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Add & Subtract
Rational Numbers

Master the rules for adding and subtracting fractions, decimals, and integers: apply properties of operations to rational number expressions.

7th Grade Math

What Are Rational Numbers?

A rational number is any number that can be written as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \neq 0\). This includes integers, fractions, terminating decimals, and repeating decimals.

Integers
\(-5,\; 0,\; 3\) can be written as \(\frac{-5}{1},\; \frac{0}{1},\; \frac{3}{1}\)
Fractions
\(\frac{2}{3},\; -\frac{7}{4}\) are already in \(\frac{a}{b}\) form
Decimals
\(0.75 = \frac{3}{4}\) and \(0.\overline{3} = \frac{1}{3}\)

Rational Numbers on a Number Line

-3 -2 -1 0 1 2 3 -1.5 0.5 1.75 The number line helps visualize the position and order of rational numbers.

Why This Matters

Rational number operations appear everywhere: bank balances (deposits and withdrawals), temperature changes, cooking with fractions, and science measurements. Mastering these rules is the foundation for algebra and beyond.

Money
You have \$12.50 and spend \$8.75: \(12.50 - 8.75 = 3.75\)
Temperature
It is \(-3°\)F and drops \(7°\): \(-3 + (-7) = -10\)

Core Rules: Adding Rational Numbers

1
Same sign: Add the absolute values and keep the shared sign. \((-4) + (-5) = -(4+5) = -9\)
2
Different signs: Subtract the smaller absolute value from the larger, then take the sign of the number with the larger absolute value. \((-7) + 3 = -(7-3) = -4\)
3
Adding zero: Any number plus zero equals itself. \(a + 0 = a\)

Core Rules: Subtracting Rational Numbers

1
Rewrite the subtraction as addition of the opposite.
2
Apply the addition rules (same sign or different signs).
3
Simplify the result.

Example: \(5 - (-3) = 5 + 3 = 8\)

Visualizing Addition on the Number Line

-5 -4 -3 -2 -1 0 1 2 3 4 Start: -2 + 3 Result: 1
\((-2) + 3 = 1\)

Subtraction vs. Negative Sign

Students often confuse the subtraction operation with a negative sign. Although they look identical, they play different roles. The key insight: both are resolved the same way using the add-the-opposite strategy.

Subtraction (operation) Negative (sign)
Symbol \(a - b\) \(-b\)
Meaning Take away \(b\) from \(a\) The opposite of \(b\)
Rewrite \(a + (-b)\) Already an addend
Example \(6 - 4 = 6 + (-4) = 2\) \(6 + (-4) = 2\)

Properties You Can Use

Commutative
\(a + b = b + a\). Order does not matter for addition.
Associative
\((a + b) + c = a + (b + c)\). Grouping does not matter.
Additive Inverse
\(a + (-a) = 0\). Every number has an opposite that sums to zero.

Step-by-Step Approach

1
Rewrite subtraction as addition: Change every \(a - b\) to \(a + (-b)\).
2
Find common denominators (if fractions). Convert all terms to the same denominator before adding.
3
Apply sign rules: Same signs: add absolute values, keep sign. Different signs: subtract absolute values, take sign of the larger.
4
Simplify: Reduce fractions or convert improper fractions to mixed numbers if needed.

Example 1: Integers

Compute \(-8 + 5 - (-3)\)

1
Rewrite subtraction: \(-8 + 5 + 3\)
2
Rearrange (commutative): \(5 + 3 + (-8)\)
3
Add positives: \(5 + 3 = 8\)
4
Combine: \(8 + (-8) = 0\)

Example 2: Fractions

Compute \(\frac{3}{4} + \left(-\frac{1}{6}\right)\)

1
Find LCD: LCD of \(4\) and \(6\) is \(12\).
2
Convert: \(\frac{3}{4} = \frac{9}{12}\) and \(-\frac{1}{6} = -\frac{2}{12}\).
3
Add (different signs): \(\frac{9}{12} + \left(-\frac{2}{12}\right) = \frac{9-2}{12} = \frac{7}{12}\).

Example 3: Decimals

Compute \(-2.5 + 4.3 - 1.8\)

1
Rewrite: \(-2.5 + 4.3 + (-1.8)\)
2
Group negatives: \(-2.5 + (-1.8) = -4.3\)
3
Combine: \(-4.3 + 4.3 = 0\)

Example 4: Mixed Numbers

Compute \(3\frac{1}{2} - 5\frac{3}{4}\)

1
Convert to improper fractions: \(\frac{7}{2} - \frac{23}{4}\)
2
Rewrite: \(\frac{7}{2} + \left(-\frac{23}{4}\right)\)
3
LCD = 4: \(\frac{14}{4} + \left(-\frac{23}{4}\right) = \frac{14-23}{4} = -\frac{9}{4}\)
4
Convert back: \(-\frac{9}{4} = -2\frac{1}{4}\)

Common Mistakes to Avoid

✕ Dropping the Negative
Writing \(5 - (-3) = 5 - 3 = 2\) instead of correctly converting to \(5 + 3 = 8\). Two negatives make a positive.
✕ Wrong Sign on Result
In \(-7 + 4\), computing \(7 - 4 = 3\) but forgetting the answer should be \(-3\) because \(|-7| > |4|\).
✕ Forgetting the LCD
Adding \(\frac{1}{3} + \frac{1}{4}\) as \(\frac{2}{7}\) instead of finding the common denominator: \(\frac{4}{12} + \frac{3}{12} = \frac{7}{12}\).

Summary of Rules

Situation Rule Example
Same sign (add) Add absolute values, keep sign \(-3+(-5) = -8\)
Different signs (add) Subtract abs. values, take larger's sign \(-7+4 = -3\)
Subtract a positive \(a - b = a + (-b)\) \(6 - 9 = 6+(-9) = -3\)
Subtract a negative \(a - (-b) = a + b\) \(2-(-5) = 2+5 = 7\)
Fractions (different denom.) Find LCD, then apply sign rules \(\frac{1}{3}-\frac{1}{2}=-\frac{1}{6}\)

Question 1 of 5

What is the value of \(-6 + (-9)\)?

Question 2 of 5

Simplify: \(7 - (-4)\)

Question 3 of 5

What is \(\frac{2}{5} + \left(-\frac{3}{10}\right)\)?

Question 4 of 5

Which property justifies rewriting \((-3) + 8 + 3\) as \((-3) + 3 + 8\)?

Question 5 of 5

A diver is at \(-12.5\) meters (below sea level). She ascends \(8.3\) meters. What is her new depth?