What is a percent?

Links and useful resources

• gr7's Prealgebra Course Outline START HERE<<<
• AoPS Online Textbook
• IXL Grade 7 index
• IXL Grade 8 index
• AoPS Alcumus
• IXL prealgebra practice index
• OpenSTAX Prealgebra-1 textbook

Concept summary and lesson

• Parts out of 100
• why we use percents
• Percents are the ultimate common denominator expressions.

Percents are (lterally) hundredths. Parts per hundred. We use them anywhere that will have fractional numbers that we need to be able to compare and remember easily, because they're "good enough" for most cases and they're easy to work with.

• You can compute a percent with a quick two-digit long division
• Percents are fractions with common denominators, so we can easily compare them (and add or subract them when they refer to the same overall quantity).
• Converting percents to fractions just requires writing them over 100 and then simplifying
• Multiplying percents is best done by either converting to fractions, or by converting back to plain decimal notation. Otherwise, remember you have to divide the result by $10,000$!

Converting to percents

To convert a number to a percentage, write it as a decimal, multiply by 100, and put a % sign after it! Yes, it works even if the number is bigger than one:

• 1/4 is 0.25, so we would write it as $0.25\cdot 100=25$%
• 2.4 as a percentage is $2.4\cdot 100=240$%

Converting from percents

To convert a percent to a regular number, divide by 100. If you want a fraction, you can then write that number over 100 and simplify:

• 37% is $\frac{37}{100}=0.37$
• 625% is $\frac{625}{100}=6.25$
• 22.4% is $\frac{22.4}{100}=0.224$

Percents and language

Most of the trouble around percents comes from the language we use when we're talking about them. There are several question types we'll have to know how to interpret:

• What is $x$% of $y$?
  • multiply $y\cdot \frac{x}{100}$
• What percent of $x$ is $y$?
  • divide $\frac{y}{x}$ and multiply by 100
• What is 10% of 30% of $x$?
  • multiply $\frac{10}{100}\cdot \frac{30}{100}\cdot x$
• How much does it cost with a 10% discount?
  • A discount means a fraction of the original price is subtracted
  • Find 10% of the original price, then subtract it from the original price
  • OR (not and, OR) just multiply the original price by 100%-discount (90% in this case)
• How much does it cost with 14% tax added?
  • Taxes and fees expressed as percents are fractions of the original price that get added to the total
  • Find 14% of the original price and add it to the original price
  • OR multiply the original price by $100\mathrm{\%}+14\mathrm{\%}=114\mathrm{\%}=1.14$
• How much does it cost with both a 10% discount and 14% tax?
  • Does it matter which one we do first?? - we'll work it out in class!
  • Hint: each step works on the result of the previous step, so it is $price\ast 0.9\ast 1.14$ if you apply the discount first, or $price\ast 1.14\ast 0.9$ if you apply the tax first.
  • In real life, this can get complicated when there are special rules like "You apply the tax to the price before you take the discount, but the discount only applies to the pre-tax price..."
    • If that happens in a problem, just work it through step by step and everything will be fine.
• What was your rate of return if your investment grew from $\mathrm{\$}98,234.00$ to $\mathrm{\$}99,331.00$?
  • Subtract the starting amount from the ending amount to get the amount of change
  • Divide the change by the starting amount and multiply by 100 to get the percent change, which can be negative!
    • Amount of change is $99,331-98,234=1093$
    • percent of original: $\frac{1093}{98,234}\cdot 100=0.0111\cdot 100=1.1\mathrm{\%}$
    • Our investment grew by 1.1%.
    • We have 101.1% of the money we started with

Media resources

• Youtube search for "Parts out of 100"
• Youtube search for "why we use percents"
• Youtube search for "Percents are the ultimate common denominator expressions."

Guided practice

• A student answered 17 out of 23 questions correctly on a test. What was the student's score as a percentage?
• That same student wants to get his score up to at least 90%. How many more answers must he get correct on his retake of the test?
• Write 75% as a fraction in simplest terms
• Write 5/8 as a percent.
• Write 0.074 as a percent.
• How much would a $40.00 item cost after a 10% discount? How about if we added a 30% import tariff?

Homework

• #hw (math) (gr7) 8.1 What is a Percent? (p. 320) 2025-02-12