Percentages
Wilkin sees some sneakers at a store in Santiago with a 20 % discount. At the register, they add 18 % ITBIS (sales tax). Will he pay more or less than the price on the tag? Percentages are everywhere (the download bar, sales, taxes) and they are just a way of writing a ratio: so many for every hundred.
What a percentage is
If the whole is divided into 100 small squares, each small square is $1\,\%$. This is how percentages are read on a grid: you count the shaded squares.
Look at the following grid representing the plan of a leisure area in a shopping center. Each small square represents $1\%$ of the total. If the orange shaded area is…
Look at the following grid representing the plan of a leisure area in a shopping center. Each small square represents $1\%$ of the total. If the orange shaded area is reserved for arcade machines, what percentage of the total does this area occupy? Express the result also as an irreducible fraction.
Work out the following percentages mentally using the division strategies learned in class (divide by $2$ for $50\%$, by $4$ for $25\%$ and by $10$ for $10\%$): a)…
Work out the following percentages mentally using the division strategies learned in class (divide by $2$ for $50\%$, by $4$ for $25\%$ and by $10$ for $10\%$):
- a) $50\%$ of $84$ viewers on a livestream.
- b) $25\%$ of a $400$ gram pizza.
- c) $10\%$ of a transit pass costing $2\,100$ pesos.
Calculating a percentage of a quantity
In an e-sports tournament, $20\%$ of the $150$ participants play on a tablet, while the rest play on a computer. Calculate the exact number of players using a tablet to…
In an e-sports tournament, $20\%$ of the $150$ participants play on a tablet, while the rest play on a computer. Calculate the exact number of players using a tablet to compete.
The loading bar of your favorite video game shows that $75\%$ of the total download has been completed. If the full file takes up $40$ GB, how many gigabytes have been…
The loading bar of your favorite video game shows that $75\%$ of the total download has been completed. If the full file takes up $40$ GB, how many gigabytes have been downloaded so far?
The battery of Lucía's tablet has a total capacity of $5000$ mAh. If after playing "Roblox" for a while she has $18\%$ battery left, how many mAh of power does she still…
The battery of Lucía's tablet has a total capacity of $5000$ mAh. If after playing "Roblox" for a while she has $18\%$ battery left, how many mAh of power does she still have? What percentage of the battery has she already used?
What percentage is a part of the whole?
Sometimes you are given the part and the whole, and asked for the percentage. Write the ratio $\dfrac{\text{part}}{\text{whole}}$ and turn it into a fraction with denominator $100$ (or divide and multiply by $100$).
This is also how you measure how much a quantity has changed: the percent increase is $\dfrac{\text{increase}}{\text{initial quantity}}$. If a value doubles, the increase is $100\,\%$.
In a survey carried out in your 7th grade class, $18$ out of the $30$ students confirmed that they prefer horror movies to comedy movies. Calculate the percentage of…
In a survey carried out in your 7th grade class, $18$ out of the $30$ students confirmed that they prefer horror movies to comedy movies. Calculate the percentage of students who prefer horror movies.
Hint: You can first write the ratio as a fraction $\frac{18}{30}$ and then find an equivalent fraction with a denominator of $100$.
In a multiple-choice mathematics exam with $40$ questions, a student answered $28$ questions correctly, got $8$ wrong and left $4$ blank. Calculate the percentage of…
In a multiple-choice mathematics exam with $40$ questions, a student answered $28$ questions correctly, got $8$ wrong and left $4$ blank. Calculate the percentage of correct answers, wrong answers and unanswered questions relative to the total. Check that the sum of the three percentages is $100\%$.
A neighbourhood had $1,200$ residents ten years ago and today has $3,000$. a) By how many residents has the population increased? b) What percentage increase does this…
A neighbourhood had $1,200$ residents ten years ago and today has $3,000$.
- a) By how many residents has the population increased?
- b) What percentage increase does this represent compared to the number of residents ten years ago? Is it a percentage greater than or less than $100\%$? Interpret the result.
Finding the whole when you know a part
Now the other way around: you know the part and its percentage, and you are looking for the whole. Since $\text{part} = \text{whole} \times \dfrac{n}{100}$, to get back to the whole you divide by $\dfrac{n}{100}$.
Another way to reason: $40\,\%$ is $1{,}800$ pesos, so $10\,\%$ is $450$ pesos and $100\,\%$ is $450 \times 10 = 4{,}500$ pesos.
