Basic Addition and Subtraction

Basic Addition and Subtraction

Addition combines quantities or describes a change. Subtraction finds a difference or removes an amount. On a number line, numbers increase to the right and decrease to the left. The distance from zero is absolute value; a number farther right is greater.

How to solve these problems

Start at the first number. Adding a positive number moves right; subtracting a positive number moves left. Adding a negative moves left; subtracting a negative moves right. Opposites are equally far from zero on opposite sides.

Read the number line

smaller ←                                  → greater
   −5  −4  −3  −2  −1   0   1   2   3   4   5   6   7
    |   |   |   |   |   |   |   |   |   |   |   |   |

3 + 4: start at 3 → 4 → 5 → 6 → 7
2 − 5: −3 ← −2 ← −1 ← 0 ← 1 ← start at 2

Each neighboring tick is one unit apart. −3 and 3 are opposites, each three units from zero. −3 is less than 3 because it lies farther left.

Worked Example 1: 3 + 4

3 + 4

Solution: Start at 3 and move 4 units right: 4, 5, 6, 7. Answer: 7.

Why it works: Addition of a positive number increases the starting value.

Worked Example 2: 6 − 4

6 − 4

Solution: Start at 6 and move 4 units left: 5, 4, 3, 2. Answer: 2.

Why it works: Subtraction of a positive number decreases the starting value.

Worked Example 3: −2 + 5

−2 + 5

Solution: Start at −2, move 2 units to zero, then 3 more units right. Answer: 3.

Why it works: The five-unit move crosses zero; the positive change is larger than the starting distance below zero.

Worked Example 4: 2 − 5

2 − 5

Solution: Start at 2, move 2 units left to zero, then 3 more left. Answer: −3.

Why it works: Removing more than the starting amount produces a negative result.

Worked Example 5: −3 + (−2)

−3 + (−2)

Solution: Start at −3 and move 2 units left to −5.

Why it works: Adding a negative represents a decrease.

Worked Example 6: −4 − (−6)

−4 − (−6)

Solution: Rewrite as −4 + 6. Move 6 units right from −4 to 2.

Why it works: Subtracting a negative undoes a decrease, so it has the same effect as adding its opposite.

Worked Example 7: Compare −5 and −1

Compare −5 and −1

Solution: −1 is to the right of −5, so −1 > −5. Their distance is 4 units.

Why it works: A number closer to zero on the negative side is greater, even though its absolute value is smaller.

Worked Example 8: 148 + 176

148 + 176

Solution: Add ones: 8 + 6 = 14; carry 1 ten. Add tens: 4 + 7 + 1 = 12; carry 1 hundred. Hundreds: 1 + 1 + 1 = 3. Answer: 324.

Why it works: Regrouping exchanges ten smaller units for one larger unit without changing the quantity.

Worked Example 9: 302 − 178

302 − 178

Solution: Regroup 302 as 2 hundreds, 9 tens, 12 ones. Compute 12 − 8 = 4, 9 − 7 = 2, and 2 − 1 = 1. Answer: 124.

Why it works: Regroup through the zero tens place. Check: 178 + 124 = 302.

Worked Example 10: The temperature is −2°C and rises 7°C. What is the new temperature?

The temperature is −2°C and rises 7°C. What is the new temperature?

Solution: Calculate −2 + 7 = 5°C. On the number line move 7 units right.

Why it works: A rise is a positive change. The result should be above the starting temperature.

Try it again

Cover the solutions and solve the ten examples on your own. Explain which rule you used, then compare your method and answer with the worked solution. Check signs, equivalent forms, and units.