Hard GCSE Simple and compound interest Questions

Plan the year-by-year table

Interest first, then the $\pounds 500$ 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 $\pounds 500$ is $\pounds 30.00$, giving $\pounds 530$.

Year $1$: add the $\pounds 500$ payment

The payment goes in after the interest, so the balance is $\pounds 1030$.

Year $2$: add the interest

$6\%$ interest on $\pounds 1030$ is $\pounds 61.80$, giving $\pounds 1091.80$.

Year $2$: add the $\pounds 500$ payment

The payment goes in after the interest, so the balance is $\pounds 1591.80$.

Year $3$: add the interest

$6\%$ interest on $\pounds 1591.80$ is $\pounds 95.51$, giving $\pounds 1687.31$.

Year $3$: add the $\pounds 500$ payment

The payment goes in after the interest, so the balance is $\pounds 2187.31$.

Year $4$: add the interest

$6\%$ interest on $\pounds 2187.31$ is $\pounds 131.24$, giving $\pounds 2318.55$.

Year $4$: add the $\pounds 500$ payment

The payment goes in after the interest, so the balance is $\pounds 2818.55$.

Year $5$: add the interest

$6\%$ interest on $\pounds 2818.55$ is $\pounds 169.11$, giving $\pounds 2987.66$.

Year $5$: add the $\pounds 500$ payment

The payment goes in after the interest, so the balance is $\pounds 3487.66$.

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 $\pounds 3000$ of her own money altogether.

State the answer

After $5$ years the account holds $\pounds 3487.66$, of which $\pounds 487.66$ is interest.

Plan the year-by-year table

Each year: interest first, then the $\pounds 100$ payment. Keep going until the balance passes $\pounds 3200$. 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 $\pounds 87.50$ is added to $\pounds 2500$, then the $\pounds 100$ payment goes in: $\pounds 2687.50$. The exact balance carries forward.

Year $2$: interest, then the payment

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

Year $3$: interest, then the payment

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

Year $4$: interest, then the payment

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

Year $5$: interest, then the payment

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

Check year $3$ against the target

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

Check year $4$ against the target

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

Check the total she has paid in

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

Say why the payments alone are not enough

Without any interest the account would hold only $\pounds 2900$ 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, $\pounds 2500$ would grow to only $\pounds 2868.81$ in $4$ years. It takes BOTH the payments and the interest to pass $\pounds 3200$ in year $4$.

Work out the interest earned in year $4$ alone

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

Confirm year $5$ stays above

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

State the answer

After $4$ complete years the account is worth more than $\pounds 3200$ 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 $1$: $\pounds 3000 \times 1.05 = \pounds 3150$. 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 $2$: $\pounds 3150 \times 1.05 = \pounds 3307.50$. 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 $3$: $\pounds 3307.50 \times 1.05 = \pounds 3472.88$. 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 $4$: $\pounds 3472.88 \times 1.05 = \pounds 3646.52$. 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 $\pounds 3646.52$.

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 $1$: $\pounds 3200 \times 1.03 = \pounds 3296$. 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 $2$: $\pounds 3296 \times 1.03 = \pounds 3394.88$. 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 $3$: $\pounds 3394.88 \times 1.03 = \pounds 3496.73$. 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 $4$: $\pounds 3496.73 \times 1.03 = \pounds 3601.63$. 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 $\pounds 3601.63$.

Compare the two accounts and find the difference

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

Explain the result

Over $4$ years Account A multiplies its money by $1.2155$ and Account B only by $1.1255$. $A \times 1.2155$ on $\pounds 3000$ beats $B \times 1.1255$ on $\pounds 3200$, 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 $\pounds 30$ $000$ to $\pounds 23400 --$ a loss of $\pounds 6600$ 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 $2$: $\pounds 23400 \times 0.88 = \pounds 20592$. Every year is worked out from the exact year-$1$ value $\pounds 23400$, so no rounding error can creep in.

Value after year $3$

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

Value after year $4$

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

Value after year $5$

Year $5$: $\pounds 15946.44 \times 0.88 = \pounds 14032.87$. Every year is worked out from the exact year-$1$ value $\pounds 23400$, 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 $\pounds 15831.96$, which is far too big -- it ignores the extra $10$ percentage points lost in the first year.

Compare the yearly losses

The van loses $\pounds 6600$ in year $1$ but only $\pounds 2808$ 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 $\pounds 15967.13$ 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 $\pounds 14032.87$.

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 $1$: $\pounds 8000 \times 1.045 = \pounds 8360$. 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 $2$: $\pounds 8360 \times 1.045 = \pounds 8736.20$. 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 $3$: $\pounds 8736.20 \times 1.045 = \pounds 9129.33$. 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 $4$: $\pounds 9129.33 \times 1.045 = \pounds 9540.15$. 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 $5$: $\pounds 9540.15 \times 1.045 = \pounds 9969.46$. 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 $\pounds 429.31$.

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 $\pounds 360$. Year $5$ earns $\pounds 429.31 --$ 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 $\pounds 360$ 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 $\pounds 429.31$.