Simple & Compound Interest, Growth & Decay Flashcards

All 20 cards in this deck

What is simple interest?

Interest calculated only on the original amount (the principal), so the same amount of interest is earned each year.

What is the formula for the simple interest earned over $n$ years?

e.g. £500 invested at 3% per annum for 4 years

$I = P \times \frac{r}{100} \times n$

e.g. $500 \times 0.03 \times 4 = £60$

True or false? With simple interest, the amount of interest earned gets bigger each year.

False. It is the same every year, because it is always a percentage of the original amount only.

True or false? At the same rate, simple interest and compound interest give the same total after 1 year.

True. They only differ from the second year onwards, once interest starts earning interest.

Over more than one year at the same rate, which earns more: simple interest or compound interest?

Compound interest, because interest is earned on the interest already added.

What are the steps to find the value of an investment after $n$ years of simple interest?

e.g. £2000 at 5% per annum for 3 years

1. One year's interest: $2000 \times 0.05 = 100$
2. Multiply by the years: $100 \times 3 = 300$
3. Add to the principal: $2000 + 300 = £2300$

What is compound interest?

Interest added to the balance each year, so the next year's interest is worked out on the new, larger amount.

What are the steps to work out the compound interest earned, as an amount and as a percentage of the original sum?

e.g. £4000 at 3% per annum for 2 years

1. Find the value: $4000 \times 1.03^2 = 4243.60$
2. Subtract the original: $4243.60 - 4000 = £243.60$
3. As a percentage: $243.60 \div 4000 \times 100 = 6.09\%$

How do you find the annual percentage rate when you know the starting value, the final value and the number of years $n$?

e.g. £250 000 grows to £325 000 in 6 years

Take the $n$th root of (final $\div$ start), then subtract 1: $\sqrt[6]{325000 \div 250000} = 1.0447\ldots$, so $x = 4.5\%$ (2 s.f.)

True or false? An account paying compound interest of 2.4% for the first year and 1.8% for each extra year gives a value after 3 years of $P \times 1.024 \times 1.018^2$.

True. Multiply the starting amount by each year's multiplier in turn.

What does it mean for a value to depreciate?

Its value falls over time, usually by a fixed percentage of its current value each year.

What is the multiplier for a depreciation of 15% per year?

$0.85$, since $100\% - 15\% = 85\%$ of the value remains.

What are the steps to find the original value when you know the value after $n$ years of depreciation?

e.g. worth £8100 after 2 years of 10% depreciation per year

1. Yearly multiplier: $1 - 0.1 = 0.9$
2. Multiplier for the period: $0.9^2 = 0.81$
3. Divide the final value by it: $8100 \div 0.81 = £10\,000$

True or false? A car whose value depreciates by 20% each year has lost 40% of its value after 2 years.

False. The multiplier is $0.8^2 = 0.64$, so it has lost 36% of its value.

In the growth and decay formula $A = P(1 \pm r)^n$, what do $P$, $r$ and $n$ stand for?

$P$ = starting amount, $r$ = rate per period as a decimal, $n$ = number of periods; use $+$ for growth and $-$ for decay.

How do you find the number of years for a value to first fall below a given amount?

e.g. a £12 500 boat losing 15% of its value each year, first below £6250

Form the inequality and try $n = 1, 2, 3, \ldots$ until it is true: $12500 \times 0.85^n < 6250$ gives $n = 5$ years.

In the decay model $V_t = 2000 \times 0.98^t$, what do 2000 and 0.98 tell you?

$2000$ is the starting amount; $0.98$ is the multiplier, a 2% decrease per time period.

In the iterative process $P_{t+1} = kP_t$, what does $k$ tell you?

The multiplier for each period: $k > 1$ means growth, $k < 1$ means decay.

e.g. $k = 1.01$ is a 1% increase each period.

What are the steps to use the iterative process $P_{t+1} = kP_t$ to find a later term?

e.g. $P_1 = 400$, $k = 1.01$, find $P_3$

1. Multiply the first term by $k$: $400 \times 1.01 = 404 = P_2$
2. Multiply that result by $k$ again: $404 \times 1.01 = 408.04 = P_3$

True or false? In an exponential decay model, a multiplier of $0.75$ means the amount falls by 75% each year.

False. It falls by 25% each year; 75% of the amount remains.