times 1.03 = £3394.88$. This is worked out from the exact starting amount as $P \times \text{multiplier}^{2}$, so no rounding error can build up.

Balance after year 3

Year 3: $£3394.88 \times 1.03 = £3496.73$. This is worked out from the exact starting amount as $P \times \text{multiplier}^{3}$, so no rounding error can build up.

Balance after year 4

Year 4: $£3496.73 \times 1.03 = £3601.63$. This is worked out from the exact starting amount as $P \times \text{multiplier}^{4}$, so no rounding error can build up.

Round Account B to the nearest penny

Account B finishes on £3601.63.

Compare the two accounts and find the difference

Account A ends up worth more -- by £44.89 -- even though it started with £200 less.

Explain the result

Over 4 years Account A multiplies its money by 1.2155 and Account B only by 1.1255. A x 1.2155 on £3000 beats B x 1.1255 on £3200, so the higher rate wins here -- but note it is close, and with fewer years B would have won.

Plan the calculation

The first year uses a different multiplier from the other four, so it must be done separately and the remaining years compounded on top of it. Exam convention: keep the full unrounded balance in the calculator all the way through and round to the nearest penny only at the very end.

Write the yearly percentage loss as a multiplier

Losing 22% each year leaves 78% of the value, so multiply by 0.78 each year.

Value after year 1

A 22% drop in the first year takes the van from £30 000 to £23400 -- a loss of £6600 in one year.

Write the yearly percentage loss as a multiplier

Losing 12% each year leaves 88% of the value, so multiply by 0.88 each year.

Value after year 2

Year 2: £23400 x 0.88 = £20592. Every year is worked out from the exact year-1 value £23400, so no rounding error can creep in.

Value after year 3

Year 3: £20592 x 0.88 = £18120.96. Every year is worked out from the exact year-1 value £23400, so no rounding error can creep in.

Value after year 4

Year 4: £18120.96 x 0.88 = £15946.44. Every year is worked out from the exact year-1 value £23400, so no rounding error can creep in.

Value after year 5

Year 5: £15946.44 x 0.88 = £14032.87. Every year is worked out from the exact year-1 value £23400, so no rounding error can creep in.

Round to the nearest penny

The exact value is rounded to the penny only here, at the end.

Check with a single calculation

One calculation gives the same value as the table, which confirms it.

Reject the wrong method

Using 12% for all five years gives £15831.96, which is far too big -- it ignores the extra 10 percentage points lost in the first year.

Compare the yearly losses

The van loses £6600 in year 1 but only £2808 in year 2. The percentage is applied to a smaller value each year, so the cash loss shrinks every year -- which is why the value never reaches zero.

Work out the total value lost

The van has lost £15967.13 of its value in 5 years.

Work out the loss as a fraction of the price

The van has lost about 53.2% of its value -- just over half -- which is a sensible size for a five-year-old van.

State the answer

After 5 years the van is worth £14032.87.

Understand what is being asked

The interest earned in year 5 alone is 4.5% of the balance at the START of year 5, which is the balance at the end of year 4. Exam convention: keep the full unrounded balance in the calculator all the way through and round to the nearest penny only at the very end.

Write the yearly percentage as a multiplier

A 4.5% increase keeps the whole 100% and adds 4.5% on top, so multiply by 1.045 each year.

Balance after year 1

Year 1: $£8000 \times 1.045 = £8360$. This is worked out from the exact starting amount as $P \times \text{multiplier}^{1}$, so no rounding error can build up.

Balance after year 2

Year 2: $£8360 \times 1.045 = £8736.20$. This is worked out from the exact starting amount as $P \times \text{multiplier}^{2}$, so no rounding error can build up.

Balance after year 3

Year 3: $£8736.20 \times 1.045 = £9129.33$. This is worked out from the exact starting amount as $P \times \text{multiplier}^{3}$, so no rounding error can build up.

Balance after year 4

Year 4: $£9129.33 \times 1.045 = £9540.15$. This is worked out from the exact starting amount as $P \times \text{multiplier}^{4}$, so no rounding error can build up.

Note the exact balance at the start of year 5

This exact value is what year 5 interest is worked out on -- rounding it to the penny first would give a slightly different answer.

Balance after year 5

Year 5: $£9540.15 \times 1.045 = £9969.46$. This is worked out from the exact starting amount as $P \times \text{multiplier}^{5}$, so no rounding error can build up.

Subtract to find the year 5 interest

The interest earned in year 5 alone is £429.31.

Check it directly as a percentage

4.5% of the start-of-year-5 balance gives the same answer, which confirms it.

Compare with year 1

In year 1 the account earned only £360. Year 5 earns £429.31 -- more, because the balance it is worked out on is bigger.

Explain the growth in the yearly interest

Each year's interest is 1.045 times the year before, because the balance it is charged on is 1.045 times bigger. After 4 years of that, year 5 interest is $1.045^4$ times the year 1 interest.

Reject the simple-interest answer

A simple-interest account would pay exactly £360 in year 5 as well. The compound answer is bigger, so the two are genuinely different -- do not confuse them.

Round to the nearest penny

The exact difference is rounded to the penny once, at the end.

State the answer

The interest earned in year 5 alone is £429.31.