Basic Math Review
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Throughout life, almost everyone forgets the math basics we learned in grade school. Like riding a bike, a quick refresher course often is all it takes to help us remember how to solve arithmetic and algebra problems. This brief review is designed to help you brush up on fractions, decimals, ratios and basic algebra.
Understanding Fractions
A fraction is a number expressed as x/y, where x and y are both integers (whole numbers, either positive or negative), but y cannot equal 0. X is called the numerator and y is the denominator.
For example: In 6/8, 6 is the numerator and 8 is the denominator.
The fraction 3/0 cannot exist because the denominator cannot equal 0.
Adding/Subtracting Fractions:
Adding or subtracting fractions is an easy task when the denominator is the same. All you have to do is add or subtract the numerators.
For example:
5/4 + 6/4 = 11/4
11/13 - 9/13 = 2/13
When you want to add or subtract fractions that don't share a common denominator, the process requires a few extra steps.
To find the least common denominator, you need to list the multiples of each and find the smallest number in common.
For example: To add 5/6 + 3/8, we need to find the least common denominator between 6 and 8.
The multiples of 6 are: 12, 18, 24, 30...
The multiples of 8 are: 16, 24, 32, 40...
The smallest number 6 and 8 share is 24. This is our least common denominator.
Back to our addition problem. We want the fractions to look like this: x/24 + y/24.
Our original denominator (6) must be multiplied by 4 to equal 24. This means we must multiply the numerator (5) by 4 as well to get an equivalent fraction.
Since 5 times 4 equals 20, our equivalent fraction for 5/6 is 20/24. If we apply the same concept to 3/8, our new fraction is 9/24.
Now, we're ready to add the fractions.
20/24 + 9/24 = 29/24
This process is the same for subtraction problems.
Multiplying/Dividing Fractions
In order to multiply two fractions, you multiply the numerators and the denominators (in this case, the denominators can be different).
For example: (6/7)*(2/5) =?
(6)(2) = 12
(7)(5) = 35
Easy, right?
To divide, we use the same process; except we invert the fraction we are dividing by first. Next, we follow the same steps for multiplication.
For example: (7/9) / (3/8) =?
(7/9)*(8/3) =?
(7)(8) = 56
(9)(3) = 27
By now, you should feel fairly comfortable with adding, subtracting, multiplying and dividing fractions.
Related: Study Tips for Math Class
Understanding Decimals
Decimals use the base 10 to represent other numbers. Each number after a decimal point corresponds to a power of 10.
For example: 34.567
3 is the tens number; the place value of 3 is 10.
4 is the units number; the place value of 4 is 1.
5 is the tenths number; the place value of 5 is 1/10.
6 is the hundredths number; the place value of 6 is 1/100.
7 is the thousandths number; the place value of 7 is 1/1000.
And so on.
Adding/Subtracting Decimals
Decimals are easy to add and subtract as long as you do one thing: line up the decimal points.
For example:
1248.96 1248.96
+ 13.41 - 13.41
1262.37 1235.55
Multiplying/Dividing Decimals
When you multiply decimals, you do not have to line up the decimal points. Just multiply the two numbers as you would any others. Then, when you have your answer, you can determine where the decimal point goes.
Take the sum of the number of decimal points in the two numbers you are multiplying, and that is where you place the decimal point in your answer.
For example:
8.9432 (4 decimal places)
x 21.47 (2 decimal places)
192010504 (6 decimal places)
192.010504
When you divide one decimal by another, you have to move the decimal point in the divisor to the right until it becomes an integer. Then, you move the decimal point in the dividend the same number of spaces.
For example:
100.51/3.98
3.98 (the divisor) must become 398. We moved the decimal 2 places.
Now 100.51 (the dividend) must also move 2 places. Our new equation looks like this:
10051/398
From here, it's just simple long division.
10051/398 = 25.253...
If you have a calculator handy, this is even easier!
Understanding Ratios
Ratios are really quite simple. A ratio is just another way of expressing division.
For example:
A ratio of 12 and 36 can be written:
12 to 36
12:36
12/36
Since a ratio is implied division, you can reduce the ratio to its lowest terms.
In other words, 12/36 can be reduced to 1/3.
Math Foundation
Understanding Basic Algebra
Basic algebra is really just advanced arithmetic. The basic difference between algebra and arithmetic is the introduction of unknown variables, often expressed as letters such as x, y, z, or a, b, c.
In algebra, you must figure out the numerical value of these variables to solve a function or equation.
When an algebraic function has multiple terms, you can simplify it by combining like variables.
For example:
3x + 9x = (3+9)x = 12x
Factoring is an important skill in basic algebra. One tool for factoring is:
X2 - y2 = (x + y)(x - y)
X2 - 16 = (x + 4)(x - 4)
Solving a Linear Equation with One Variable
When solving for a single variable in a linear equation, there are two key rules you must remember about balancing equations:
1) When you add or subtract the same constant to both sides of an equation, you create an equivalent equation.
2) When you multiply or divide both sides of an equation by the same nonzero constant, you create an equivalent equation.
For example:
8x - 4 = 20
8x - 4 + 4 = 20 + 4
8x = 24
8x = 24
8 8
x = 3
Solving a Linear Equation with Two Variables
If you want to solve a linear equation with two variables, you must determine how to express one variable in terms of the other.
For example:
Let's say you need to solve for a system of two equations with two variables.
2x + y = 4
3x + 4y = 36
Let's express the first equation in terms of y.
y = 4 - 2x
Next, let's substitute that equivalent for y in the next equation.
3x + 4(4 - 2x) = 36
That simplifies to:
3x + 16 - 8x = 36
16 - 5x = 36
To solve for x:
16 - 5x - 16 = 36 - 16
-5x = 20
x = -4
Now that we know x = -4, we can substitute -4 in the first equation to solve for y.
2(-4) + y = 4
-8 + y = 4
-8 + y + 8 = 4 + 8
y = 12
Obviously, there is still a lot more out there to learn about math, but these lessons will provide you with a good foundation for continued learning. Now that you've mastered this quick math review, you're ready to tackle any problems involving fractions, ratios, decimals and basic algebra.
