Percentages
The download bar of a video game is at 75%. During El Buen Fin, a pair of sneakers has 20% off. On the restaurant bill you see 16% VAT (IVA) and you're asked how much tip to leave. Percentages are everywhere, and they are just a way of writing a ratio: so many per hundred. At the end you'll see how to find the price of a cell phone before VAT from what you paid.
What a percentage is
A $10 \times 10$ grid has 100 small squares, so each one is $1\,\%$. Counting the shaded squares gives the percentage directly.
Since every percentage is a fraction with denominator 100, converting is easy. For example, $\dfrac{30}{100} = \dfrac{3}{10}$ (simplest form) $= 0.3$.
Calculate 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)…
Calculate 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 $700$ pesos.
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.
Calculating a percentage of a quantity
A percentage can be greater than $100\,\%$ or less than $1\,\%$. $200\,\%$ of a quantity is double it; $150\,\%$ is one and a half times it. And $0.5\,\%$ is $\dfrac{0.5}{100} = 0.005$.
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.
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?
What percentage is a part of the whole?
Now the other way around: you know the part and the whole, and you want the percentage. Write the ratio $\dfrac{\text{part}}{\text{whole}}$ and convert it to a fraction with denominator 100 (or to a decimal by dividing, and multiplying by 100).
How much it changed, as a percentage
To find what percentage a quantity increased or decreased, compare the change with the original value: $\dfrac{\text{change}}{\text{original}}$.
If the change is greater than the original value, the percentage exceeds $100\,\%$: the quantity more than doubled.
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 from a part
If you know that a part is $n\,\%$ of the whole, divide by $n$ to find $1\,\%$ and multiply by 100. It's a unit value, as in the rule of three.
This idea also works: if the part is $70\,\%$, then $\text{total} = \dfrac{\text{part}}{0.7}$.
When a problem gives percentages and a remainder as a quantity, first calculate what percentage the remainder is: $100\,\% - 45\,\% - 35\,\% = 20\,\%$. That $20\,\%$ corresponds to the given quantity, and from there you get the total.
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 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, discounts, and tips
In Mexico the general VAT (IVA) is $16\,\%$, so adding it is an increase with factor $1.16$.
VAT is not always 16%: along the northern border it is 8%, and some exercises use another rate. The method is the same: only the factor changes.
The tip
At a restaurant it is customary to leave a tip of between $10\,\%$ and $15\,\%$ of the bill. It's just another increase: you calculate that percentage of the bill and add it.
A pair of sneakers that cost $1\,200$ 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 $1\,200$ 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 $9\,000$ pesos excluding taxes. If $21\%$ VAT (Value Added Tax) must be applied, what will be the final price we will have to pay for…
The price of a laptop in a store is $9\,000$ pesos excluding taxes. If $21\%$ VAT (Value Added Tax) must be applied, what will be the final price we will have to pay for the laptop?
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?
Chained percentages and percentages of what's left
When several percentages are applied one after another, each one is applied to the previous result.
The same thing happens when a percentage is applied to "what's left." If a 128 GB cell phone has $10\,\%$ taken up by the system, and then you fill $50\,\%$ of the free space, the second percentage is calculated on the free space, not on the 128 GB.
In a trendy shoe store, a pair of sneakers costing $\$1\,700$ has a discount of $20\%$. However, as it is the last day of the sale, an additional discount of $10\%…
In a trendy shoe store, a pair of sneakers costing $\$1\,700$ 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 Álex 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 Álex 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?
VAT in reverse: finding the price before tax
When you know the final price with a percentage already included, to undo it you divide by the factor instead of multiplying.
Back to the beginning: on a bill with 16% VAT included, the price before VAT is found by dividing the total by $1.16$.
Alex bought a computer to play his favorite video games. The final price he paid, including $21\%$ VAT, was $\$29\,040…
Alex bought a computer to play his favorite video games. The final price he paid, including $21\%$ VAT, was $\$29\,040$. His friend tells him that the computer cost $\$24\,000$ before taxes. Is his friend right? Perform the necessary calculations to find the base price (excluding VAT) of the computer.