Basic Operations - Trivium Test Prep Online Courses

Basic Operations

Basic Operations

Arithmetic Operations

The four basic arithmetic operations are addition, subtraction, multiplication, and division.

  • Add to combine two or more quantities (6 + 5 = 11).
  • Subtract to find the difference of two or more quantities (10 − 3 = 7).
  • Multiply to add a quantity multiple times (4 × 3 = 12 ↔ 3 + 3 + 3 + 3 = 12).
  • Divide to determine how many times one quantity goes into another (10 ÷ 2 = 5).

Word problems contain clue words that help determine which operation to use.

Operations Word Problems
Clue Words
Addition sum, together, (in) total, all, in addition, increased, give
Leslie has 3 pencils. If her teacher gives her 2 pencils, how many does she now have in total?
3 + 2 = 5 pencils
minus, less than, take away, decreased, difference, how many left, How many more/less
Sean has 12 cookies. His sister takes 2 cookies. How many cookies does Sean have left?
12 − 2 = 10 cookies
product, times, of, each/every, groups of, twice
A hospital department has 10 patient rooms. If each room holds 2 patients, how many patients can stay in the department?
10 × 2 = 20 patients
divided, per, each/every, distributed, average, How many for each
A teacher has 150 stickers to distribute to her class of 25 students. If each student gets the same number of stickers, how many stickers will each student get?
150 ÷ 25 = 6 stickers

Positive and Negative Numbers

Positive numbers are greater than zero, and negative numbers are less than zero. Use the rules in the table below to determine the sign of the answer when performing operations with positive and negative numbers.

Operations with Positive and Negative Numbers
Multiplication and Division
positive + positive = positive
4 + 5 = 9
positive × positive = positive
5 × 3 = 15
negative + negative = negative
−4 + (−5) = −9 → −4 − 5 = −9
negative × negative = positive
−6 × (−5) = 30
negative + positive = sign of the larger number
−15 + 9 = −6
negative × positive = negative
−5 × 4 = −20

Subtraction is the same as adding a negative value.

5 – 7 = 5 + (−7) = −2

A number line shows numbers increasing from left to right (usually with zero in the middle). When adding positive and negative numbers, a number line can be used to find the sign of the answer. When adding a positive number, count to the right. When adding a negative number, count to the left.

Order of Operations

Operations in a mathematical expression are always performed in a specific order, which is described by the acronym PEMDAS:

  1. Parentheses
  2. Exponents
  3. Multiplication
  4. Division
  5. Addition
  6. Subtraction

Perform the operations within parentheses first, and then address any exponents. After those steps, perform all multiplication and division. These are carried out from left to right as they appear in the problem. Finally, do all required addition and subtraction, also from left to right as each operation appears in the problem.

When working with complicated expressions, underline or highlight the operation being performed in each step to avoid confusion.

Comparison of Rational Numbers

Rational numbers can be ordered from least to greatest, or greatest to least, by placing the numbers in the order in which they appear on a number line. 

  • When comparing a set of fractions try to convert each value to an equivalent fraction with a common denominator. Then, it is only necessary to compare the numerators of each fraction.
  • When working with numbers in multiple forms (for example, a group that contains fractions and decimals), convert the values so that the set contains only fractions or only decimals. 
  • When ordering negative numbers, remember that the negative number with the greatest absolute value is furthest from 0 and is therefore the least number. (For example, −75 is less than −25.)

Order the numbers from least to greatest:


Convert each fraction to a decimal: 




Plot each decimal value on a number line.

Order from least to greatest (left to right on the number line) using the original given values.


Drawing a number line can help when comparing numbers. The final list should be ordered from left to right (least to greatest) or right to left (greatest to least) on the line.

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