To organize the end-of-year trip, 7th grade students have sold raffle tickets. If they have already sold $360$ tickets, which represents $75\%$ of the total they had,…
To organize the end-of-year trip, 7th grade students have sold raffle tickets. If they have already sold $360$ tickets, which represents $75\%$ of the total they had, how many tickets did they have in total to sell? How many do they still have left to sell?
Marta has posted a video on TikTok. In the first two hours, she has received $180$ likes. If she knows that $12\%$ of the people who watch the video like it, how many…
Marta has posted a video on TikTok. In the first two hours, she has received $180$ likes. If she knows that $12\%$ of the people who watch the video like it, how many views has the video had in total during those two hours?
In a survey carried out at your school about social media use, $45\%$ of the students prefer Instagram, $35\%$ prefer TikTok and the remaining 48 students choose other…
In a survey carried out at your school about social media use, $45\%$ of the students prefer Instagram, $35\%$ prefer TikTok and the remaining 48 students choose other platforms. How many students are there in total at the school? How many of them prefer Instagram?
The remainder and the complement
If a part is $n\,\%$, what is left is $(100 - n)\,\%$. To chain parts (for example, "of what was left...") calculate step by step, always on the correct quantity.
When a piece of data is given as a difference (for example, "28 more votes than..."), that difference is equivalent to a difference of percentages, and it lets you find the quantities: if C has $15\,\%$ more (of the total) than B and that is 30 votes, the total is $30 \div 0.15 = 200$ votes. Look carefully at what each percentage represents: here the $15\,\%$ is the difference between the percentages of C and B, and it is applied to the total number of voters.
In the vote to choose the activities for the school festival, $1200$ students took part. $65\%$ chose the sports tournament and the rest preferred the robotics workshop.…
In the vote to choose the activities for the school festival, $1200$ students took part. $65\%$ chose the sports tournament and the rest preferred the robotics workshop. How many students voted for the robotics workshop?
A soft drink brand has decided to reduce the sugar content of its main drink. Previously, a $330 \text{ ml}$ can contained $35 \text{ g}$ of sugar. Now they have…
A soft drink brand has decided to reduce the sugar content of its main drink. Previously, a $330 \text{ ml}$ can contained $35 \text{ g}$ of sugar. Now they have launched a version with a $40\%$ reduction in sugar and, in addition, they have increased the size of the can by $10\%$.
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How many grams of sugar does the new can have?
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What percentage of sugar does the weight of the sugar represent relative to the total volume in the new can? (Assume $1 \text{ ml} = 1 \text{ g}$ for the soft drink).
In the student council elections, candidate A obtained $30\%$ of the votes, candidate B obtained $25\%$, and candidate C received the remaining votes. If candidate C won…
In the student council elections, candidate A obtained $30\%$ of the votes, candidate B obtained $25\%$, and candidate C received the remaining votes. If candidate C won with 27 votes more than candidate B, how many students voted in total? How many votes did each candidate receive?
Increases and discounts
A pair of sneakers that cost $3\,600$ pesos has a $15\%$ discount today. How many pesos do you save by buying them today? What is the final price you will pay for them?
A pair of sneakers that cost $3\,600$ pesos has a $15\%$ discount today. How many pesos do you save by buying them today? What is the final price you will pay for them?
The price of a laptop in a store is $27\,000$ pesos excluding taxes. If $21\%$ sales tax must be applied, what will be the final price we will have to pay for the laptop?
The price of a laptop in a store is $27\,000$ pesos excluding taxes. If $21\%$ sales tax must be applied, what will be the final price we will have to pay for the laptop?
Chained percentages
This answers Wilkin's question: with a $20\,\%$ discount followed by $18\,\%$ ITBIS, you multiply by $0.80 \times 1.18 = 0.944$. He pays a little less than the tag price.
In a trendy shoe store, a pair of sneakers costing $\text{RD\$}\ 5\,100$ has a discount of $20\%…
In a trendy shoe store, a pair of sneakers costing $\text{RD\$}\ 5\,100$ has a discount of $20\%$. However, as it is the last day of the sale, an additional discount of $10\%$ is applied to the already reduced price. What is the final price of the sneakers? Is it the same as applying a single discount of $30\%$? Justify your answer by calculating both cases.
The storage of a cell phone is $128 \text{ GB}$. The operating system takes up $12.5\%$ of the total. If Alex has already used up $60\%$ of the remaining free space, how…
The storage of a cell phone is $128 \text{ GB}$. The operating system takes up $12.5\%$ of the total. If Alex has already used up $60\%$ of the remaining free space, how many gigabytes does he still have available to install new apps?
