GMAT Quantitative Skills Course: Basic Operations with Integers
- Goalisb
- Jul 11
- 8 min read
Mastering Integer Arithmetic: The Foundation of Quantitative Reasoning
After laying the groundwork with integers and the number line, our next critical step is to master the fundamental operations that govern their interactions: addition, subtraction, multiplication, and division. While these might seem like elementary school concepts, a firm and nuanced grasp of integer arithmetic, especially concerning positive and negative signs, is absolutely essential.
Missteps here can ripple through entire GMAT problems, leading to incorrect answers even when the underlying strategy is sound. This lesson will provide a thorough exploration of each operation, delve into crucial properties, clarify the all-important order of operations, and equip you with the precision needed to navigate any quantitative challenge.
I. Addition of Integers: Combining Quantities and Directions
Adding integers isn't always about "getting bigger"; it's about combining directed movements on the number line.
Rule 1: Adding Integers with the Same Sign
Concept: If both integers are positive or both are negative, add their absolute values and keep the original sign.
Intuition: Think of walking on the number line. If you walk right then right, you end up further right. If you walk left then left, you end up further left.
Examples:
Positive + Positive: 5 + 3 = 8 (Walk 5 right, then 3 more right; end up at 8).
Negative + Negative: -5 + (-3) = -8 (Walk 5 left from 0, then 3 more left from -5; end up at -8).
Common Pitfall: Students sometimes incorrectly think -5 + (-3) = 8. Remember, you're going "deeper into debt."
Rule 2: Adding Integers with Different Signs
Concept: Subtract the smaller absolute value from the larger absolute value. The result takes the sign of the integer with the larger absolute1 value.
Intuition: You're walking in opposite directions. The result depends on which "side" (positive or negative) had the longer walk.
Examples:
Positive + Negative: 7 + (-4) = 3 (|7| = 7, |-4| = 4. 7 - 4 = 3. Since |7| is larger and positive, the result is positive 3).
Think: You gained $7, then lost $4. Your net gain is $3.
Negative + Positive: -10 + 6 = -4 (|-10| = 10, |6| = 6. 10 - 6 = 4. Since |-10| is larger and negative, the result is negative 4).
Think: You lost $10, then gained $6. Your net loss is $4.
When absolute values are equal: -5 + 5 = 0 (You moved left 5, then right 5; you're back at 0).
II. Subtraction of Integers: The "Add the Opposite" Rule
Subtraction of integers is often the trickiest for beginners. The most reliable method is to transform every subtraction problem into an addition problem.
The Rule: a - b = a + (-b)
Concept: Change the subtraction sign to an addition sign, and change the sign of the number immediately following the subtraction sign to its opposite. Then, follow the rules for integer addition.
Intuition: Subtracting a positive quantity is like adding a negative quantity. Subtracting a negative quantity is like adding a positive quantity (a double negative becomes a positive).
Examples:
7 - 3 = 7 + (-3) = 4 (Standard subtraction, but confirms the rule)
3 - 7 = 3 + (-7) = -4
-5 - 2 = -5 + (-2) = -7 (You start at -5 and move 2 more units left)
-4 - (-6) = -4 + 6 = 2 (This is the most common pitfall: subtracting a negative makes it positive. You're "removing" a debt, which is like gaining money.)
9 - (-2) = 9 + 2 = 11
III. Multiplication and Division of Integers: Consistent Sign Rules
Multiplication and division of integers follow the exact same set of sign rules.
Rule 1: Same Signs -> Positive Result
Concept: If the two integers being multiplied or divided have the same sign (both positive OR both negative), the result is always positive.
Examples:
Positive Positive: 4 3 = 12
Negative Negative: -4 -3 = 12 (A debt taken away 3 times results in a positive effect)
Positive / Positive: 10 / 2 = 5
Negative / Negative: -10 / -2 = 5
Rule 2: Different Signs -> Negative Result
Concept: If the two integers being multiplied or divided have different signs (one positive and one negative), the result is always negative.
Examples:
Positive Negative: 5 -2 = -10
Negative Positive: -5 2 = -10
Positive / Negative: 12 / -3 = -4
Negative / Positive: -12 / 3 = -4
Multiplication/Division with Zero:
Any integer multiplied by zero is zero: 5 0 = 0, -7 0 = 0.
Zero divided by any non-zero integer is zero: 0 / 5 = 0.
Division by Zero is UNDEFINED: You cannot divide any number by zero. This is a crucial concept. 5 / 0 is undefined. 0 / 0 is also undefined (or indeterminate, depending on context, but on the GMAT, it's simply undefined).
IV. Order of Operations: The Hierarchy of Calculations (PEMDAS/BODMAS)
When an expression involves more than one operation, the order of operations dictates the sequence in which calculations must be performed to ensure a consistent and correct result. A widely used acronym is PEMDAS (or BODMAS).
PEMDAS:
Parentheses (or Brackets)
Exponents (or Orders/Powers)
Multiplication and Division (from left to right)
Addition and Subtraction (from left to right)
Key Considerations for PEMDAS:
Multiplication & Division: These operations have equal precedence. Perform them from left to right as they appear in the expression.
Addition & Subtraction: These operations also have equal precedence. Perform them from left to right as they appear.
