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Solve Problems With
Rational Numbers

Master the four operations with fractions, decimals, and negative numbers to tackle real-world problems with confidence.

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
\(-5,\; 0,\; 12\) can all be written as fractions: \(\frac{-5}{1},\; \frac{0}{1},\; \frac{12}{1}\)
Fractions
\(\frac{3}{4},\; -\frac{7}{2},\; \frac{11}{5}\) are already in \(\frac{a}{b}\) form
Decimals
\(0.75 = \frac{3}{4}\) and \(0.\overline{3} = \frac{1}{3}\) are rational

Rational Numbers on the Number Line

-3 -2 -1 0 1 2 3 -1.5 0.25 2.5
\( = -\frac{3}{2} \)
\( = \frac{1}{4} \)
\( = \frac{5}{2} \)

Why Rational Number Operations Matter

You encounter rational numbers everywhere: temperatures drop below zero, bank accounts gain and lose fractional amounts, and recipes scale with mixed numbers. Being fluent with all four operations lets you solve any of these real-world problems.

Finance
A bank balance of \(\$125.50\) after a \(-\$47.75\) withdrawal
Cooking
Tripling a recipe that calls for \(\frac{2}{3}\) cup of flour
Science
Temperature change from \(-4.5^\circ\)C to \(2.8^\circ\)C

The Four Operations with Rational Numbers

1
Addition: Same sign: add absolute values, keep the sign. Different signs: subtract absolute values, keep the sign of the larger.
2
Subtraction: Rewrite as addition of the opposite: \(a - b = a + (-b)\), then follow the addition rules.
3
Multiplication: Multiply absolute values. Same signs \(\rightarrow\) positive. Different signs \(\rightarrow\) negative.
4
Division: Same sign rules as multiplication. Rewrite as multiplying by the reciprocal: \(a \div b = a \times \frac{1}{b}\).

Sign Rules for Multiplication and Division

Positive Result
\( (+) \times (+) = (+) \)
\( (-) \times (-) = (+) \)
 Negative Result
\( (+) \times (-) = (-) \)
\( (-) \times (+) = (-) \)
 Quick Rule
\( \text{Same signs} \rightarrow \text{Positive} \)
\( \text{Different signs} \rightarrow \text{Negative} \)

Fractions vs. Decimals: When to Use Which

Fractions Decimals
Best for Exact answers, ratios Estimation, money
Addition/Subtraction Need common denominator Line up decimal points
Multiplication Multiply across: \(\frac{a}{b} \times \frac{c}{d}\) Multiply, count decimal places
Division Multiply by reciprocal Move decimal, then divide
Example \(\frac{1}{3} = 0.\overline{3}\) (exact) \(0.333...\) (approximate)

How to Solve Any Rational Number Problem

1
Read carefully. Identify which operation (or operations) the problem requires.
2
Convert if needed. Turn mixed numbers to improper fractions, or keep everything in decimals.
3
Determine the sign. Use sign rules before computing the absolute values.
4
Compute. Perform the operation with the absolute values (find common denominators, multiply across, etc.).
5
Attach the sign and simplify. Reduce fractions or round decimals as the problem requires.

Example 1: Adding Rational Numbers

Problem: The temperature at noon was \(3.7^\circ\)C. By midnight it dropped \(8.2^\circ\)C. What is the midnight temperature?

1
Set up the expression: \(3.7 + (-8.2)\)
2
Different signs: Subtract the smaller absolute value from the larger: \(8.2 - 3.7 = 4.5\)
3
Keep the sign of the larger absolute value: \(|-8.2| > |3.7|\), so the result is negative.

Example 2: Multiplying Fractions

Problem: A recipe calls for \(\frac{2}{3}\) cup of sugar. You want to make \(1\frac{1}{2}\) times the recipe. How much sugar do you need?

1
Convert: \(1\frac{1}{2} = \frac{3}{2}\)
2
Multiply across: \(\frac{2}{3} \times \frac{3}{2} = \frac{2 \times 3}{3 \times 2} = \frac{6}{6}\)
3
Simplify: \(\frac{6}{6} = 1\)

Example 3: Multi-Step Problem

Problem: A scuba diver descends \(12.5\) meters, then ascends \(4.8\) meters, then descends another \(6.3\) meters. What is the diver's final depth?

0 m -12.5 +4.8 -6.3 -14 m Start Final

Dividing Fractions: Keep, Change, Flip

Keep the first fraction
\( -\frac{3}{4} \)
 → Change division to multiplication
\( \div \rightarrow \times \)
 → Flip the second fraction
\( \frac{2}{5} \rightarrow \frac{5}{2} \)

Order of Operations Still Applies

When a problem has multiple operations, follow PEMDAS even with negative and fractional numbers. The sign rules are applied at each step.

P
Parentheses first
E
Exponents (including negative bases)
MD
Multiplication and Division left to right
AS
Addition and Subtraction left to right

Example 4: Complex Expression

Evaluate: \(-\frac{1}{2} + \frac{3}{4} \times \left(-\frac{2}{3}\right)\)

1
Multiplication first: \(\frac{3}{4} \times \left(-\frac{2}{3}\right) = -\frac{6}{12} = -\frac{1}{2}\)
2
Then addition: \(-\frac{1}{2} + \left(-\frac{1}{2}\right) = -\frac{1}{2} - \frac{1}{2}\)
3
Same sign, add: \(-\frac{2}{2} = -1\)

Common Mistakes to Avoid

✕ Forgetting Sign Rules
Saying \((-4) \times (-3) = -12\) instead of \(+12\). Two negatives multiply to a positive.
✕ Subtracting Wrong Way
Computing \(3 - 7\) as \(4\) instead of \(-4\). Always subtract from the larger absolute value.
✕ Forgetting the Reciprocal
Dividing fractions by multiplying straight across instead of flipping the second fraction first.
✕ Skipping PEMDAS
Adding before multiplying in expressions like \(2 + 3 \times (-4)\), getting \(-20\) instead of \(-10\).

Operations Summary

Operation Fractions Decimals Sign
Add Common denominator, add numerators Align decimal points See add rules
Subtract Add the opposite Add the opposite See add rules
Multiply \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\) Multiply, count places Same → +, Diff → −
Divide \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\) Multiply by reciprocal Same → +, Diff → −

Question 1 of 5

What is \(-\frac{3}{5} + \frac{7}{10}\)?

Question 2 of 5

A stock loses \(\$2.75\) per day for \(4\) days, then gains \(\$5.50\). What is the total change?

Question 3 of 5

What is \(-\frac{2}{3} \div \left(-\frac{4}{9}\right)\)?

Question 4 of 5

Evaluate: \((-2)^3 + 5 \times (-1.5)\)

Question 5 of 5

A submarine is at \(-120\) feet. It rises \(\frac{2}{5}\) of that distance. What is its new depth?