Percentage increases and decreases
A computer costs 847 € including 21 % VAT, after a 30 % discount on its list price. What was that list price? And if your salary goes up by 20 % one year and down by 20 % the next, do you gain, lose or end up exactly the same? Both questions are solved with the same tool: the multiplier.
The multiplier
The price of a ticket for the "Tecno-Vibes" music festival is $55\text{ €}$. If the organizers decide to raise the price by $12\text{ \%}$ due to high demand, what will…
The price of a ticket for the "Tecno-Vibes" music festival is $55\text{ €}$. If the organizers decide to raise the price by $12\text{ \%}$ due to high demand, what will the new ticket price be? Also state the change index that was applied.
The salary of a young programmer increases by $5\text{ \%}$ at the end of their first year under contract. However, in the second year, due to a crisis in the company,…
The salary of a young programmer increases by $5\text{ \%}$ at the end of their first year under contract. However, in the second year, due to a crisis in the company, their salary is reduced by $5\text{ \%}$. Do you think their final salary is the same as the initial one? Justify your answer by calculating the overall index of variation.
Chained changes: several increases or decreases in a row
When two or more percentage changes are applied one after another, the multipliers are multiplied together: you never add or subtract the percentages directly.
A small local shop decides to update its product prices in line with the cost of living. At the beginning of the year, it raises all prices by $4$ %. In summer, due to a…
A small local shop decides to update its product prices in line with the cost of living. At the beginning of the year, it raises all prices by $4$ %. In summer, due to a special promotion, it applies a discount of $10$ % to the updated prices.
- a) Calculate the overall index of variation.
- b) What was the total percentage change compared to the original price at the end of last year?
- c) If a product originally cost $25$ €, what is its price after both changes?
In an urban fashion store, successive discounts are applied. First, a general discount of $20\%$ is announced on all items. During the "clearance week", an additional…
In an urban fashion store, successive discounts are applied. First, a general discount of $20\%$ is announced on all items. During the "clearance week", an additional discount of $30\%$ is applied to the already discounted price. A customer has a VIP coupon that gives an extra discount of $10\%$ on the latest price obtained.
Is this process of three discounts equivalent to a single discount of $60\%$? Justify your answer by calculating the overall index of variation and determine what the initial price of a hoodie was if it ends up costing $39,20$ € after the three discounts.
Undoing a percentage: VAT, income tax and prices that have already been reduced
If you know the final amount and need the initial one, you divide by the multiplier instead of multiplying.
An electronic component has a retail price of $242\text{ €}$ with $IVA$ $21\text{ \%}$ VAT already included. Determine the price of the product before tax (taxable base).
An electronic component has a retail price of $242\text{ €}$ with $IVA$ $21\text{ \%}$ VAT already included. Determine the price of the product before tax (taxable base).
The price of a premium subscription to a streaming platform is $15,99$ € per month, VAT included. Knowing that the VAT rate applied to these digital services in Spain is…
The price of a premium subscription to a streaming platform is $15,99$ € per month, VAT included. Knowing that the VAT rate applied to these digital services in Spain is $21$ %, calculate:
- a) The taxable base of the bill (the price before tax).
- b) The exact amount paid in VAT.
- c) If the platform announces an increase of $5$ % in the taxable base for next year, what will be the new monthly price including VAT?
Changes in opposite directions: do an increase and a decrease of the same size cancel out?
"It goes up by 20 % and then down by 20 %: I'm back where I started." This is the most common mistake with percentages, and it is never true unless the percentage is $0\,\%$.
An investor buys shares of a technology company. During the first month, the value of the shares rises by $20$ %. During the second month, the value falls by $20$ %. The…
An investor buys shares of a technology company. During the first month, the value of the shares rises by $20$ %. During the second month, the value falls by $20$ %. The investor claims: "I'm back where I started, I've neither gained nor lost".
- a) Prove mathematically whether the investor is right or not, using variation indices.
- b) If he initially invested $5.000$ €, what is the value of his investment after the two months?
- c) Calculate the overall percentage change of the investment.
An investor splits their savings into two financial products. In Product A, they invest $60\%$ of their capital and make a profit of $15\%$. In Product B, they invest…
An investor splits their savings into two financial products. In Product A, they invest $60\%$ of their capital and make a profit of $15\%$. In Product B, they invest the rest and suffer a loss of $8\%$.
Calculate the overall index of variation for their total capital and the net percentage gain or loss. If at the end of the year their total capital amounts to $53.100$ €, what was their initial capital and how much money did they lose specifically in Product B?
Back to the start: The computer had a discount of $30\,\%$ ($I_{v1} = 0{,}70$) and a VAT surcharge of $21\,\%$ ($I_{v2} = 1{,}21$), giving an overall multiplier of $0{,}70 \cdot 1{,}21 = 0{,}847$; dividing the final price by this multiplier gives the list price: $847 : 0{,}847 = 1\,000$ €.