Iteration, growth and decay – Year 11 GCSE maths worksheet with answers to download as PDF

A percentage multiplier is the number you multiply by to change an amount by a percentage. For example, an increase of 30% has multiplier 1.30.

1. Write the multiplier for an increase of 12% and for a decrease of 8%. [2 marks]
2. Write the multiplier for an increase of 2.5% and a decrease of 15%. [2 marks]
3. Increase £240 by 35% using a single multiplier. [2 marks]
4. A price of £50 is decreased by 24%. Work out the new price. [2 marks]

Anna invests £2 000 in an account paying 4% compound interest each year.

1. Write the multiplier for 4% interest. [1 mark]
2. Work out the amount in the account after 3 years. [3 marks]
3. Work out the total interest earned. [1 mark]
4. Write a formula for the amount after \(n\) years. [1 mark]

A new car costs £15 000. Its value falls by 20% each year.

1. Write the multiplier. [1 mark]
2. Work out the value after 3 years. [3 marks]
3. By how much has the car lost value? [1 mark]
4. Work out the percentage loss over the 3 years. [2 marks]

Ben invests £4 000. The interest rates are 3% in the first year, 2.5% in the second year and 2% in the third year.

1. Write the multiplier for each year. [2 marks]
2. Work out the value of the investment after 3 years. [3 marks]
3. Work out the single percentage increase that gives the same final amount. [2 marks]

Work out the original amount in each case.

1. After a 20% increase, a price is £90. [2 marks]
2. After a 15% decrease, a value is £680. [2 marks]
3. After a 4% increase, a salary is £31 200. [2 marks]

The equation \(x^3=x+3\) has a solution close to 1.7. It can be solved by the iteration \(x_{n+1}=\sqrt[3]{x_n+3}\) with \(x_0=1.5\).

1. Work out \(x_1\), \(x_2\) and \(x_3\) to 3 decimal places. [3 marks]
2. Show that the equation has a solution between 1.6 and 1.7. [2 marks]
3. Write the solution to 2 decimal places. [1 mark]

Maya puts £1 500 into a savings account that pays 4.5% compound interest per year.

1. How much is in the account after 5 years? [3 marks]
2. How much interest has she earned? [1 mark]
3. She wants at least £2 000. Does she have enough after 7 years? [3 marks]
4. Explain why compound interest gives more than simple interest. [1 mark]

A loaf of bread costs £3.20. Because of inflation, prices rise by 3% each year.

1. Work out the price after 4 years, to the nearest penny. [3 marks]
2. Write a formula for the price after \(n\) years. [1 mark]
3. What is the percentage increase over the 4 years, to 1 decimal place? [2 marks]
4. A newspaper costs £1.40 today. Which will cost more after 4 years, the loaf or the newspaper? Show how you know. [2 marks]

The population of a town is 80 000. It falls by 2% each year.

1. Work out the population after 10 years to the nearest whole number. [3 marks]
2. Write the percentage fall over 10 years to 1 decimal place. [2 marks]
3. Use trial and improvement to find the first year in which the population is below 70 000. [3 marks]

A scientist grows bacteria. There are 200 bacteria at the start. The number increases by 50% each hour.

1. Work out the number after 4 hours. [3 marks]
2. Write a formula for the number after \(t\) hours. [1 mark]
3. After how many whole hours is the number first more than 2 000? [3 marks]

A new phone costs £600. It loses 30% of its value in the first year and then 20% of its value each following year.

1. Work out the value after 3 years. [3 marks]
2. What is the percentage lost over 3 years? [2 marks]
3. The seller says "It has lost less than half its value." Is the seller correct? [2 marks]

A shop raises all its prices by 10%. A month later it reduces them by 10%. Maria says "The prices are back to where they started."

1. A jacket costs £100 at the start. Work out its price after the rise and after the reduction. [3 marks]
2. Is Maria correct? Explain. [2 marks]
3. Write the single multiplier for the two changes and the overall percentage change. [2 marks]

Two banks offer 5% interest per year on £1 000 for 10 years. Bank S pays simple interest and bank C pays compound interest.

1. Work out the amount in Bank S after 10 years. [2 marks]
2. Work out the amount in Bank C after 10 years. [2 marks]
3. How much more is earned in Bank C? [1 mark]
4. Explain why the difference grows each year. [2 marks]

Isaac invests £5 000 at 7% compound interest per year.

1. Work out the value after 10 years. [2 marks]
2. After how many whole years has the money first more than doubled? [3 marks]
3. Is it true that money doubles in 14 years at 7% simple interest? Explain. [3 marks]

A sum of money has grown to £2 315.25 after 3 years at 5% compound interest per year.

1. Write the overall multiplier for the 3 years. [2 marks]
2. Work out the original sum. [3 marks]
3. Work out the amount after the first year. [2 marks]
4. Check that your answer works by growing it forward. [2 marks]

£4 000 is invested at a fixed compound interest rate and grows to £4 494.40 in 2 years.

1. Show that the overall multiplier is 1.1236. [2 marks]
2. Work out the yearly percentage rate. [3 marks]
3. Check your answer. [2 marks]
4. How much would it be after 5 years at this rate? [2 marks]

A sample of 80 g of a radioactive substance loses 25% of its mass each day.

1. Work out the mass after 3 days. [2 marks]
2. Write a formula for the mass after \(n\) days. [1 mark]
3. After how many whole days is the mass first below 10 g? Show your working. [4 marks]
4. Roughly how many days does it take for the mass to halve? [2 marks]

Bank A pays 3.2% compound interest per year. Bank B pays 2.5% in the first year and 3.5% in each of the next three years. Zara invests £1 000 for 4 years.

1. Work out the amount in Bank A. [2 marks]
2. Work out the amount in Bank B. [3 marks]
3. Which bank is better and by how much? [2 marks]
4. Work out the average yearly percentage for each bank (the single yearly rate that gives the same final amount) to 2 decimal places. [4 marks]

The equation \(x^2-x-1=0\) can be rearranged as \(x=1+\dfrac1x\). Use the iteration \(x_{n+1}=1+\dfrac1{x_n}\) with \(x_0=1\).

1. Show that the rearrangement is correct. [2 marks]
2. Work out \(x_1\), \(x_2\), \(x_3\), \(x_4\). [3 marks]
3. Show that a root lies between 1 and 2. [2 marks]
4. Write the root to 2 decimal places and check it in the original equation. [3 marks]

A sum of money doubles in exactly 12 years with compound interest at a fixed yearly rate.

1. Write an equation for the yearly multiplier \(m\) and solve it. [3 marks]
2. Write the rate as a percentage to 1 decimal place. [1 mark]
3. A rule of thumb says that the doubling time is about \(72\div r\) years for a rate of \(r\%\). Use \(r=6\) to test it. [3 marks]
4. Explain why this rule works less well at a rate of 30%. [3 marks]