Percentages
A store in Santiago advertises: "50% off plus an additional 20% off: a total markdown of 70%!" And your receipt shows the 18% ITBIS sales tax already included. Is the 70% claim true? How do you find out what the product cost before the tax? Percentages are at the corner store, at the bank, and in cell phone deals. Mastering them keeps you from being fooled.
Calculating a percentage
Converting a percentage to a decimal is just dividing by 100, that is, moving the decimal point two places to the left: $35\,\% = 0.35$, $4\,\% = 0.04$, $120\,\% = 1.2$, $0.5\,\% = 0.005$. As a fraction: $35\,\% = \dfrac{35}{100} = \dfrac{7}{20}$.
Mental shortcuts: $50\,\%$ is half, $25\,\%$ is a quarter, $10\,\%$ is dividing by 10, and $75\,\%$ is $50\,\% + 25\,\%$.
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 $800$.
Calculate: a) $150\%$ of $60$. b) $0.5\%$ of $800$.
Calculate:
a) $150\%$ of $60$.
b) $0.5\%$ 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) $120\%$
What percent is a part of the whole?
If you want to know what percent a part is of the whole, divide the part by the whole and multiply by $100$.
Look at the grid in the figure, made up of $100$ squares. a) What percentage of the squares are shaded? b) What percentage are left unshaded?
Look at the grid in the figure, made up of $100$ squares.
a) What percentage of the squares are shaded?
b) What percentage are left unshaded?
What percentage does $45$ represent out of $180$?
What percentage does $45$ represent out of $180$?
A basketball player has scored $17$ of the $25$ free throws she took this season. What is her success percentage?
A basketball player has scored $17$ of the $25$ free throws she took this season. What is her success percentage?
Percent change
When a quantity goes from an initial value to a final one, the percent change is calculated relative to the initial value.
Percentages can go above $100\,\%$. $100\,\%$ is the whole amount, $200\,\%$ is double, and $150\,\%$ is one and a half times. An increase of $100\,\%$ doubles the quantity, and an increase of $200\,\%$ triples it.
The price of a comic has gone from $\text{RD\$}\ 720$ to $\text{RD\$}\ 900$. What percentage has it gone up by?
The price of a comic has gone from $\text{RD\$}\ 720$ to $\text{RD\$}\ 900$. What percentage has it gone up by?
The chart shows the comics that a shop sold between Monday and Friday last week. a) What percentage of the week's sales were made on Friday? b) By what percentage did…
The chart shows the comics that a shop sold between Monday and Friday last week.
a) What percentage of the week's sales were made on Friday?
b) By what percentage did sales on Friday increase compared to Monday?
State whether the following statements are true or false, justifying each answer with a numerical example: a) An amount cannot increase by more than $100\%$. b) $200\%$…
State whether the following statements are true or false, justifying each answer with a numerical example:
a) An amount cannot increase by more than $100\%$.
b) $200\%$ of an amount is double that amount.
c) If the price of a product goes from $\text{RD\$}\ 3\,000$ to $\text{RD\$}\ 9\,000$, it has increased by $200\%$.
d) It is impossible to reduce a price by $110\%$.
Increases and discounts in a single calculation
The ITBIS in the Dominican Republic is $18\,\%$. To go from a price before ITBIS to the price with ITBIS, you multiply by $1.18$.
A pair of sneakers costing $\text{RD\$}\ 3\,600$ is reduced by $15\%$. How many pesos do you save? How much do you pay?
A pair of sneakers costing $\text{RD\$}\ 3\,600$ is reduced by $15\%$. How many pesos do you save? How much do you pay?
A pair of headphones costs $\text{RD\$}\ 3\,000$ excluding ITBIS. If ITBIS is $21\%$, what is the final price?
A pair of headphones costs $\text{RD\$}\ 3\,000$ excluding ITBIS. If ITBIS is $21\%$, what is the final price?
A jacket that cost $\text{RD\$}\ 4\,494$ is reduced by $30\%$. How many pesos are discounted? What is the final price?
A jacket that cost $\text{RD\$}\ 4\,494$ is reduced by $30\%$. How many pesos are discounted? What is the final price?
Chained percentages
When two or more percentages are applied one after another, each one is calculated on the previous result. That is why you multiply the factors.
