Growth & decay

£8000 is invested at a rate of r % per year simple interest. After 7 years the value of the investment is £10 352.

Find the value of r.

The population of fish, P, in a pond is predicted by the formula

P = 640 × 1.025ⁿ

where n is the number of years after 1st January 2020.

Write down the population of fish on 1st January 2020.

Write down the percentage increase per year used in the formula.

Show that the formula predicts that the population of fish will reach 1000 between

1st January 2038 and 1st January 2039.

Give one reason why this might not happen.

Amit invests some money at a rate of 4% per year compound interest. After 2 years the value of Amit’s investment is £1352.

Work out how much money Amit invested.

A pot of tea is made using boiling water and left to cool. The temperature, T °C, of the pot of tea after n minutes is given by the formula

T = abⁿ.

The graph shows some information about the temperature of the pot of tea.

Find the value of a and the value of b.

At the start of 2020, the value of a particular caravan was £25 000. The value of the caravan then decreased by 12% each year.

Calculate the total loss in value of the caravan at the end of 3 years.

Darcie invests £ x at a rate of 1.5% per year simple interest for 5 years. Ivan also invests £ x but at a rate of 1.1% per year simple interest for 6 years.

Darcie earns £108 more interest than Ivan.

Work out the value of x. You must show your working.

A person invests £8000 at a rate of 5% per year compound interest.

Calculate the total amount of interest earned after 3 years.

The expected value of a painting, £ P, is given by the formula

P = 2500 × 1.2ⁿ

where n is the number of years after it was bought and 0 ⩽ n ⩽ 4.

Write down the value of the painting when it was bought.

Write down the annual percentage increase in the expected value of the painting.

The table shows the expected value of the painting n years after it was bought.

On the page opposite, draw a suitable graph to show the expected value of the painting n years after it was bought, where 0 ⩽ n ⩽ 4.

Years after the painting was bought (n)

An art collector correctly works out 2500 × 1.2¹⁰ as 15479.

They say,

The expected value of the painting 10 years after it was bought is £15479.

What assumption has the art collector made.

Charlie invests £1200 at a rate of 5% per year compound interest.

Show that the total amount of interest Charlie will have received after 2 years is £123.

Riley invests some money in a savings account that pays 4% simple interest per year. After 5 years the total interest that Riley’s investment has earned is £360.

Find the total value of Riley’s investment after 5 years. You must show your working.

Sasha invests £1000 at a rate of 5% per year compound interest. Sasha says

After one year, my investment will get £50 in interest and will be worth £1050. Therefore, after two years, my investment will get another £50 in interest and will be worth £1100.

Is Sasha correct? Give a reason for your answer.

Sasha buys a car. The value, £ V, of the car after n years is given by the formula

V = a × bⁿ.

The graph shows some information about the value of the car.

Find the value of a and the value of b.

Nina invests £540 at a simple interest rate of 2% per year. Kareem invests £540 at a compound interest rate of 2% per year.

Work out the difference in value between the two investments at the end of 5 years. You must show your working.

A biologist assumes the population, P, of birds on an island can be predicted using the formula

P = 3800 × 1.042ⁿ

where n is the number of years after the start of 2020.

Write down the percentage increase per year that is used in the formula.

Calculate the predicted population at the start of 2024.

Show that the number of birds is predicted to exceed 7000 during 2034.

A researcher says that between 2022 and 2030 the percentage increase per year in the population will be 2.8%.

If the researcher is correct, explain how this new information will affect the answer in part (c)(i).

Hiro invests £2500 for 2 years in a bank account paying r % per year compound interest. At the end of 2 years, the amount in the bank account is £2704.

Calculate r.

Azmi invests £700 at a rate of 2% per year simple interest.

Work out the interest Azmi receives after one year.

Work out the value of Azmi’s investment after 3 years.

Alex invests £4500 at a rate of 7.5% per year simple interest.

Find the value of the investment at the end of 4 years.

At the end of t years, the value of the investment is over £13 500 for the first time.

Find the value of t.

The price of a seat on a flight, £ P, is given by

P = 49 × 1.009ⁿ

where n is the number of seats already sold on this flight.

Write down the percentage increase in price of the second seat sold compared to the first seat sold.

Show that the price of the 40th seat sold is less than £70.

Rowan invests £4000 at a rate of 3.5% per year compound interest.

Calculate the value of Rowan’s investment after 5 years. Give your answer correct to the nearest penny.

Kai invests £600 at a simple interest rate of r % each year. After 5 years, Kai’s investment is worth £690.

