Simple & Compound Interest, Growth & Decay Flashcards

All 21 cards in this deck

What is simple interest?

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

What is the formula for the total simple interest earned?

e.g. £3500 invested for 6 years at 2.5% per year

$\text{Interest} = \dfrac{P \times R \times T}{100}$

$\dfrac{3500 \times 2.5 \times 6}{100} = £525$

How do you find the total amount in an account after simple interest?

e.g. £2000 for 4 years at 3% simple interest per year

Add the total interest to the principal.

$2000 + (4 \times 60) = £2240$

What are the steps to work out the annual rate of simple interest from the total interest?

e.g. £600 invested for 5 years gives £75 total simple interest

1. Divide the total interest by the number of years: $75 \div 5 = £15$ per year
2. Divide by the principal: $15 \div 600 = 0.025$
3. Multiply by 100: rate $= 2.5\%$

How do you find the number of years an amount was invested, given the total simple interest?

e.g. £500 at 2% per year gives £50 total interest

Divide the total interest by one year's interest.

One year $= £10$, so $50 \div 10 = 5$ years

What is compound interest?

Interest worked out on the principal plus the interest already earned, so each year's interest is based on the new total.

What is the formula for the total amount in an account after compound interest?

e.g. £4500 for 2 years at 3.4% per year compound interest

$\text{Amount} = P \times \left(1 + \dfrac{r}{100}\right)^n$

$4500 \times 1.034^2 = £4811.20$

What are the steps to work out the amount of interest only for compound interest?

e.g. £4500 for 3 years at 1.8% per year compound interest

1. Write the multiplier: $1.018$
2. Find the total amount: $4500 \times 1.018^3 = £4747.40$
3. Subtract the principal: $4747.40 - 4500 = £247.40$

What are the steps to compare a simple interest account with a compound interest account?

e.g. £1000 for 3 years at 2% simple or 2% compound

1. Total for simple: $1000 + 3 \times 20 = £1060$
2. Total for compound: $1000 \times 1.02^3 = £1061.21$
3. Compare: compound is better, by £1.21

True or false? Investing at 5% compound interest for 2 years gives the same total as a single 10% increase.

False. The second year's interest is worked out on the larger amount, so compound interest gives more.

What does depreciation mean?

A fall in the value of an item over time, usually by a given percentage of its current value each year.

What multiplier represents a decrease of 4% per year?

$0.96$, from $\dfrac{100-4}{100}$.

What are the steps to find the number of years for a value to fall below a given amount?

e.g. a £8000 car depreciating 20% per year — when is it worth less than £4000?

1. Write the multiplier: $0.8$
2. Multiply repeatedly, counting the years: $6400$, $5120$, $4096$, $3276.80$
3. First value below £4000 is at year 4, so 4 years

True or false? An item that loses 10% of its value each year for 3 years has lost 30% of its original value.

False. Each 10% is taken off a smaller amount, so the total loss is about 27.1%.

What is exponential growth?

A quantity multiplied by the same factor greater than 1 in each time period.

What is exponential decay?

A quantity multiplied by the same factor between 0 and 1 in each time period.

What multiplier do you use for a repeated increase of $r\%$ per time period?

e.g. a population growing 6% per year

$1 + \dfrac{r}{100}$

For 6% per year the multiplier is $1.06$.

How do you find a population after several years of growth at a fixed percentage per year?

e.g. 25 000 workers growing at 6% per year, after 3 years

Multiply the starting number by the growth multiplier once for each year.

$25\,000 \times 1.06^3 = 29\,775$ workers

True or false? A population that decays by 20% each year will have reached zero after 5 years.

False. Each year it falls to 80% of the previous year's value, so it keeps getting smaller without reaching zero.

What are the steps to find the value of an investment when the rate of compound interest changes part-way through?

e.g. £7000 for 2 years: 3% in year 1, then 1.5% in year 2

1. Apply year 1's multiplier: $7000 \times 1.03 = 7210$
2. Apply year 2's multiplier to that answer: $7210 \times 1.015 = 7318.15$
3. Value at the end: £7318.15

True or false? If an account pays 3% in year 1 and 1.5% in year 2, you can add the rates and use a single multiplier of $1.045$.

False. You multiply the two multipliers: $1.03 \times 1.015 = 1.04545$.