An online store sells a backpack for $\text{RD\$}\ 5\,700$ excluding tax. It applies a discount of $20\%$ and then adds $21\%…
An online store sells a backpack for $\text{RD\$}\ 5\,700$ excluding tax. It applies a discount of $20\%$ and then adds $21\%$ tax to the discounted price. What is the final price you pay?
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 $\text{RD\$}\ 7\,200…
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 $\text{RD\$}\ 7\,200$. Explain the error in the advertisement, calculate the actual percentage discount applied and the correct final price of the sneakers.
Two shops sell the same rucksack for $\text{RD\$}\ 4\,800$. Shop A reduces it directly by $25\%$. Shop B first applies a discount of $10\%…
Two shops sell the same rucksack for $\text{RD\$}\ 4\,800$. Shop A reduces it directly by $25\%$. Shop B first applies a discount of $10\%$ and then, on the resulting price, another discount of $15\%$.
a) Calculate the final price of the rucksack in each shop.
b) In which shop is it better to buy it? Explain, showing your calculations, why the two offers are not equivalent.
Finding the whole or undoing a percentage
If you know the result after the percentage and you want the starting value, divide by the factor.
It also works when you know a part and the percentage it represents: whole = part ÷ (percentage as a decimal). If $360$ students are $37.5\,\%$ of a high school, the total is $360 : 0.375 = 960$.
With chained percentages, you divide by the product of the factors.
I paid $\text{RD\$}\ 3\,570$ for a pair of trousers discounted by $15\%$. What was the price before the discount?
I paid $\text{RD\$}\ 3\,570$ for a pair of trousers discounted by $15\%$. What was the price before the discount?
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?
Lucía bought a pair of headphones on a website for $\text{RD\$}\ 5\,808$. This price already included a discount of $20\%$ on the base price and a $21\%…
Lucía bought a pair of headphones on a website for $\text{RD\$}\ 5\,808$. This price already included a discount of $20\%$ on the base price and a $21\%$ tax rate applied after the discount. What was the base price of the headphones, without discount or tax?
Recovering from a decrease: percentages are not symmetric
| Drop | To recover it has to go up |
|---|---|
| $10\,\%$ | $11.11\,\%$ |
| $20\,\%$ | $25\,\%$ |
| $50\,\%$ | $100\,\%$ |
| $90\,\%$ | $900\,\%$ |
A video game that cost $\text{RD\$}\ 3\,600$ went up by $10\%$ during the Christmas campaign, and after the holidays, the new price dropped by $10\%…
A video game that cost $\text{RD\$}\ 3\,600$ 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?
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.
Percent of a percent and bills
To find "$25\,\%$ of $40\,\%$," calculate in two steps or multiply the percentages as decimals: $0.25 \times 0.40 = 0.10$, that is, $10\,\%$ of the total.
On a bill, each surcharge is calculated on the base it indicates. Read carefully which sum each percentage is applied to, and go step by step: first the sum of the base, then the percentage, and finally the total.
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 recycling bin collects $800$ kg of packaging. $45\%$ of the weight is plastic and, of that plastic, only $60\%$ can be recycled. a) How many kilograms of plastic are…
A recycling bin collects $800$ kg of packaging. $45\%$ of the weight is plastic and, of that plastic, only $60\%$ can be recycled.
a) How many kilograms of plastic are in the bin?
b) How many kilograms are actually recycled?
c) What percentage of the total bin contents is recycled?
Here is Alex's family's electricity bill for one month: Consumption: $\text{RD\$}\ 3\,744$ Contracted power: $\text{RD\$}\ 1\,092$ Electricity tax: $5.11\%$ on the sum…
Here is Alex's family's electricity bill for one month:
-
Consumption: $\text{RD\$}\ 3\,744$
-
Contracted power: $\text{RD\$}\ 1\,092$
-
Electricity tax: $5.11\%$ on the sum of consumption and contracted power
-
Sales tax: $21\%$ 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?
Deals and price comparison
To compare two deals, calculate the total cost of each over the same period and compare.
A "3×2" deal: you pay for $2$ out of every $3$ items. The discount is $\dfrac{1}{3} \approx 33.33\,\%$ off the regular price. "Second item at $50\,\%$": you pay $1 + 0.5 = 1.5$ units for $2$, that is, a total discount of $25\,\%$.
