Percentage Increases and Decreases
During Buen Fin, a computer costs $\$10{,}150$ with 16% VAT (IVA) already included, after a 30% discount off its catalog price. What was that catalog price? And if your salary goes up 20% one year and down 20% the next, do you gain, lose, or end up the same? Both questions are solved with the same tool: the variation index.
The variation index
You can also work backward: if you know the initial and final prices, the index is $I_v = \dfrac{\text{final}}{\text{initial}}$, and the percentage change is $(I_v - 1) \times 100$.
The price of a ticket for the "Tecno-Vibes" music festival is $\$55$. If the organizers decide to raise the price by $12\text{ \%}…
The price of a ticket for the "Tecno-Vibes" music festival is $\$55$. 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.
A sportswear store announces a seasonal clearance sale with a discount of $25\text{ \%}$ on all its sneakers. If a pair of running shoes originally cost $\$2\,200…
A sportswear store announces a seasonal clearance sale with a discount of $25\text{ \%}$ on all its sneakers. If a pair of running shoes originally cost $\$2\,200$, how much will we pay for them after the discount? Calculate the corresponding index of variation.
In the year 2022, a monthly subscription to a streaming platform cost $\$250$. In 2023, the price rose to $\$295$. Calculate the index of variation and the percentage…
In the year 2022, a monthly subscription to a streaming platform cost $\$250$. In 2023, the price rose to $\$295$. Calculate the index of variation and the percentage increase that the subscription experienced.
Chained variations: several increases or decreases in a row
When two or more percentage changes are applied one after another, you multiply the indices: you never add or subtract the percentages directly.
A cinema ticket costs $\$180$. If you pay with a youth card, you get a discount of $20\text{ \%}…
A cinema ticket costs $\$180$. If you pay with a youth card, you get a discount of $20\text{ \%}$. But if it is also "cheap Wednesday", the already discounted price is reduced by an additional $10\text{ \%}$.
- a) Calculate the final price of the ticket on a Wednesday for a youth card holder.
- b) Calculate the total index of variation and the actual percentage discount on the initial price.
A tech store launches a new model of wireless headphones at a price of $120$ pesos. Due to high demand, the following month they decide to increase its price by $15$ %.…
A tech store launches a new model of wireless headphones at a price of $120$ pesos. Due to high demand, the following month they decide to increase its price by $15$ %. However, when the "Black Friday" week arrives, they apply a discount of $20$ % to the increased price. Calculate the final price of the headphones and determine the overall index of variation of the price.
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 $500$ pesos, 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 $784$ pesos after the three discounts.
Undoing a percentage: VAT and already discounted prices
If you know the final amount and need the initial one, you divide by the index instead of multiplying.
After applying a discount of $15\text{ \%}$, a laptop costs $\$680…
After applying a discount of $15\text{ \%}$, a laptop costs $\$680$. What was the original price of the laptop before the promotion? Use the concept of index of variation to solve it.
In the January sales, a pair of sneakers has a discount of $35$ %. If the discounted price on the tag is $1\,144$ pesos, what was the price of the sneakers before the…
In the January sales, a pair of sneakers has a discount of $35$ %. If the discounted price on the tag is $1\,144$ pesos, what was the price of the sneakers before the sale started? State also the total savings made in pesos.
The price of a premium subscription to a streaming platform is $319.80$ pesos per month, VAT included. Knowing that the VAT rate applied to these digital services in…
The price of a premium subscription to a streaming platform is $319.80$ pesos per month, VAT included. Knowing that the VAT rate applied to these digital services in Mexico 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?
Variations in opposite directions: do an increase and a decrease of the same size cancel out?
"It goes up 20% and then down 20%: I end up where I started." This is the most common mistake with percentages, and it is never true unless the percentage is $0\,\%$.
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.
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 $5,000$ pesos, 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 $53,100$ pesos, what was their initial capital and how much money did they lose specifically in Product B?
Back to the beginning: the computer had a $30\,\%$ discount ($I_{v1} = 0.70$) and $16\,\%$ VAT ($I_{v2} = 1.16$). The overall index is $0.70 \times 1.16 = 0.812$. Dividing the final price by that index: $10{,}150 \div 0.812 = 12{,}500$, so the catalog price was $\$12{,}500$. And for the salary: $1.20 \times 0.80 = 0.96$, you lose $4\,\%$.