Compound interest and percentage change
Some shares rise by 40 %, then fall by 30 % and then rise by 10 %. Have you gained or lost? Intuitively it seems you have gained 20 %… but you haven't. To answer correctly, and to understand why compound interest makes savings grow, you need a tool: the multiplier.
The multiplier
$$\text{cantidad final} = \text{cantidad inicial} \cdot I_v$$And the other way round: if the multiplier is $1{,}45$, the quantity has increased by 45 %; if it is $0{,}7$, it has decreased by 30 %. If it is $2{,}5$, it has increased by 150 % (percentages can go above 100).
Write down the multiplier corresponding to each of these percentage changes: a) An increase of $25\ \%$. b) A decrease of $15\ \%$. c) An increase of $3\ \%$. d) A…
Write down the multiplier corresponding to each of these percentage changes:
- a) An increase of $25\ \%$.
- b) A decrease of $15\ \%$.
- c) An increase of $3\ \%$.
- d) A decrease of $60\ \%$.
- e) An increase of $130\ \%$.
What percentage increase or decrease corresponds to each of these multipliers? a) $1,45$ b) $0,82$ c) $2,1$ d) $0,05$ e) $1,06$
What percentage increase or decrease corresponds to each of these multipliers?
- a) $1,45$
- b) $0,82$
- c) $2,1$
- d) $0,05$
- e) $1,06$
Apply the corresponding multiplier to calculate the final value: a) A festival ticket costing $80\ \text{€}$ goes up by $15\ \%$. b) A pair of trainers costing…
Apply the corresponding multiplier to calculate the final value:
- a) A festival ticket costing $80\ \text{€}$ goes up by $15\ \%$.
- b) A pair of trainers costing $64\ \text{€}$ is discounted by $25\ \%$.
- c) The rent of $950\ \text{€}$ per month for premises goes up by $4\ \%$.
After a discount of $25\ \%$ is applied, a coat costs $54\ \text{€}$. a) What is the multiplier for the discount? b) Calculate the initial price by dividing by the…
After a discount of $25\ \%$ is applied, a coat costs $54\ \text{€}$.
- a) What is the multiplier for the discount?
- b) Calculate the initial price by dividing by the multiplier.
- c) Check the result by applying the discount to the initial price you obtained.
Successive percentage changes
When there are several changes one after another, you multiply the multipliers:
A shop increases the price of its products by $20\ \%$ and, one month later, discounts them by $20\ \%$. a) A pair of headphones initially cost $50\ \text{€}$. How much…
A shop increases the price of its products by $20\ \%$ and, one month later, discounts them by $20\ \%$.
- a) A pair of headphones initially cost $50\ \text{€}$. How much do they cost after the price increase? And after the discount?
- b) Calculate the overall multiplier by multiplying the individual multipliers.
- c) Have the prices returned to their initial value? What was the overall percentage change?
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. a) Calculate the multiplier for…
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.
- a) Calculate the multiplier for each year and the overall multiplier for the three years.
- b) If a share portfolio was worth $2.500\ \text{€}$ at the beginning, how much is it worth at the end of the third year?
- c) What was the overall percentage change? Why does it not match the sum of the three percentages?
Compound interest
With compound interest, each year's interest is added to the capital, and the following year it also earns interest. It is a succession of percentage changes: each year the amount is multiplied by the same multiplier.
Compare simple and compound interest: at first you hardly notice the difference, but over the years it shoots up (compound interest is a geometric progression).
A saver deposits $2.000\ \text{€}$ in a deposit account at $4\ \%$ per annum compound interest; that is, each year the interest is added to the principal and also…
A saver deposits $2.000\ \text{€}$ 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.
- a) What does the principal become at the end of the first year?
- b) What about at the end of the second year?
- c) Why is the increase in the second year greater than that in the first year, if the percentage is the same?
Grandparents deposit $3.000\ \text{€}$ in a savings account at $5\ \%$ per annum compound interest for their granddaughter. a) Write the expression that gives the final…
Grandparents deposit $3.000\ \text{€}$ in a savings account at $5\ \%$ per annum compound interest for their granddaughter.
- a) Write the expression that gives the final amount after $4$ years using a power.
- b) Calculate the final amount using a calculator.
- c) How many euros correspond to interest?
Compare two ways of investing $1.000\ \text{€}$ for $3$ years at $6\ \%$ per annum, as shown in the figure: Simple interest: each year the same interest is generated on…
Compare two ways of investing $1.000\ \text{€}$ 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.
- a) Calculate the final amount with simple interest.
- b) Calculate the final amount with compound interest.
- c) What is the difference? Where does this difference come from?
Back to the beginning: With the shares, you combine the changes by multiplying their multipliers: $1{,}40 \cdot 0{,}70 \cdot 1{,}10 = 1{,}078$. As the result is greater than 1, you have gained, but only 7,8 % (and not the 20 % that intuition suggested).