Rational Numbers and Operations
Rational numbers include integers, fractions and terminating or repeating decimals. They can be positive, negative or zero. Learn how to recognise, compare and calculate with rational numbers using reliable methods and clear worked examples.
Learning time: approximately 35–40 minutes · Developing level · Includes practice, quiz and printable worksheet
What You Will Learn
- Recognise rational numbers in fraction, decimal and integer form.
- Understand positive and negative rational values.
- Compare and order rational numbers.
- Add and subtract rational numbers.
- Multiply and divide rational numbers.
- Use operation rules to solve everyday problems and check answers.
1. What Is a Rational Number?
A rational number is any number that can be written as a fraction a/b, where a and b are integers and b is not zero.
All of these are rational numbers because each can be written as a fraction of integers.
- 3/4 is already a fraction.
- −2 can be written as −2/1.
- 0.6 = 6/10 = 3/5.
- 0 = 0/1.
- 0.333... = 1/3, so a repeating decimal is rational.
A decimal that terminates or repeats can be written as a fraction. A non-terminating, non-repeating decimal is not rational.
2. Positive, Negative and Zero
Rational numbers can appear on either side of zero on a number line.
- Positive: greater than zero, such as 1/2 or 2.4.
- Negative: less than zero, such as −3/4 or −1.2.
- Zero: neither positive nor negative.
Opposites have the same distance from zero but different signs. For example, 3/5 and −3/5 are opposites.
3. Compare and Order Rational Numbers
To compare rational numbers, it can help to write them as fractions with a common denominator or convert them to decimals. On a number line, the number farther right is greater.
Use a common denominator of 12:
- 2/3 = 8/12
- 3/4 = 9/12
Since 8/12 is less than 9/12, 2/3 < 3/4.
Convert to decimals: −1/2 = −0.5 and −3/4 = −0.75. Since −0.5 is farther right on the number line, −1/2 > −3/4.
4. Add Rational Numbers
When adding fractions, use a common denominator. With signed numbers, remember that adding a negative is like moving left on the number line.
Same sign
Different signs
Unlike denominators
- Use 12 as the common denominator.
- 1/3 = 4/12 and 1/4 = 3/12.
- 4/12 + 3/12 = 7/12.
5. Subtract Rational Numbers
Subtracting a number is the same as adding its opposite. Keep the first number, change subtraction to addition, and change the sign of the second number.
6. Multiply Rational Numbers
To multiply fractions, multiply the numerators together and the denominators together. You can simplify before or after multiplying.
For signed numbers:
- Positive × positive = positive.
- Negative × negative = positive.
- Positive × negative = negative.
- Negative × positive = negative.
7. Divide Rational Numbers
To divide by a fraction, multiply by its reciprocal. The reciprocal swaps the numerator and denominator. A number cannot be divided by zero.
- Keep the first fraction: 2/3.
- Change division to multiplication and flip the second fraction: 5/4.
- Multiply: 2/3 × 5/4 = 10/12 = 5/6.
8. Use the Order of Operations
When an expression has more than one operation, use the agreed order: parentheses first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right.
- Multiply first: 2 × 1/4 = 2/4 = 1/2.
- Add: 1/2 + 1/2 = 1.
Do not automatically work from left to right when operations have different priorities.
9. Rational Numbers in Everyday Situations
Temperature change
The temperature is −2.5°C and rises by 4°C. Calculate −2.5 + 4 = 1.5°C.
Sharing a quantity
You have 3/4 of a pizza and share it equally among 3 people. Calculate 3/4 ÷ 3 = 3/4 × 1/3 = 1/4 of a pizza each.
Distance below sea level
A diver is at −2.4 metres and descends another 1.3 metres. The new position is −2.4 + (−1.3) = −3.7 metres.
10. Common Mistakes to Avoid
11. Practice Exercises
Try these questions independently before opening the worked answers.
- Is −7/8 a rational number? Explain briefly.
- Which is greater: −2/3 or −3/4?
- Calculate 2/5 + 1/5.
- Calculate 3/4 − 1/2.
- Calculate (−2/3) × (3/5).
- Calculate 3/4 ÷ 1/2.
- Evaluate 1/3 + 2 × 1/3 using the order of operations.
- A temperature of −1.5°C rises by 3°C. What is the new temperature?
Show practice answers and working
- Yes. It is already a fraction of two integers with a non-zero denominator.
- −2/3 is greater. As decimals, approximately −0.667 is greater than −0.75.
- 2/5 + 1/5 = 3/5.
- 3/4 − 2/4 = 1/4.
- (−2/3) × (3/5) = −6/15 = −2/5.
- 3/4 × 2/1 = 6/4 = 3/2.
- Multiply first: 2 × 1/3 = 2/3. Then 1/3 + 2/3 = 1.
- −1.5 + 3 = 1.5°C.
12. Interactive Knowledge Check
Choose one answer for each question, then select Check Answers. Each question is worth one point.
13. Free Printable Worksheet
Print this worksheet or save it as a PDF. Show your working and simplify fraction answers where possible.
Learnora — Grade 7 Mathematics
Rational Numbers and Operations Worksheet
Name: __________________________ Date: _______________
- Write 0.75 as a fraction in simplest form.
- Which is greater: −2/5 or −3/5?
- Calculate 3/8 + 1/8.
- Calculate 5/6 − 1/3.
- Calculate (−3/5) × (5/6).
- Calculate 4/5 ÷ 2/3.
- Evaluate 1/2 + 3 × 1/6 using the order of operations.
- A temperature of −4°C rises by 6.5°C. Find the new temperature.
Keep practising — every step builds confidence!
Learnora — Learn Better. Practice Smarter.
Teacher / Parent Answer Key
- 0.75 = 75/100 = 3/4.
- −2/5 is greater because it is closer to zero.
- 3/8 + 1/8 = 4/8 = 1/2.
- 5/6 − 2/6 = 3/6 = 1/2.
- (−3/5) × (5/6) = −15/30 = −1/2.
- 4/5 × 3/2 = 12/10 = 6/5.
- Multiply first: 3 × 1/6 = 1/2; then 1/2 + 1/2 = 1.
- −4 + 6.5 = 2.5°C.
14. Lesson Summary
- A rational number can be written as a fraction of integers with a non-zero denominator.
- Rational numbers may be positive, negative or zero.
- Use common denominators to add or subtract fractions.
- Multiply numerators and denominators to multiply fractions.
- Divide by a fraction by multiplying by its reciprocal.
- Apply sign rules and the order of operations carefully.
Educational note: This is a curriculum-neutral Grade 7 learning resource. Topic order and terminology may vary between schools and countries.