A stationer's has a "3 for 2" offer (buy three, pay for two) on felt-tip pens that cost $\text{RD\$}\ 4.50$ each. If you buy $3…
A stationer's has a "3 for 2" offer (buy three, pay for two) on felt-tip pens that cost $\text{RD\$}\ 4.50$ each. If you buy $3$ felt-tip pens:
a) How much do you pay in total?
b) What percentage discount does the offer represent compared to the price without discount? Round to two decimal places.
Two gyms have the following offers: Gym A: $\text{RD\$}\ 1\,800$ per month, with a $20\%$ discount if you pay for the full year in advance. Gym B: $\text{RD\$}\ 1\,500$…
Two gyms have the following offers:
-
Gym A: $\text{RD\$}\ 1\,800$ per month, with a $20\%$ discount if you pay for the full year in advance.
-
Gym B: $\text{RD\$}\ 1\,500$ per month, plus a one-off joining fee of $\text{RD\$}\ 2\,400$.
a) Calculate how much a full year costs at each gym.
b) Which one would you choose for one year? What if you only want to join for $6$ months (in which case Gym A does not apply the annual discount)? Justify both answers.
Nora's family is comparing three cell phone plans: | Plan | Monthly price | Data | Conditions | |---|---|---|---| | A | $\text{RD\$}\ 777$ | $50$ GB | $25\%…
Nora's family is comparing three cell phone plans:
| Plan | Monthly price | Data | Conditions |
|---|---|---|---|
| A | $\text{RD\$}\ 777$ | $50$ GB | $25\%$ discount for the first $6$ months |
| B | $\text{RD\$}\ 1\,197$ | $120$ GB | $3$ free months per year |
| C | $\text{RD\$}\ 597$ | $20$ GB | Fixed price for $2$ years |
a) Calculate the cost of a full year for each plan, taking into account the conditions.
b) Nora uses around $45$ GB per month. Which plan suits her best? Justify your answer using your calculations from the previous section.
c) Her father uses $110$ GB per month. Which plan would you recommend to him? How much does he save per year compared to the most expensive one that meets his needs?
Simple interest and repeated increases
When you deposit money in a credit union or a bank (Banreservas, Banco Popular), they pay you interest: a percentage of the money deposited.
If you are asked for the principal, the rate, or the time, you solve from $I = C \cdot i \cdot t$:
$$C = \dfrac{I}{i \cdot t} \qquad i = \dfrac{I}{C \cdot t} \qquad t = \dfrac{I}{C \cdot i}$$Repeated increases. If a quantity goes up by the same percentage several periods in a row, the factor is repeated. With $5\,\%$ each year for $3$ years, you multiply by $1.05 \times 1.05 \times 1.05 = 1.05^3 \approx 1.1576$. The total increase is $15.76\,\%$ and not $15\,\%$.
To buy an electric bike, Andrés deposits $\text{RD\$}\ 72\,000$ into an account paying a simple interest rate of $3\%$ per annum for $2…
To buy an electric bike, Andrés deposits $\text{RD\$}\ 72\,000$ into an account paying a simple interest rate of $3\%$ per annum for $2$ years. How much interest will he receive? How much money will he have in total at the end?
A savings account offers simple interest of $0.4\%$ per month. If you deposit $\text{RD\$}\ 180\,000…
A savings account offers simple interest of $0.4\%$ per month. If you deposit $\text{RD\$}\ 180\,000$:
a) How much interest do you earn each month?
b) How many full months must pass to accumulate $\text{RD\$}\ 10\,800$ in interest?
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 $\text{RD\$}\ 16\,200$ in interest?
b) At what annual percentage rate must he invest $\text{RD\$}\ 150\,000$ for $2$ years to earn $\text{RD\$}\ 12\,000$ in interest?
c) For how long must he keep $\text{RD\$}\ 108\,000$ at $5\%$ per annum to earn $\text{RD\$}\ 21\,600$ in interest?
Back to the beginning: a 50% markdown plus an additional 20% is calculated as $0.50 \times 0.80 = 0.40$: you pay $40\,\%$, that is, a real discount of $60\,\%$, not $70\,\%$. And to find the price without ITBIS, you divide the final price by $1.18$.