Basic Math and Pre-Algebra All-in-One For Dummies Cheat Sheet

A sketchy knowledge of multiplication can really hold back an otherwise good math student. Here’s a quick quiz: the ten toughest problems from the multiplication table.

8 x 7 = __

7 x 9 = __

6 x 6 = __

7 x 7 = __

8 x 8 = __

9 x 9 = __

6 x 8 = __

8 x 9 = __

9 x 6 = __

7 x 6 = __

Can you do this, 10 for 10, in 20 seconds? If so, you’re a multiplication whiz. If not, you may want to get a set of flash cards and practice with them a bit until you nail the multiplication table once and for all!

It’s easy to get confused when adding and subtracting negative numbers. To begin, think of adding a number as moving up and subtracting a number as moving down. For example:

So if you go up 2 steps, then up 1 more step, and then down 6 steps, you’ve gone a total of 3 steps down; therefore, 2 + 1 – 6 = –3.

Here’s another example:

This time, go down 3 steps, then up 8 steps, and then down 1 step, for a total of 4 steps up; therefore, –3 + 8 – 1 = 4.

You can turn every problem involving negative numbers into an up-and-down example. The way to do this is by combining adjacent signs:

• Combine a plus and minus as a minus sign.
• Combine two minus signs as a plus sign.

When you multiply or divide a positive number by a negative number (or vice versa), the answer is always negative. For example:

2 x (–4) = –8

14 ÷ (–7) = –2

–3 x 5 = –15

–20 ÷ 4 = –5

When you multiply two negative numbers, remember this simple rule: Two negatives always cancel each other out and equal a positive. For example:

–8 x (–3) = 24

–30 ÷ (–5) = 6

Lots of students get factors and multiples confused because they’re so similar. Both are related to the concept of divisibility. When you divide one number by another and the answer has no remainder, the first number is divisible by the second. For example:

12 ÷ 3 = 4 → 12 is divisible by 3.

When you know that 12 is divisible by 3, you know two other things as well:

3 is a factor of 12.

12 is a multiple of 3.

In the positive numbers, the factor is always the smaller of the two numbers and the multiple is always the larger.

Math teachers usually request their students to use the smallest-possible version of a fraction — that is, to simplify fractions.

To simplify a fraction, divide the numerator (top number) and denominator (bottom number) by a common factor, a number that they’re both divisible by. For example, 50 and 100 are both divisible by 10, so

That resulting fraction can be further simplified, because both 5 and 10 are divisible by 5:

When you can no longer make the numerator and denominator smaller by dividing by a common factor, the result is a fraction that’s fully simplified.

Adding and subtracting fractions that have the same denominator is pretty simple: Perform the operation (adding or subtracting) on the two numerators and keep the denominators the same, as shown here:

When two fractions have different denominators, you can add or subtract them without finding a common denominator by using cross-multiplication, as shown here:

To multiply fractions, multiply their two numerators to get the numerator of the answer, and multiply their two denominators to get the denominator. For example:

To divide two fractions, turn the problem into multiplication by taking the reciprocal of the second fraction — that is, by flipping it upside-down. For example:

Now multiply the two resulting fractions:

The main idea of algebra is simply that an equation is like a balance scale: Provided that you do the same thing to both sides, the equation stays balanced. For example, consider the following equation:

8x – 12 = 5x + 9

To find x, you can do anything to this equation as long as you do it equally to both sides. For example:

Add 2:   8x – 12 = 5x + 9 becomes 8x – 10 = 5x + 11

Subtract 5x:   8x – 12 = 5x + 9 becomes 3x – 12 = 9

Multiply by 10:   8x – 12 = 5x + 9 becomes 80x – 120 = 50x + 90

Each of these steps is valid. One, however, is more helpful than the others, because it simplifies the equation, as you see in the next section.

The best way to find x is to isolate it — that is, get x on one side of the equation with a number on the other side. To do this while keeping the equation balanced requires great cunning and finesse. Here’s an example, using the equation from the preceding section:

Original problem:   8x – 12 = 5x + 9

Subtraction 5x:  3x – 12 = 9

Add 12:  3x = 21

Divide by 3:  x = 7

As you can see, the final step isolates x, giving you the solution: x = 7.