Percentages and Decimals

Definition

Decimals are used to indicate the fractional component of a number. For example, 52\frac{5}{2} in decimal form is 2.52.5.

There are three types of decimals:

  • A terminating decimal does not go on forever. Example: 13.598213.5982.
  • A repeating decimal goes on forever but the digits repeat. Example: 0.3525252520.352525252\dots, which can be rewritten more succinctly as 0.3520.3\overline{52} by putting a line over the repeating digits.
  • An irrational decimal goes on forever but the digits have no repeating pattern. Example: π=3.14159\pi = 3.14159\dots.

A percentage is a number displayed as a fraction of 100, often denoted with a percent sign %, such as 98%98\%, which is 98 parts of 100.

Technique

You will often have to manipulate numbers between their decimal, percentage, and fraction forms.

To convert between percentage and decimal, you simply move the decimal point left or right two places.

17%=0.17189%=1.890.4%=0.004 \begin{aligned} 17\% &= 0.17 \\ 189\% &= 1.89 \\ 0.4\% &= 0.004 \end{aligned}

Percentages can be rewritten as fractions by placing them over a denominator of 100 100 .

17%=17100189%=1891000.4%=0.4100=41000=2500 \begin{aligned} 17\% &= \frac{17}{100} \\ 189\% &= \frac{189}{100} \\ 0.4\% &= \frac{0.4}{100} = \frac{4}{1000} = \frac{2}{500} \end{aligned}

To convert a terminating decimal into a fraction, first count the number of decimal places. Then divide the decimal's digits over 1 followed by the number of zeroes equal to the number of decimal places.

0.125=1251000=183.72=372100=9325\begin{aligned} 0.125 &= \frac{125}{1000} = \frac{1}{8} \\ 3.72 &= \frac{372}{100} = \frac{93}{25} \end{aligned}

Decimals that are irrational cannot be converted into fractions, but repeating decimals can. Let's look at a technique for converting repeating decimals into fractions:

Express 0.3520.3\overline{52} as a fraction.

Let's have 0.3520.3\overline{52} be equal to the fraction xx.

There are two repeating digits in this fraction, so we multiply xx twice by 1010 to get 100x100x. Now we have the equations
100x=35.252x=0.352 \begin{array}{rcr} 100x &=& 35.2\overline{52} \\ x &=& 0.3\overline{52} \end{array}
If we subtract them, we get 99x=34.999x = 34.9. Solving for xx gives us the answer:
x=34.999=349990x = \frac{34.9}{99} = \frac{349}{990} \quad _\square

Note that the number of 10's you multiply with depends on the number of repeating digits. For example, if x=0.12345x = 0.\overline{12345}, you would multiply by 1010 five times to get 100000x100000x. The rest of the steps remain the same.

Application and Extensions

Susan and Sally have just finished dinner at a nice sushi restaurant. Their waiter gives them their bill and they see that their (pre-tax and pre-tip total) is $50. If tax is 5% and they decide to give a 19% tip, how much money should Susan and Sally pay?

Tax and tip are calculated based on the original price, so tax is $50×0.05=$2.50\$50 \times 0.05 = \$2.50 and tip is $50×0.19=$9.50\$50 \times 0.19 = \$9.50. Their total bill is therefore $50+$2.50+$9.50=$62\$50 + \$2.50 + \$9.50 = \$62. _\square

 

A clothing store is having a clearance sale. Their advertisement says "Take 50% off already already reduced prices!" You decide to buy a clearance jacket with a 15%-off label, and pay $34 for it. What was the jacket's original price?

Let PP be the original price of the jacket. A 15%-off sale means that 15% of the original price is subtracted from the original price, or 100%-15% = 85% of the original price. Clearance takes another 50% off the sale price. So,
(P×0.85)×0.5=34 (P \times 0.85) \times 0.5 = 34
Solving for PP gives us the original price of $80\$80. _\square

Note by Arron Kau
5 years, 7 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

There are no comments in this discussion.

×

Problem Loading...

Note Loading...

Set Loading...