Pre-Algebra

Although it's not at all hard to increase or decrease a number by a percent (review the Percent Increase/Decrease lesson if this doesn't sound familiar), it can be tricky to find the percent a number was increase or decreased by.

Finding a percent of a number means finding a part or portion of a number. Increasing/decreasing a number by percent means finding a percent of a number and adding or subtracting that amount from the original number. We deal with percent increases (mark-ups, tips, sales tax) and decreases (sales, discounts) all the time. Calculating percent increases and decreases mostly involves finding a percent (if you need to review that, review the Finding a Percent lesson) and then adding it to, or subtracting it from, your original number. 

Percents are one of the types of math that we use the most in real life, which is probably why they lend themselves so nicely to word problems!

If you know how to find percents from all different angles (see our Percents Lessons), word problems are easy.  You can use a simple formula, or a proportion, fill in the items you know from the word problem, and solve.

Remember, "of" means multiplication in math.

Sometimes you are given a percent and a number, and asked to find the whole number that that percent is based on.  

There are several ways to find a whole number based on a percent.  EdBoost's preferred way to to use a proportion (not necessarily because it's the easiest way to do this operation, but because there are several related operations, and using a proportion makes it clear how to solve all kinds of percent problems).

Example: 20 is 80% of what number?

Finding a percent of a number means finding a part or portion of a number. A percent is another way to write a decimal or a fraction, so finding a percent of a number is the same as finding a fraction or decimal of a number. One great math rule to remember: the word "of" means multiply in math.

Finding a percent based on a portion of a number means figuring out what percent a specific portion represents.  When thinking about percents, remember that percent means "per 100" (think, per (each) and cent (100 as in century)).  Remember that portions do not have to be written as percents.  They can also be written as decimals or fractions (which are fairly easily converted to percents).  So 15% can also be written as .15 or $\dfrac{15}{100}$.  

When you multiply a square root by itself, the square root disappears.

$\sqrt{7} \times \sqrt{7}=7$

This leads to a pattern when working with $i$ (which equals $\sqrt{-1}$):

$i^1=\sqrt{-1}$

$i^2=\sqrt{-1} \times \sqrt{-1} =-1$

$i^3=\sqrt{-1} \times \sqrt{-1} \times \sqrt{-1} =-\sqrt{-1}  $

$i^4=\sqrt{-1} \times \sqrt{-1} \times \sqrt{-1} \times \sqrt{-1}= -1 \times -1 = 1$

In math, you can divide just about everything, including radicals!

But, let's remember, we don't put fractions or decimals under radicals (in final answers), so we only divide radicals when the number that will end up under the radical is a whole number.

When radicals are divided by other radicals, you can divide the numbers and keep the product under the radical. If you have a hard time remembering this rule, you can always test the rule with square roots that you know. 

In math, you can multiply just about everything, including radicals!

Let's review what a radical is: a radical is a root.  If there's no superscript number in front of the radical, it's a square root (the 2 in front of the radical is assumed if there is no other number), which asks "what number, times itself, equals this number?"

Because a radical is a square root, a radical squared is the number under the radical:

$\sqrt{12}^2=(\sqrt{12})(\sqrt{12})=12$

You already know that when a number is under a radical (e.g., $\sqrt{49}$), you're supposed to find the square root of the number (the number that, when multiplied by itself, equals the number under the radical).

You also know that most numbers are not perfect squares, so it's nearly impossible for the human mind to calcluate most square roots.

However, we can simplify many square roots -- even if we can't calculate them precisely.