In a theater with a capacity of $750$ people, $12\%$ of the tickets were left unsold today. Of the tickets sold, $65\%$ were for adults and the rest for minors under 18…
In a theater with a capacity of $750$ people, $12\%$ of the tickets were left unsold today. Of the tickets sold, $65\%$ were for adults and the rest for minors under 18 years old. How many minors under 18 attended the theater today?
ITBIS: finding the price before tax
The ITBIS (Tax on the Transfer of Industrialized Goods and Services) is the tax added to the price of many products. Its general rate is $18\,\%$, so the final price is the base price times $1.18$. With ITBIS, the government collects money to pay for public services such as schools and roads. Some exercises use a different rate: it is always given to you in the problem, and the method is the same.
In restaurants and bars, besides ITBIS, the law sets a legal tip of $10\,\%$ on the amount consumed. Both percentages are calculated on the base price of the bill.
When you know the final price with the tax, to undo it you divide by the factor instead of multiplying.
The same works for any discount or increase: if the final price is $P$ and the factor is $f$, the initial price is $P \div f$.
Alex bought a computer to play his favorite video games. The final price he paid, including $21\%$ tax, was $\text{RD\$}\ 87\,120…
Alex bought a computer to play his favorite video games. The final price he paid, including $21\%$ tax, was $\text{RD\$}\ 87\,120$. His friend tells him that the computer cost $\text{RD\$}\ 72\,000$ before taxes. Is his friend right? Perform the necessary calculations to find the base price (excluding tax) of the computer.
A movie streaming service costs $\text{RD\$}\ 9.99$ per month. If you decide to pay for the whole year at once, they offer a $15\%…
A movie streaming service costs $\text{RD\$}\ 9.99$ per month. If you decide to pay for the whole year at once, they offer a $15\%$ discount. However, this month there is a special promotion, and on top of that already reduced annual price, they offer a further fixed discount of $\text{RD\$}\ 10$. How much will you save in total per year if you take advantage of both offers compared to paying month by month?
Percentages greater than 100 % and less than 1 %
A percentage does not have to be between $0$ and $100$.
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Greater than $100\,\%$: it is more than the whole. $250\,\% = 2.5$, so $250\,\%$ of 30 is $75$. $100\,\%$ is the quantity itself and $200\,\%$ is double.
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Less than $1\,\%$: it is less than one hundredth. $0.2\,\% = \dfrac{0.2}{100} = 0.002$, so $0.2\,\%$ of $5{,}000$ is $10$.
Calculate, showing your reasoning at each step: a) $200\%$ of $45$. (Think: $200\%$ of an amount is double its value.) b) $150\%$ of $60$. c) $0.5\%$ of $800$. d)…
Calculate, showing your reasoning at each step:
- a) $200\%$ of $45$. (Think: $200\%$ of an amount is double its value.)
- b) $150\%$ of $60$.
- c) $0.5\%$ of $800$.
- d) $300\%$ of $12$.
Percentages can also be smaller than $1\%$: a) A saline solution contains $0.9\%$ salt. How many grams of salt are in a $500$ g bag of saline solution? b) A batch of…
Percentages can also be smaller than $1\%$:
- a) A saline solution contains $0.9\%$ salt. How many grams of salt are in a $500$ g bag of saline solution?
- b) A batch of $2,000$ kg of flour contains $0.05\%$ impurities. How many kilograms of impurities does it contain?
The number of members in a youth book club has increased this year by $150\%$ compared to last year, when there were $24$ members. a) A student says: "an increase of…
The number of members in a youth book club has increased this year by $150\%$ compared to last year, when there were $24$ members.
- a) A student says: "an increase of $150\%$ means that there are now $150$ members". Explain their mistake.
- b) Calculate how many members there are now. Remember: an increase of $150\%$ means adding $150\%$ of the initial number to it.
- c) If next year the club wants to triple last year's member count, what percentage increase does it need compared to last year?
A look at simple interest
When you take out a loan from a credit union or save money in a bank, the money earns interest. With simple interest, it is always calculated on the initial principal.
Summary
Back to the beginning: with a $20\,\%$ discount and $18\,\%$ ITBIS, Wilkin multiplies the price by $0.80 \times 1.18 = 0.944$, so he pays a little less than the tag price.