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Arithmetic & Algebra Study Guide

Fractions

What is a Fraction? A fraction is a number that expresses part of a group. Fractions are written in the form or a/b, where a and b are whole numbers, and the number b is not 0. The number a is called the numerator, and the number b is called the denominator. The fraction 1/2 , which means 1 divided by 2, can represent such things as 10 pencils out of a box of 20, or 50 cents out of a dollar. The number a is called the numerator, and the number b is called the denominator.

More Examples of Fraction Operations can be found here and here.

Decimals

Decimals: A linear array of digits that represents a real number, every decimal place indicating a multiple of a negative power of 10. For example, the decimal 0.1 = 1/10 , 0.12 = 12/100 , 0.003 = 3/1000 . Also called decimal fraction.

More examples about decimals and decimal’s operations:

Learn how to:

Percentages and Proportions

The purpose of this study guide is to help students better understand percentages and proportions. The study guide includes material that posted on the web by different web sources. The material is linked to the corresponding web sources.

Percentage

What is a percentage? Percentage - a proportion in relation to a whole (which is usually the amount per hundred).

More about Percentage

Percentage is the result obtained by multiplying a quantity by a percent. So 10 percent of 50 apples is 5 apples: the 5 apples is the percentage.

Basic "Percent of" Word Problems

When you learned how to translate simple English statements into mathematical expressions, you learned that "of" can indicate "times". This frequently comes up when using percentages.

Proportions

What is a proportion? A proportion is an equation written in the form = stating that two ratios are equivalent. In other words, two sets of numbers are proportional if one set is a constant times the other.

A proportion is simply a statement that two ratios are equal. It can be written in two ways: as two equal fractions a/b = c/d; or using a colon, a:b = c:d. The following proportion is read as "twenty is to twenty-five as four is to five"; 20:25 = 4:5.

Cross multiplication to solve proportions. When the terms of a proportion are cross multiplied, the cross products are equal. Cross multiplication is the multiplication of the numerator of the first ratio by the denominator of the second ratio and the multiplication of the denominator of the first ratio by the numerator of the second ratio.

More examples on solving proportions

Signed Numbers

An integer is a whole number that can be either greater than 0, called positive, or less than 0, called negative. Zero is neither positive nor negative.

Two integers that are the same distance from zero in opposite directions are called opposites.

Every integer on the number line has an absolute value, which is its distance from zero.

The Number Line

The number line is a line labeled with positive and negative numbers in increasing order from left to right, that extends in both directions. The number line shown below is just a small piece of the number line from -4 to 4.

About Positive and Negative Numbers

Positive numbers are any numbers greater than zero, for example: 1, 2.9, 3.1000, 10000, and 0.005. For each positive number, there is a negative number that is its opposite. We write the opposite of a positive number with a negative or minus sign in front of the number, and call these numbers negative numbers. The opposites of the numbers in the list above would be: -1, -2.9, -3.14159, -40000, and -0.0005. Negative numbers are less than zero. Similarly, the opposite of any negative number is a positive number. For example, the opposite of -12.3 is 12.3.
We do not consider zero to be a positive or negative number.
The sum of any number and its opposite is 0.
The sign of a number refers to whether the number is positive or negative, for example, the sign of -3.2 is negative, and the sign of 442 is positive.
We may also write positive and negative numbers as fractions or mixed numbers.

In arithmetic we cannot subtract a larger number from a smaller:

2 - 3

But in algebra we can. And to do it, we invent "negative" numbers.

2 - 3 = -1 ("Minus 1" or "Negative 1").

That is, since

2 - 2 = 0,

2 - 3 is one "less" than 0.

We call it -1. -1 is a signed number. Its algebraic sign is - ("minus")

Its algebraic sign , + or -, and its absolute value , which is simply the arithmetical value, that is, the number without its sign.

The algebraic sign of +3 ("plus 3" or "positive 3") is +, and its absolute value is 3.

The algebraic sign of -3 ("negative 3" or "minus 3") is -. The absolute value of -3 is also 3.

For better or for worse, the minus sign '-' is not only the sign of a negative number. It is also the sign for the operation of subtraction. Those are two completely different concepts.

As for the algebraic sign +, normally we do not write it. The algebraic sign of 2, for example, is understood to be +.