Find the value of r.

An estimate for the number of seals on an island is given by the formula

P = 5200 × 1.02^t

where P is the number of seals t years after the start of year 2015.

Write down the annual percentage increase in the number of seals on the island.

Use the formula to show that there may have been about 4700 seals on the island at the start

of year 2010.

The number of bees, P, in a colony is given by the formula

P = ab^x

where x is the number of months after the start of July.

At the start of July, there were 25 000 bees in the colony. After one month, there were 23 500 bees in the colony.

Find the value of a and the value of b. Give the value of b as a decimal.

Mr Fox invests £400 in a savings account that pays 3% simple interest per year.

Work out the total amount of interest Mr Fox will have earned at the end of the 5th year.

1600 fish are released into a new lake which has no fish. The number of fish is expected to increase by 5% each year.

The table shows the expected number of fish in the lake at the end of 1 year and at the end of 2 years.

Complete the table. Round your answers to the nearest integer.

Use the table to draw a suitable graph to show the expected number of fish in the lake.

A maximum of 2000 fish can live in the lake.

What effect would you expect this to have on the shape of your graph after 4 years?

TURN OVER FOR QUESTIONS 25 AND 26

In January 2018, an art collector bought an antique painting. In January 2020, he sold it for £17 640.

Assume the value of the painting increased by 5% each year.

Calculate the art collector’s profit. You must show your working.

Li bought a house at the start of 2016. Li assumes the value of the house, £ V, can be predicted using the formula

V = 185000 × 1.035ⁿ

where n is the number of years after the start of 2016.

Explain how you know that the value of the house is predicted to increase each year.

Write down the percentage increase per year that is used in the formula.

Write down the value of the house at the start of 2016.

Calculate the predicted value of the house at the start of 2020, giving your answer correct to 4 significant figures.

Compared with its value at the start of 2016, show that the formula predicts the house

will have doubled in value at some point during 2036.

Give one reason why this may not happen.

1600 fish are released into a new lake which has no fish. The number of fish is expected to increase by 5% each year.

The table shows the expected number of fish in the lake at the end of 1 year and at the end of 2 years.

Complete the table. Round your answers to the nearest integer.

Use the table to draw a suitable graph to show the expected number of fish in the lake.

A maximum of 2000 fish can live in the lake.

What effect would you expect this to have on the shape of your graph after 4 years?

Charlie invests £9000 at a rate of 0.7% per year compound interest.

Calculate the total amount of interest Charlie will have earned after 5 years. Give your answer correct to the nearest penny.

A worker received a 10% pay increase in 2017 and a further 10% pay increase in 2018. The worker says

Over these two years, my pay increased by 10% + 10% = 20%.

The worker is incorrect.

Work out the correct percentage increase. You must show your working.

Jamie invests £6000 at a simple interest rate of r % each year. After 6 years the value of their investment is £7170.

Find the value of r.

The formula

P = 6800 × 1.045ⁿ

is used to predict the population, P, of an island n years after 2018.

Write down the population of the island in 2018.

Write down the percentage growth rate used in the formula.

Work out the population predicted by the formula for the year 2030.

Give one reason why the answer to (c)(i) may not be reliable.

A worker received a 10% pay increase in 2017 and a further 10% pay increase in 2018. The worker says

Over these two years, my pay increased by 10% + 10% = 20%.

The worker is incorrect.

Work out the correct percentage increase. You must show your working.

Theo invests £500 at a rate of 6% per year simple interest.

Work out the interest he receives in one year.

Work out the value of his investment after 5 years.

The price, £ P, of a car is £20 000 in 2019. The price is expected to decrease by 5% each year after 2019.

Jasmine says

This means the price in 2021 is expected to be £18 000.

She is incorrect. Explain her error and work out the correct answer.

The correct answer is £........

Write a formula for P in terms of n, where n is the number of years after 2019.

Circle the graph that best represents the price, £ P, of the car n years after 2019.

Write down the multiplier for an increase of 140%. Give your answer as a decimal.

Ali invests £1500 in October. The investment increases in value by 10% in November. It then decreases in value by 20% in December.

Ali says

10% – 20% = - 10%, so the £1500 has lost exactly 10% of its value.

Explain what Ali has done wrong.

Work out the correct percentage loss.

Kay invests £1500 in an account paying 3% compound interest per year. Neil invests £1500 in an account paying r % simple interest per year.

At the end of the 5th year, Kay and Neil’s accounts both contain the same amount of money.

Calculate r. Give your answer correct to 1 decimal place.

