Subtraction: Complete Guide
What is Subtraction?
Subtraction is one of the four basic operations in arithmetic, alongside addition, multiplication, and division. It represents taking away one quantity from another to find the difference.
The formula is: a - b = c, where:
- a is the minuend (the number you're subtracting from)
- b is the subtrahend (the number being subtracted)
- c is the difference (the result)
Types of Subtraction
1. Basic Subtraction
Simple taking away of one number from another.
Example: 8 - 3 = 5
2. Subtraction with Regrouping (Borrowing)
When the digit being subtracted is larger than the digit it's being subtracted from.
borrow 1
43
- 1
2
becomes 14
114
- 7
7
Example: 43 - 17 = 26
3. Subtraction with Zeros
When borrowing needs to go through one or more zeros.
10
09
09
54
1005
- 237
768
Example: 1005 - 237 = 768
4. Subtraction with Decimals
Subtracting numbers with decimal points.
Example: 5.75 - 2.38 = 3.37
Key rule: Align the decimal points!
5. Subtraction with Negative Numbers
When subtracting involves negative values.
Examples:
- Positive - Positive: 8 - 3 = 5
- Positive - Negative: 8 - (-3) = 8 + 3 = 11
- Negative - Positive: -8 - 3 = -11
- Negative - Negative: -8 - (-3) = -8 + 3 = -5
Remember: Subtracting a negative is the same as adding a positive!
6. Subtraction with Fractions
Subtracting one fraction from another.
Same Denominator: Just subtract the numerators.
\( \frac{5}{8} - \frac{3}{8} = \frac{2}{8} = \frac{1}{4} \)
Different Denominators: Find a common denominator first.
\( \frac{3}{4} - \frac{1}{3} = \frac{9}{12} - \frac{4}{12} = \frac{5}{12} \)
7. Mental Subtraction Strategies
- Counting Back: 25 - 7 (Count: 24, 23, 22, 21, 20, 19, 18) = 18
- Counting Up: 43 - 27 (From 27 → 30 = 3, 30 → 40 = 10, 40 → 43 = 3; Total: 3 + 10 + 3 = 16)
- Using Place Value: 82 - 36 = (80 - 30) + (2 - 6) = 50 - 4 = 46
- Using Known Facts: 73 - 8 = 73 - 3 - 5 = 70 - 5 = 65
Real-world Applications
- Finding change when shopping: $50 - $37.50 = $12.50
- Calculating time differences: 5:45 PM - 3:15 PM = 2 hours 30 minutes
- Weight loss tracking: 185 lbs - 170 lbs = 15 lbs lost
- Budget management: $2000 (income) - $1850 (expenses) = $150 (savings)
- Temperature changes: 75°F - 62°F = 13°F decrease
Subtraction Properties
Not Commutative
Changing order changes result
a - b ≠ b - a
Example: 8 - 3 = 5, but 3 - 8 = -5
Not Associative
Grouping matters
(a - b) - c ≠ a - (b - c)
Example: (10 - 5) - 2 = 3, but 10 - (5 - 2) = 7