Percentages
A sporting goods store advertises: "50% off plus an additional 20% off: 70% off in total!" Is that true? And if a cell phone costs $\$4{,}640$ with 16% VAT (IVA) included, how much did it cost before VAT? Percentages show up in the Buen Fin sale, on receipts, and in tips. Mastering them keeps you from getting fooled.
Calculating a percentage
Calculate mentally: a) $25\%$ of $240$. b) $10\%$ of $380$. c) $50\%$ of $96$. d) $75\%$ of $800$.
Calculate mentally:
a) $25\%$ of $240$.
b) $10\%$ of $380$.
c) $50\%$ of $96$.
d) $75\%$ of $800$.
Express each percentage as a decimal and as an irreducible fraction: a) $35\%$ b) $4\%$ c) $120\%$
Express each percentage as a decimal and as an irreducible fraction:
a) $35\%$
b) $4\%$
c) $120\%$
In a survey of $400$ young people, $64\%$ state that they do sport every week. How many young people do sport?
In a survey of $400$ young people, $64\%$ state that they do sport every week. How many young people do sport?
What percent is it?
To find what percent a part is of the whole, divide the part by the whole and multiply by $100$.
This also works for measuring how much something changed: the percent change is $\dfrac{\text{change}}{\text{initial value}} \times 100$. If a comic book goes from $\$240$ to $\$300$, the change is $60$ and $\dfrac{60}{240}=0.25$, so it went up $25\,\%$.
Of the $30$ students in a class, $18$ are girls. What percentage of the class are girls? And boys?
Of the $30$ students in a class, $18$ are girls. What percentage of the class are girls? And boys?
What percentage does $45$ represent out of $180$?
What percentage does $45$ represent out of $180$?
The price of a comic has gone from $\$240$ to $\$300$. What percentage has it gone up by?
The price of a comic has gone from $\$240$ to $\$300$. What percentage has it gone up by?
Discounts, increases, and VAT
VAT (IVA) in Mexico is $16\,\%$: the price with VAT is the price before VAT multiplied by $1.16$.
A pair of sneakers costing $\$1\,200$ is reduced by $15\%$. How many pesos do you save? How much do you pay?
A pair of sneakers costing $\$1\,200$ is reduced by $15\%$. How many pesos do you save? How much do you pay?
Complete in your notebook: a) If you get a $30\%$ discount, you pay $\%$ of the price. b) If a price goes up by $12\%$, you pay $\%$ of the original price. c) If you…
Complete in your notebook:
a) If you get a $30\%$ discount, you pay ____ $\%$ of the price.
b) If a price goes up by $12\%$, you pay ____ $\%$ of the original price.
c) If you have used $45\%$ of the battery, you have ____ $\%$ left.
A jacket that cost $\$1\,498$ is reduced by $30\%$. How many pesos are discounted? What is the final price?
A jacket that cost $\$1\,498$ is reduced by $30\%$. How many pesos are discounted? What is the final price?
Chained percentages
When two percentages are applied one after the other, the second is calculated on the result of the first.
A video game that cost $\$1\,200$ went up by $10\%$ during the Christmas campaign, and after the holidays, the new price dropped by $10\%…
A video game that cost $\$1\,200$ went up by $10\%$ during the Christmas campaign, and after the holidays, the new price dropped by $10\%$.
a) What was the price after the increase?
b) What is the price after the drop?
c) Has it returned to the original price? If not, what is the total percentage change?
In a survey of $1\,500$ young people, $40\%$ state that they play team sports, and of those, $25\%$ play basketball. How many young people play basketball? What…
In a survey of $1\,500$ young people, $40\%$ state that they play team sports, and of those, $25\%$ play basketball. How many young people play basketball? What percentage of the total do they represent?
A sports store advertises: "Sale: $50\%$ off plus an extra $20\%$ off: a total discount of $70\%$!". A pair of sneakers at the store costs $\$2\,400…
A sports store advertises: "Sale: $50\%$ off plus an extra $20\%$ off: a total discount of $70\%$!". A pair of sneakers at the store costs $\$2\,400$. Explain the error in the ad, calculate the actual percentage discount applied and the correct final price of the sneakers.
Undoing a percentage
If you know the final price with a percentage already included, don't subtract the percentage: divide by the factor.
The same idea works for finding the whole when you know a part: if $360$ students are $37.5\,\%$, the total is $360 \div 0.375 = 960$.
A cooking channel increased its followers by $25\%$ in the last year and now has $1\,800$ followers. How many followers did it have a year ago?
A cooking channel increased its followers by $25\%$ in the last year and now has $1\,800$ followers. How many followers did it have a year ago?
An online store sells a backpack for $\$1\,900$ excluding sales tax. It applies a discount of $20\%$ and then adds $21\%…
An online store sells a backpack for $\$1\,900$ excluding sales tax. It applies a discount of $20\%$ and then adds $21\%$ sales tax to the discounted price. What is the final price you pay?
In a secondary school, $360$ students play a federated sport, which represents $37.5\%$ of the total. a) How many students are there in the school? b) How many do not…
In a secondary school, $360$ students play a federated sport, which represents $37.5\%$ of the total.
a) How many students are there in the school?
b) How many do not play a federated sport?
Recovering from a decrease
The ticket price for a water park dropped by $20\%$ at the end of the summer. When the next season arrives, the park wants to restore the exact original price. By what…
The ticket price for a water park dropped by $20\%$ at the end of the summer. When the next season arrives, the park wants to restore the exact original price. By what percentage must the reduced price increase? Explain why the answer is not $20\%$.
A bookshop's sales fell by $35\%$ in February compared to January. In March, sales recovered to reach the January level again. By what percentage did March sales…
A bookshop's sales fell by $35\%$ in February compared to January. In March, sales recovered to reach the January level again. By what percentage did March sales increase compared to February? Round your answer to two decimal places.
Simple interest
When you keep money in an account, the bank pays you a percentage called the interest rate. With simple interest, the percentage is always calculated on the initial principal.
If you are missing a value, solve for it: $C = \dfrac{I}{r\,t}$, $\;r = \dfrac{I}{C\,t}$, $\;t = \dfrac{I}{C\,r}$.
Martín wants to buy a mountain bike and is studying several simple interest deposit options: a) What principal must he deposit at $4\%$ per annum for $3$ years to get…
Martín wants to buy a mountain bike and is studying several simple interest deposit options:
a) What principal must he deposit at $4\%$ per annum for $3$ years to get $\$5\,400$ in interest?
b) At what annual percentage rate must he invest $\$50\,000$ for $2$ years to earn $\$4\,000$ in interest?
c) For how long must he keep $\$36\,000$ at $5\%$ per annum to earn $\$7\,200$ in interest?
Here is Alex's family's electricity bill for one month: Consumption: $\$1\,248$ Contracted power: $\$364$ Electricity tax: $5.11\%$ on the sum of consumption and…
Here is Alex's family's electricity bill for one month:
-
Consumption: $\$1\,248$
-
Contracted power: $\$364$
-
Electricity tax: $5.11\%$ on the sum of consumption and contracted power
-
VAT: $21\%$ on the previous total
a) Calculate the amount of electricity tax, rounding to the nearest cent.
b) Calculate the total amount of the bill.
c) The family installs solar panels and their consumption drops by $40\%$. If the contracted power and taxes are calculated in the same way, what will the total amount of the new bill be?
Back to the beginning: a 50% discount plus an additional 20% means paying $0.50 \times 0.80 = 0.40$, which is a real discount of 60%, not 70%. And to find the price before VAT, divide the final price by $1.16$.