As for 0, it is useful to say that it has both signs: -0 = +0 = 0.

When we place a number within vertical lines, |-3|, that imply its absolute value .

|-3| = 3

|3| = 3

Adding Positive and Negative Numbers

1) When adding numbers of the same sign, we add their absolute values, and give the result the same sign.

Examples:

2 + 5.7 = 7.7
(-7.3) + (-2.1) = -(7.3 + 2.1) = -9.4
(-100) + (-0.05) = -(100 + 0.05) = -100.05

2) When adding numbers of the opposite signs, we take their absolute values, subtract the smaller from the larger, and give the result the sign of the number with the larger absolute value.

Example:

7 + (-3.4) = ?
The absolute values of 7 and -3.4 are 7 and 3.4. Subtracting the smaller from the larger gives 7 - 3.4 = 3.6, and since the larger absolute value was 7, we give the result the same sign as 7, so 7 + (-3.4) = 3.6.

Example:

8.5 + (-17) = ?
The absolute values of 8.5 and -17 are 8.5 and 17. Subtracting the smaller from the larger gives 17 - 8.5 = 8.5, and since the larger absolute value was 17, we give the result the same sign as -17, so 8.5 + (-17) = -8.5.

Example:

-2.2 + 1.1 = ?
The absolute values of -2.2 and 1.1 are 2.2 and 1.1. Subtracting the smaller from the larger gives 2.2 - 1.1 = 1.1, and since the larger absolute value was 2.2, we give the result the same sign as -2.2, so -2.2 + 1.1 = -1.1.

Example:

6.93 + (-6.93) = ?
The absolute values of 6.93 and -6.93 are 6.93 and 6.93. Subtracting the smaller from the larger gives 6.93 - 6.93 = 0. The sign in this case does not matter, since 0 and -0 are the same. Note that 6.93 and -6.93 are opposite numbers. All opposite numbers have this property that their sum is equal to zero. Two numbers that add up to zero are also called additive inverses.


Subtracting Positive and Negative Numbers

Subtracting a number is the same as adding its opposite.

Examples:

In the following examples, we convert the subtracted number to its opposite, and add the two numbers.
7 - 4.4 = 7 + (-4.4) = 2.6
22.7 - (-5) = 22.7 + (5) = 27.7
-8.9 - 1.7 = -8.9 + (-1.7) = -10.6
-6 - (-100.6) = -6 + (100.6) = 94.6

Note that the result of subtracting two numbers can be positive or negative, or 0.


Multiplying Positive and Negative Numbers

To multiply a pair of numbers if both numbers have the same sign, their product is the product of their absolute values (their product is positive). If the numbers have opposite signs, their product is the opposite of the product of their absolute values (their product is negative). If one or both of the numbers is 0, the product is 0.

Examples:

In the product below, both numbers are positive, so we just take their product.
0.5 × 3 = 1.5

In the product below, both numbers are negative, so we take the product of their absolute values.
(-1.1) × (-5) = |-1.1| × |-5| = 1.1 × 5 = 5.5

In the product of (-3) × 0.7, the first number is negative and the second is positive, so we take the product of their absolute values, which is |-3| × |0.7| = 3 × 0.7 = 2.1, and give this result a negative sign: -2.1, so (-3) × 0.7 = -2.1

In the product of 21 × (-3.1), the first number is positive and the second is negative, so we take the product of their absolute values, which is |21| × |-3.1| = 21 × 3.1 = 65.1, and give this result a negative sign: -65.1, so 21 × (-3.1) = -65.1.


To multiply any number of numbers:

1. Count the number of negative numbers in the product.
2. Take the product of their absolute values.
3. If the number of negative numbers counted in step 1 is even, the product is just the product from step 2, if the number of negative numbers is odd, the product is the opposite of the product in step 2 (give the product in step 2 a negative sign). If any of the numbers in the product is 0, the product is 0.

Example:

2 × (-1.1) × 5 (-1.2) × (-9) = ?
Counting the number of negative numbers in the product, we see that there are 3 negative numbers: -1.1, -1.2, and -9. Next, we take the product of the absolute values of each number: 2 × |-1.1| × 5 × |-1.2| × |-9| = 2 × 1.1 × 5 × 1.2 × 9 = 118.8
Since there were an odd number of numbers, the product is the opposite of 118.8, which is -118.8, so 2 × (-1.1) × 5 (-1.2) × (-9) = -118.8.


