Percentages and Decimals

Definition

Decimals are used to indicate the fractional component of a number. For example, $\frac{5}{2}$ in decimal form is $2.5$.

There are three types of decimals:

• A terminating decimal does not go on forever. Example: $13.5982$.
• A repeating decimal goes on forever but the digits repeat. Example: $0.352525252\dots$, which can be rewritten more succinctly as $0.3\overline{52}$ by putting a line over the repeating digits.
• An irrational decimal goes on forever but the digits have no repeating pattern. Example: $\pi = 3.14159\dots$.

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

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

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

$\begin{aligned} 17\% &= \frac{17}{100} \\ 189\% &= \frac{189}{100} \\ 0.4\% &= \frac{0.4}{100} = \frac{4}{1000} = \frac{2}{500} \end{aligned}$

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.

$\begin{aligned} 0.125 &= \frac{125}{1000} = \frac{1}{8} \\ 3.72 &= \frac{372}{100} = \frac{93}{25} \end{aligned}$

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 $0.3\overline{52}$ as a fraction.

Let's have $0.3\overline{52}$ be equal to the fraction $x$.

There are two repeating digits in this fraction, so we multiply $x$ twice by $10$ to get $100x$. Now we have the equations
$\begin{array}{rcr} 100x &=& 35.2\overline{52} \\ x &=& 0.3\overline{52} \end{array}$
If we subtract them, we get $99x = 34.9$. Solving for $x$ gives us the answer:
$x = \frac{34.9}{99} = \frac{349}{990} \quad _\square$

Note that the number of 10's you multiply with depends on the number of repeating digits. For example, if $x = 0.\overline{12345}$, you would multiply by $10$ five times to get $100000x$. 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 $\$50 \times 0.05 = \$2.50$ and tip is $\$50 \times 0.19 = \$9.50$. Their total bill is therefore $\$50 + \$2.50 + \$9.50 = \$62$. $_\square$

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 $P$ 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,
$(P \times 0.85) \times 0.5 = 34$
Solving for $P$ gives us the original price of $\$80$. $_\square$