The value of a house, £ H, is given by the formula

H = 165 000 × 1.03^t

where t is the number of years after 1st January 2010.

Write down the value of the house on 1st January 2010.

Write down the annual percentage increase in the value of the house.

Show that the value of the house is over £200 000 on 1st January 2017.

Kay invests £1500 in an account paying 3% compound interest per year. Neil invests £1500 in an account paying r % simple interest per year.

At the end of the 5th year, Kay and Neil’s accounts both contain the same amount of money.

Calculate r. Give your answer correct to 1 decimal place.

Claudia invests £25 000 at a rate of 2% per year compound interest.

Calculate the total amount of interest she will have earned after 5 years. Give your answer correct to the nearest penny.

Claudia invests £25 000 at a rate of 2% per year compound interest.

Calculate the total amount of interest she will have earned after 5 years. Give your answer correct to the nearest penny.

In 2017, the value of a house increased by 4%. In 2018, the value of the house then decreased by 3%.

Teresa says

Over the two years the value of the house increased by exactly 1% because 4 – 3 = 1.

Show that Teresa is wrong.

At the start of 2018, the population of a town was 17 150. At the start of 2019, the population of the town was 16 807.

It is assumed that the population of the town is given by the formula

where P is the population of the town t years after the start of 2018.

Write down the value of a.

Show that r = 0.98.

Show that the population is predicted to be less than 16 000 at the start of 2022.

Use the formula to work out what the population might have been at the start of 2017.

Here are the interest rates for two bank accounts.

Mia puts £6400 in each account.

Calculate the difference in value between the two accounts after 8 years. Give your answer correct to the nearest penny.

Here are the interest rates for two bank accounts.

Mia puts £6400 in each account.

Calculate the difference in value between the two accounts after 8 years. Give your answer correct to the nearest penny.

The number of gannets on an island is assumed to follow this exponential growth model.

N is the number of gannets, in thousands. x is the number of years after 1^st January 2010.

Complete the table for N = 0.45 × 1.07^x.

Draw the graph of N = 0.45 × 1.07^x.

Number of years, x, after 1^st January 2010

Use the graph to find the year when the gannet population is predicted to reach 1000.

The price of a computer was £750.

In a sale the price is reduced by 20%. On the final day the sale price is reduced by a further 12%.

How much is saved in total by buying the computer on the final day of the sale?

The value of a car, £ V, is given by

V = 16 500 × 0.82ⁿ

where n is the number of years after it is bought from new.

Write down the value of the car when new.

Write down the annual percentage decrease in the value of the car.

Show that the value of the car after 4 years is less than half its value when new.

Ratna invests £1200 for 2 years in a bank account paying r % per year compound interest. At the end of 2 years, the amount in the bank account is £1379.02.

Calculate r.

Luka invests £1500. At the end of the first year, 2% interest is added. At the end of the second year, after interest has been added, the investment is worth £1606.50.

Show that 5% interest has been added at the end of the second year.

Amelia buys a new car. The expected future value of this car, £ V, is given by

where t is the age of the car in complete years.

Write down the value of the car when new.

Write down the annual percentage decrease in the expected value of the car.

Show that the expected value of the car when 2 years old is £9000.

Amelia sketches a graph to show the expected value of her car as it gets older.

Explain how you know that Amelia’s graph is incorrect.

Amelia assumes that her car will have no value at all after 20 years.

Explain why her assumption is mathematically incorrect.

Gustavo invests £520 for 6 years in a bank account paying simple interest. At the end of 6 years, the amount in the bank account is £629.20.

Calculate the annual rate of interest.

At the start of 2014 Priya’s house was worth £240 000. The value of her house increased by 5% every year.

Work out the value of her house at the start of 2017.

Corinne invests £8400 at a simple interest rate of 12% per year.

Work out the value of the investment after 3 years.

On 1^st November 2015 there were 4200 trees planted in a wood. On 1^st November 2016, only 3948 of these trees were still alive.

It is assumed that the number of trees still alive is given by

N = ar^t

where N is the number of trees still alive t years after 1^st November 2015.

Write down the value of a.

Show that r is 0.94.

Show that on 1^st November 2030 the number of trees still alive is predicted to have decreased

by over 60% compared with 1^st November 2015.

Rashid invests money into an account which pays a fixed rate of compound interest each year. The value, £ V, of his investment after t years is given by the formula

How much money did Rashid invest?

What rate of compound interest is paid each year?

Circle the graph that best represents the growth in Rashid’s account.