18.3: Benchmark Percentages
Lesson
Let's contrast percentages and fractions.
Exercise \(\PageIndex{1}\): What Percentage is Shaded?
What percentage of each diagram is shaded?
Exercise \(\PageIndex{2}\): Liters, Meters, and Hours
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- How much is 50% of 10 liters of milk?
- How far is 50% of a 2,000-kilometer trip?
- How long is 50% of a 24-hour day?
- How can you find 50% of any number?
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- How far is 10% of a 2,000-kilometer trip?
- How much is 10% of 10 liters of milk?
- How long is 10% of a 24-hour day?
- How can you find 10% of any number?
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- How long is 75% of a 24-hour day?
- How far is 75% of a 2,000-kilometer trip?
- How much is 75% of 10 liters of milk?
- How can you find 75% of any number?
Exercise \(\PageIndex{3}\): Nine is ...
Explain how you can calculate each value mentally.
- 9 is 50% of what number?
- 9 is 25% of what number?
- 9 is 10% of what number?
- 9 is 75% of what number?
- 9 is 150% of what number?
Exercise \(\PageIndex{4}\): Matching the Percentage
Match the percentage that describes the relationship between each pair of numbers. One percentage will be left over. Be prepared to explain your reasoning.
- 7 is what percentage of 14?
- 5 is what percentage of 20?
- 3 is what percentage of 30?
- 6 is what percentage of 8?
- 20 is what percentage of 5?
- 4%
- 10%
- 25%
- 50%
- 75%
- 400%
Are you ready for more?
- What percentage of the world’s current population is under the age of 14?
- How many people is that?
- How many people are 14 or older?
Summary
Certain percentages are easy to think about in terms of fractions.
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25% of a number is always \(\frac{1}{4}\) of that number.
For example, 25% of 40 liters is \(\frac{1}{4}\cdot 40\) or 10 liters. -
50% of a number is always \(\frac{1}{2}\) of that number.
For example, 50% of 82 kilometers \(\frac{1}{2}\cdot 82\) or 41 kilometers. -
75% of a number is always \(\frac{3}{4}\) of that number.
For example, 75% of 1 pound is \(\frac{3}{4}\) pound. -
10% of a number is always \(\frac{1}{10}\) of that number.
For example, 10% of 95 meters is 9.5 meters. -
We can also find multiples of 10% using tenths.
For example, 70% of a number is always \(\frac{7}{10}\) of that number, so 70% of 30 days is \(\frac{7}{10}\cdot 30\) or 21 days.
Glossary Entries
Definition: Percent
The word percent means “for each 100.” The symbol for percent is %.
For example, a quarter is worth 25 cents, and a dollar is worth 100 cents. We can say that a quarter is worth 25% of a dollar.
Definition: Percentage
A percentage is a rate per 100.
For example, a fish tank can hold 36 liters. Right now there is 27 liters of water in the tank. The percentage of the tank that is full is 75%.
Practice
Exercise \(\PageIndex{5}\)
- How can you find 50% of a number quickly in your head?
- Andre lives 1.6 km from school. What is 50% of 1.6 km?
- Diego lives \(\frac{1}{2}\) mile from school. What is 50% of \(\frac{1}{2}\) mile?
Exercise \(\PageIndex{6}\)
There is a 10% off sale on laptop computers. If someone saves $35 on a laptop, what was its original cost? If you get stuck, consider using the table.
| savings (dollars) | percentage |
|---|---|
| \(35\) | \(10\) |
| \(?\) | \(100\) |
Exercise \(\PageIndex{7}\)
Explain how to calculate these mentally.
- 15 is what percentage of 30?
- 3 is what percentage of 12?
- 6 is what percentage of 10?
Exercise \(\PageIndex{8}\)
Noah says that to find 20% of a number he divides the number by 5. For example, 20% of 60 is 12, because \(60\div 5=12\). Does Noah’s method always work? Explain why or why not.
Exercise \(\PageIndex{9}\)
Diego has 75% of $10. Noah has 25% of $30. Diego thinks he has more money than Noah, but Noah thinks they have an equal amount of money. Who is right? Explain your reasoning.
(From Unit 3.4.1)
Exercise \(\PageIndex{10}\)
Lin and Andre start walking toward each other at the same time from opposite ends of 22-mile walking trail. Lin walks at a speed of 2.5 miles per hour. Andre walks at a speed of 3 miles per hour.
Here is a table showing the distances traveled and how far apart Lin and Andre were over time. Use the table to find how much time passes before they meet.
| elapsed time (hour) | Lin's distance (miles) | Andre's distance (miles) | distance apart (miles) |
|---|---|---|---|
| \(0\) | \(0\) | \(0\) | \(22\) |
| \(1\) | \(2.5\) | \(3\) | \(16.5\) |
(From Unit 3.3.4)