Dividing Positive and Negative Numbers

To divide a pair of numbers if both numbers have the same sign, divide the absolute value of the first number by the absolute value of the second number.
To divide a pair of numbers if both numbers have different signs, divide the absolute value of the first number by the absolute value of the second number, and give this result a negative sign.

Examples:

In the division below, both numbers are positive, so we just divide as usual.
7 ÷ 2 = 3.5

In the division below, both numbers are negative, so we divide the absolute value of the first by the absolute value of the second.
(-2.4) ÷ (-3) = |-2.4| ÷ |-3| = 2.4 ÷ 3 = 0.8

In the division (-1) ÷ 2.5, both number have different signs, so we divide the absolute value of the first number by the absolute value of the second, which is |-1| ÷ |2.5| = 1 ÷ 2.5 = 0.4, and give this result a negative sign: -0.4, so (-1) ÷ 2.5 = -0.4.

In the division 9.8 ÷ (-0.7), both number have different signs, so we divide the absolute value of the first number by the absolute value of the second, which is |9.8| ÷ |-0.7| = 9.8 ÷ 0.7 = 14, and give this result a negative sign: -14, so 9.8 ÷ (-0.7) = -14.


Exponents

The purpose of this study guide is to help students better understand exponents and exponents’ basic rules. The study guide includes material that posted on the web by different web sources. The material is linked to the corresponding web sources.

What is an exponent? The exponent of a number shows you how many times the number is to be used in a multiplication. It is written as a small number to the right and above the base number.

Example: 52 = 5 × 5 = 25

Exponents: Basic Rules

More examples could be looked at here and here .

Polynomials, Like Terms, Order of Operations

The purpose of this study guide is to help students better understand order of operations rule (PEMDA), meaning of like terms, polynomials and basic operations with polynomials. The study guide includes material that posted on the web by different web sources. The material is linked to the corresponding web sources.

Order of Operations

What is order of operations in mathematics? Order of operation comes into play when a mathematical expression has more than one arithmetical operation.

Order of operations refers to the precedence of performing one arithmetical operation over another while working on a mathematical expression.

Here are the rules:

1. Evaluate expressions inside parentheses.

2. Evaluate all powers.

3. Perform all multiplications and/or divisions from left to right.

4. Perform all additions and/or subtractions from left to right.

And more on PEMDA...

Like Terms

What is like terms? Like terms are terms that contain the same variables raised to the same power. Only the numerical coefficients are different. In an expression, only like terms can be combined. We combine like terms to shorten and simplify algebraic expressions, so we can work with them more easily. To combine like terms, we add the coefficients and keep the variables the same. We can't combine unlike terms because that's like trying to add apples and oranges!

Examples how to combine like terms

Polynomials

What is a polynomial? Polynomial comes form poly- (meaning "many") and -nomial (in this case meaning "term") ... so it says "many terms"

More examples on combining like terms, multiplication of polynomials, factoring, factoring by grouping, sums and differences of cubes

Practice on your own for Operations with Polynomials

Equations and Functions

The purpose of this study guide is to help students better understand equations and functions. The study guide includes explanations and examples as well as links to additional web sources.

Equations

What is an equation? An expression of the condition of equality between two algebraic quantities or sets of quantities, the sign = being placed between them; as, a binomial equation; a quadratic equation; an algebraic equation; a transcendental equation; an exponential equation; a logarithmic equation; a differential equation, etc.

More sample for solving equations This sections illustrates the process of solving equations of various forms. It also shows you how to check your answer three different ways: algebraically, graphically, and using the concept of equivalence. The following table is a partial list of typical equations.

Functions

What is a function? A function is a special relationship between values: Each of its input values gives back exactly one output value.
It is often written as "f(x)" where x is the value you give it.
Example: f(x) = x/2 ("f of x is x divided by 2") is a function, because for every value of "x" you get another value "x/2". So:
* f(2) = 1
* f(16) = 8
* f(-10) = -5

Some examples of Functions

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

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