MATH.
LUCY
(AMC) 8.1 What is a Percent?
A percent is really just a special way of writing a fraction. The word "percent" comes from the Latin per
centum, meaning "per hundred." (This is also related to why the French word for "hundred" is "cent,"
and why there are 100 cents in a dollar.) When we write a percent, we are really writing a fraction with a
hidden denominator of 100. For example:
27 59 80 200 −50 0
27% = 100 59% = 100 80% = 100 200% = 100 −50% = 100
0% = 100
More generally, we can write
𝑥
𝑥% =
100
where 𝑥 is any number. So percents are really nothing new:
Concept: A percent is just a fraction with a hidden denominator of 100.
The usage of the word "per" in "percent" might also make you think of a rate or a ratio. Indeed, we
often think about a percent as a ratio of some quantity out of 100. For example, if we say "37% of all
teenagers like to play video games," it means that the ratio of teenagers that like to play video games to
37
all teenagers is 37: 100, or that the fraction of teenagers that like to play video games is 100 of all
teenagers.
Like many other concepts in math, percent is a flexible idea and can be used in lots of different
situations. Even the word is flexible: many people use the word "percentage" instead of "percent." We
prefer "percent," but we will also occasionally use "percentage" so that you get accustomed to seeing it
written that way.
Problems
Problem 8.1: Write the following percents as integers, fractions, or mixed numbers.
(a)19% (b) 60% (c) 350% (d) −95% (e) −250% (f) 100%
Problem 8.2: Write the following numbers as percentages.
71 3 8 1 1
(a) 100 (b) 1 (c) 4 (d)5 (e)−2 10 (f)3
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MATH.LUCY
Problem 8.3:
(a) Write 26% as a decimal. (e) Write 0.03% as a decimal.
(b) Write 7% as a decimal. (f) Write 0.34 as a percent.
(c) Write 55.2% as a decimal. (g) Write 0.081 as a percent.
(d) Write 246% as a decimal. (h) Write -2.19 as a percent.
Problem 8.4:
(a) What is 25% of 200? (c) What is 300% of 15?
(b) What is 222% of 40? 1
(d) What is 4% of 1000?
Problem 8.5:
(a) What percent of 100 is 63? (c) What percent of 1000 is 2.47?
(b) 40 is what percent of 200? (d) −12 is what percent of 3?
Problem 8.6:
(a) 80 is 20% of what number?
(b) 2 is −50% of what number?
1
(c) 4 is 250% of what number?
Problem 8.7: If 20% of 𝑥 is 𝑦, then 35% of 𝑥 is what percent of 𝑦?
Exercises
8.1.1 Write the following percents as fractions, integers, or mixed numbers:
(a) 37% (b) 80% (c) 250% (d) -25% (e) -200% (f) 1810%
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MATH.LUCY
8.1.2 Write the following numbers as percents:
33 1 2
(a) (c) 3 (e) 0 (g)
50 4 7
2 3
(b) 5 (d) −2 8 (f) -192.5 (h) 0.319
8.1.3 Compute the following numbers:
(a) 30% of 200 (e) 15% of 380
(b) 55% of 120 (f) -100% of 617
(c) 225% of 16 (g) 0% of 2,827,192
(d) -80% of 35 (h) 3% of 2000
8.1.4
(a) What percent is 20 of 80? 1
(d) What percent is of 5?
2
(b) What percent of 30 is −60? 5 2
(e) What percent of 6
is 5?
(c) What percent of 17 is 51?
(f) What percent is 7 of −35?
8.1.5
(a) 11 is 20% of what number? (c) 3 is - 40% of what number?
(b) 3 is 30% of what number? (d) 3 is 2% of what number?
7
8.1.6 Which is greater, 9 of 180 or 75% of 200?
8.1.7 What is the sum of 60% of 75 and 75% of 60?
8.1.8 Express in simplest form: 40% of 70% of 10.
8.1.9* Two percent of half a number is 5. What is the number?
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