Fractions, Decimals, and Percents Study Guide for the Math Basics

Decimals

Another way to split up numbers into parts is to use decimals. Because we have 10 fingers (or digits) on our hands, we use 10 as the base of our number system. Decem is the latin word for 10, thus we use the decimal system in which each digit differs by a power of 10 from the next digit.

Decimal Place Value

Reference this table if you need to determine place values of decimal numbers. We’ll use the number 123.456 as an example.

\[\begin{array}{|c|c|c|c|c|c|c|} \hline \textbf{Place Value} & \text{hundreds} & \text{tens} & \text{ones} & \text {tenths} & \text{hundredths} & \text{thousandths} \\ \hline \textbf{Power of 10} & 10^2 & 10^1 & 10^0 & 10^{-1} & 10^{-2} & 10^{-3} \\ \hline \textbf{Number} & 1 & 2 & 3. & 4 & 5 & 6 \\ \hline \end{array}\]

Operations with Decimals

So we know what decimals are, let’s now figure out how to use them!

Addition and Subtraction

To add and subtract, all you have to remember is to line up the decimal points. Examples:

\[\begin{align} 34.5 & \\ \underline{+\quad 21.7}& \\ 56.2 &\\ \end{align}\] \[\begin{align} 13.575 & \\ \underline{-\quad 0.137}& \\ 13.438 &\\ \end{align}\]

Multiplication

To multiply decimals, the process is roughly the same as multiplying whole numbers.

1) Count how many digits are after the decimal points in both factors. For this example, there are 3 digits.

\[\begin{align} 1.57 & \\ \underline{\times\quad 2.1}& \\ \end{align}\]

2) Multiply normally.

\[\begin{align} 1.57 & \\ \underline{\times\quad 2.1}& \\ 157 &\\ \underline{+ \; 314 \color{red}{0}} &\\ 3297 &\\ \end{align}\]

3) Remember the number you counted to in step 1? Place the decimal point so that there are that many digits to the right of it in the product. In this case, there should be 3 digits to the right of the decimal point, so place it between the 3 and the 2.

\[\begin{align} 1.57 & \\ \underline{\times\quad 2.1}& \\ 157 &\\ \underline{+ \; 314 \color{red}{0}} &\\ 3.297 &\\ \end{align}\]

Division

Remember the parts of a division problem: \(\text{dividend} \div \text{divisor} = \text{quotient}\)

When dividing decimals, start by moving the decimal point in the divisor to the right until it gets to the end. Move the decimal that same number to the right in the dividend (if necessary, add zeroes). For \(5.6 \div 0.02\) you have to move the decimal to the right two places:

\(0.02 \rightarrow 2\) so \(5.6 \rightarrow 560\)

Thus \(5.6 \div 0.02 = 560 \div 2 = 280\)

If the divisor is still a decimal, use long division and place the decimal point directly above in the quotient.

\[\require{enclose} \begin{array}{r} 1.4 \\ 12 \enclose{longdiv}{16.8} \\[-3pt] \underline{-12}\phantom{8} \\[-3pt] 48 \\[-3pt] \underline{-48} \\[-3pt] 0 \\ \end{array}\]

Percents

Percent is a combination of two things: per and cent meaning divide by 100. So 25 percent is equivalent to:

\[25 \div 100 = \frac{25}{100} = \frac{1}{4}\]

Surprisingly that’s about it. Percent is just another way to write a fraction (or a part of a whole).

Determining the % of a Number

Many tests have problems like “What is 30 percent of 50?” To answer problems like this, remember this proportion:

\[\frac{\text{is}}{\text{of}} = \frac{\text{percent}}{100}\]

So, what is 30 percent of 50? There are 3 unknowns in the proportion above: is, of, and percent. In this problem,

\[30 = \text{"percent"}\] \[50 = \text{"of"}\]

leaving the unknown \(x = \text{"is"}\)

\[\frac{x}{50}=\frac{30}{100}\] \[100x = 1500\] \[x=15\]

So 15 is 30 percent of 50.

Here’s another example: 45 is what percent of 90? First:

\[45 = \text{"is"}\] \[90 = \text{"of"}\]

leaving the unknown \(x = \text{"percent"}\). So:

\[\frac{45}{90} = \frac{x}{100}\] \[4500 = 90x\] \[50 = x\]

So, 45 is 50 percent of 90.

Calculating % Increase or Decrease

Consider the following problem based on an everyday occurance. Your favorite department store has a sale on a hat you really want. If the hat is listed at $50 and the sale is 15% off, what’s the final price?

To answer questions like this, remember the following proportion:

\[\frac{\text{change}}{\text{original}} = \frac{\text{percent of change}}{100}\]

Now the change could be an increase (like tax or tip) or decrease (like sales), so you might prefer either of the following proportions:

\[\dfrac{\text{increase}}{\text{original}} = \dfrac{\text{percent of increase}}{100}\]

or

\[\dfrac{\text{decrease}}{\text{original}} = \dfrac{\text{percent of decrease}}{100}\]

Now, in the problem about the sale, the original price is $50, and the percent of decrease is 15%. Let \(d\) stand for decrease.

\[\dfrac{d}{50}=\dfrac{15}{100}\] \[100d = 750\] \[d=\$7.50\]

But don’t stop now! Go back to the question and make sure you’re answering it. You are being asked to find the final price (the original price - the decrease). So,

\(\$50 - \$7.50 = \$42.50\).