Exercises: Compound interest and percentage change

Write down the multiplier corresponding to each of these percentage changes:

1. a) An increase of $25\ \%$.

1. b) A decrease of $15\ \%$.

1. c) An increase of $3\ \%$.

1. d) A decrease of $60\ \%$.

1. e) An increase of $130\ \%$.

What percentage increase or decrease corresponds to each of these multipliers?

1. a) $1.45$

1. b) $0.82$

1. c) $2.1$

1. d) $0.05$

1. e) $1.06$

Apply the corresponding multiplier to calculate the final value:

1. a) A festival ticket costing $1{,}600$ pesos goes up by $15\ \%$.

1. b) A pair of sneakers costing $1{,}280$ pesos is discounted by $25\ \%$.

1. c) The monthly rent of $19{,}000$ pesos for a commercial space goes up by $4\ \%$.

A saver deposits $\$40\,000$ in a deposit account at $4\ \%$ per annum compound interest; that is, each year the interest is added to the principal and also generates interest.

1. a) What does the principal become at the end of the first year?

1. b) What about at the end of the second year?

1. c) Why is the increase in the second year greater than that in the first year, if the percentage is the same?

Percentages can be greater than $100\ \%$:

1. a) The population of a new neighbourhood has increased by $150\ \%$ in ten years. What is the multiplier? If there were $1,200$ residents, how many are there now?

1. b) A club's point balance has tripled. What percentage increase does this correspond to?

1. c) Is "increasing by $200\ \%$" the same as "doubling"? Reason your answer with an example.

Percentages can also be less than $1\ \%$:

1. a) An institution charges a fee of $0.3\ \%$ for each urgent transfer. How much does it charge for a transfer of $90{,}000$ pesos?

1. b) In an Olympic swimming pool containing $2{,}500{,}000\ \text{L}$ of water, $0.02\ \%$ corresponds to the water renewed daily due to chlorination and balancing. How many liters is that?

1. c) Express $0.5\ \%$ as a multiplier and as a fraction.

A shop increases the price of its products by $20\ \%$ and, one month later, discounts them by $20\ \%$.

1. a) A pair of headphones initially cost $\$1\,000$. How much do they cost after the price increase? And after the discount?

1. b) Calculate the overall multiplier by multiplying the individual multipliers.

1. c) Have the prices returned to their initial value? What was the overall percentage change?

Grandparents deposit $60{,}000$ pesos in a savings account at $5\ \%$ per annum compound interest for their granddaughter.

1. a) Write the expression that gives the final amount after $4$ years using a power.

1. b) Calculate the final amount using a calculator.

1. c) How many pesos correspond to interest?

After a discount of $25\ \%$ is applied, a coat costs $\$1\,080$.

1. a) What is the multiplier for the discount?

1. b) Calculate the initial price by dividing by the multiplier.

1. c) Check the result by applying the discount to the initial price you obtained.

Compare two ways of investing $20{,}000$ pesos for $3$ years at $6\ \%$ per annum, as shown in the figure:

• Simple interest: each year the same interest is generated on the initial principal.

• Compound interest: each year the interest is added to the principal.

1. a) Calculate the final amount with simple interest.

1. b) Calculate the final amount with compound interest.

1. c) What is the difference? Where does this difference come from?

A bank offers a deposit account of $\$24\,000$ at $6\ \%$ per annum with monthly interest payments (monthly compound interest).

1. a) What percentage of interest is applied each month?

1. b) Write the expression for the final amount after $2$ years, taking into account that there are $24$ monthly periods.

1. c) Calculate the final amount and compare it with the amount that would be obtained with annual interest payments: $24\,000 \times 1.06^2$. Which is greater?

The shares of a technology company rose by $40\ \%$ in the first year, fell by $30\ \%$ in the second and rose by $10\ \%$ in the third.

1. a) Calculate the multiplier for each year and the overall multiplier for the three years.

1. b) If a share portfolio was worth $\$50\,000$ at the beginning, how much is it worth at the end of the third year?

1. c) What was the overall percentage change? Why does it not match the sum of the three percentages?

The price of a city transport pass has decreased by $20\ \%$ thanks to a public subsidy.

1. a) If a single ticket cost $\$31$, how much does it cost now?

1. b) If the subsidy is withdrawn and the price returns to the initial level, by what percentage must the current price increase to return to the original price?

1. c) Explain, using multipliers, why the answer is not $20\ \%$, and state the general rule: after a decrease of $r\ \%$, what percentage increase returns the price to the start?

Two banks offer a one-year deposit for $\$5,000$:

• Bank A: $4.9\ \%$ annual interest with semiannual interest payments (half of the percentage each half-year, that is, $2.45\ \%$, with compound interest).

• Bank B: $5\ \%$ annual interest with a single interest payment at the end of the year.

1. a) Calculate the final capital in bank A, taking into account that $2.45\ \%$ is applied each half-year.

1. b) Calculate the final capital in bank B.

1. c) The APR (Annual Percentage Rate, known in Spain as the TAE, Tasa Anual Equivalente) is the real annual percentage that a deposit yields, taking into account the frequency of payments. What is the APR of each offer? Which bank is better?

Decision-making task. Inés has saved $18{,}000$ pesos working on weekends and wants to invest it for $3$ years to buy a computer when she finishes secundaria. Compare three options, as shown in the figure:

• Option 1: youth account at $2\ \%$ simple interest per annum.

• Option 2: fixed deposit at $1.8\ \%$ compound interest per annum.

• Option 3: CETES (Mexican federal treasury certificates) at $2.5\ \%$ compound interest per annum, but with a penalty of $300$ pesos if withdrawn before $3$ years.

1. a) Calculate the final amount for each option.

1. b) What is the difference in pesos between the best and the worst option?

1. c) Inés thinks she might need the money before $3$ years for a summer course. Does that change your recommendation? Justify your decision.

The price of the set menu in a neighbourhood has followed inflation: it rose by $6\ \%$ in 2023, $3.5\ \%$ in 2024, and $2\ \%$ in 2025. In 2022, the set menu cost $\$220$.

1. a) Calculate the price of the menu at the end of each year using multiplier factors.

1. b) What is the overall percentage increase of the menu over the three years?

1. c) Workers' wages in the neighbourhood have only increased by $9\ \%$ over that period. Have they gained or lost purchasing power to pay for the menu? Justify your answer by comparing multiplier factors.