Example: Evaluate 15 - 3 * (8 - 6)^2 + 10 / 5
Parentheses: (8 - 6) = 2 Expression becomes: 15 - 3 * (2)^2 + 10 / 5
Exponents: (2)^2 = 4 Expression becomes: 15 - 3 * 4 + 10 / 5
Multiplication & Division (left to right):
3 * 4 = 12
10 / 5 = 2 Expression becomes: 15 - 12 + 2
Addition & Subtraction (left to right):
15 - 12 = 3
3 + 2 = 5 Final Result: 5
V. Properties of Integer Operations
Understanding these properties can simplify calculations and provide conceptual shortcuts.
1. Commutative Property (Order Doesn't Matter)
Addition: a + b = b + a
Example: (-3) + 7 = 4, and 7 + (-3) = 4.
Multiplication: a b = b a
Example: (-2) 5 = -10, and 5 (-2) = -10.
Does NOT apply to Subtraction or Division: 5 - 3 ≠ 3 - 5. 10 / 2 ≠ 2 / 10.
2. Associative Property (Grouping Doesn't Matter)
Addition: (a + b) + c = a + (b + c)
Example: (2 + (-3)) + 5 = (-1) + 5 = 4. Also, 2 + ((-3) + 5) = 2 + 2 = 4.
Multiplication: (a b) c = a (b c)
Example: (3 -4) 2 = -12 2 = -24. Also, 3 (-4 2) = 3 -8 = -24.
Does NOT apply to Subtraction or Division: (10 - 5) - 2 ≠ 10 - (5 - 2). (20 / 4) / 2 ≠ 20 / (4 / 2).
3. Distributive Property (Multiplication over Addition/Subtraction)
Rule: a (b + c) = (a b) + (a * c)
Example: 4 (5 + (-2)) = 4 3 = 12.
Also: (4 5) + (4 -2) = 20 + (-8) = 12.
Rule: a (b - c) = (a b) - (a * c)
Example: 3 (7 - 2) = 3 5 = 15.
Also: (3 7) - (3 2) = 21 - 6 = 15.
Importance: This property is crucial for algebraic manipulation and simplifying expressions on the GMAT.
VI. Why Mastering Integer Operations Matters for the GMAT
Accuracy: Small errors with signs or order of operations are among the most common reasons for getting quantitative questions wrong. Precision here is paramount.
Efficiency: Knowing the rules perfectly means you don't hesitate, saving precious seconds per question.
Foundation for Advanced Topics: Everything from exponents and roots to algebra, functions, and word problems will involve manipulating integers. A weak foundation will constantly undermine your efforts in these areas.
Data Sufficiency: Often, GMAT Data Sufficiency questions test your understanding of integer properties and how operations affect them (e.g., "Is x positive?").
Interactive Check Your Understanding:
Calculate: -18 + 7 - (-5)
Calculate: -6 * (-3) + 15 / (-5)
Which property is illustrated by (x + 5) + y = x + (5 + y)?
If a = -2, b = 3, c = -4, evaluate a * (b - c).
Practice Questions:
Evaluate the following expressions:
a) 12 + (-8) - 4
b) -10 - (-5) + (-2)
c) 7 * (-4) / 2
d) (-30 / 6) * (-2)
Evaluate: 25 - [ (10 - 3) * 2 - 4 ]
A temperature starts at -5°C. It drops by 8°C, then rises by 15°C. What is the final temperature?
If x = -4 and y = -1, find the value of (x * y) - (x + y).
Determine whether the following statements are TRUE or FALSE:
a) (-1)^3 is a positive integer.
b) a / (b / c) = (a / b) / c for all non-zero integers a, b, c.
c) 5 * (x - 3) = 5x - 15 is an example of the distributive property.
Solutions to Practice Questions:
Evaluate Expressions:
a) 12 + (-8) - 4 = 4 - 4 = 0
b) -10 - (-5) + (-2) = -10 + 5 + (-2) = -5 + (-2) = -7
c) 7 (-4) / 2 = -28 / 2 = *-14**
d) (-30 / 6) (-2) = -5 (-2) = 10
Evaluate 25 - [ (10 - 3) * 2 - 4 ]:
Parentheses (innermost): (10 - 3) = 7 Expression: 25 - [ 7 * 2 - 4 ]
Bracket (Multiplication first): 7 * 2 = 14 Expression: 25 - [ 14 - 4 ]
Bracket (Subtraction): 14 - 4 = 10 Expression: 25 - 10
Final Subtraction: 25 - 10 = 15
Final Temperature:
Start: -5°C
Drops by 8°C: -5 - 8 = -13°C
Rises by 15°C: -13 + 15 = 2°C
The final temperature is 2°C.
Value of (x * y) - (x + y) if x = -4, y = -1:
x y = (-4) (-1) = 4
x + y = (-4) + (-1) = -5
(x y) - (x + y) = 4 - (-5) = 4 + 5 = *9**
TRUE or FALSE Statements:
a) (-1)^3 is a positive integer.
(-1)^3 = (-1) (-1) (-1) = 1 * (-1) = -1.
-1 is not a positive integer. FALSE.
b) a / (b / c) = (a / b) / c for all non-zero integers a, b, c.
Let a=20, b=10, c=2.
Left side: 20 / (10 / 2) = 20 / 5 = 4.
Right side: (20 / 10) / 2 = 2 / 2 = 1.
Since 4 ≠ 1, this property (associativity for division) does not hold. FALSE.
c) 5 * (x - 3) = 5x - 15 is an example of the distributive property.
This correctly applies the multiplication of 5 to both terms inside the parenthesis. TRUE.
