Practice Maths

Operations with Rational Numbers

Key Terms

rational number
Any number expressible as pq where p and q are integers and q ≠ 0; includes integers, fractions, and terminating/recurring decimals.
common denominator
A shared multiple of two denominators; required before adding or subtracting fractions.
reciprocal
The multiplicative inverse of a fraction; the reciprocal of pq is qp. Used when dividing fractions.
improper fraction
A fraction where the numerator is greater than the denominator, e.g. 73. Convert mixed numbers to improper fractions before operating.

Worked Examples

Add: ⅓ + ¼ = 4⁄12 + 3⁄12 = 7⁄12

Multiply: 3⁄4 × 2⁄5 = 6⁄20 = 3⁄10

Divide: 3⁄4 ÷ 3⁄8 = 3⁄4 × 8⁄3 = 24⁄12 = 2

Mixed number: 1½ + 2⅓ = 3⁄2 + 7⁄3 = 9⁄6 + 14⁄6 = 23⁄6 = 35⁄6

What Is a Rational Number?

A rational number is any number that can be expressed as a fraction p/q, where p and q are integers and q ≠ 0. This broad category includes all integers (since 5 = 5/1), positive fractions (3/4), negative fractions (−2/7), and decimals that either terminate (0.375 = 3/8) or recur (0.333... = 1/3). Irrational numbers like √2 and π cannot be written this way — they're in a separate category. Understanding rational numbers prepares you for working with all real numbers in later years.

Adding and Subtracting Fractions

To add fractions, you need a common denominator. For 3/4 + 5/6: find the LCM of 4 and 6, which is 12. Convert: 9/12 + 10/12 = 19/12 = 1 and 7/12. For subtraction: 5/6 − 3/8. LCM of 6 and 8 is 24. Convert: 20/24 − 9/24 = 11/24. With mixed numbers, convert to improper fractions first: 2⅓ − 1¾ = 7/3 − 7/4 = 28/12 − 21/12 = 7/12.

Finding the LCM is the key step. For simple denominators, listing multiples works. For larger ones, use prime factorisation: LCM of 12 and 18 is 22 × 32 = 36. Always simplify the final answer by finding the HCF of numerator and denominator.

Key Tip: Never add or subtract numerators when denominators differ. 1/3 + 1/4 ≠ 2/7. You must convert to a common denominator first: 4/12 + 3/12 = 7/12. Think of 1/3 and 1/4 as different "sizes" of pieces — you can't count them together until they're the same size.

Multiplying and Dividing Fractions

Multiplication is the easiest operation: multiply numerators together and denominators together. 3/4 × 2/5 = 6/20 = 3/10. Cancel (simplify) before multiplying where possible to keep numbers small: 3/4 × 8/9 → cancel the 4 and 8 (divide both by 4): 3/1 × 2/9 → cancel the 3 and 9 (divide both by 3): 1/1 × 2/3 = 2/3.

Division: "multiply by the reciprocal" (flip the second fraction and multiply). 3/5 ÷ 6/7 = 3/5 × 7/6 = 21/30 = 7/10. Why does this work? Dividing by a fraction is the same as multiplying by its reciprocal because (a/b) × (b/a) = 1. For mixed numbers: convert to improper fractions first, then multiply by the reciprocal.

Key Tip: For division, flip the second fraction (the divisor), then multiply. Never flip the first fraction. A helpful reminder: "Keep Change Flip" — Keep the first fraction, Change ÷ to ×, Flip the second fraction. Then multiply normally.

Operations with Negative Fractions

Negative fractions follow the same sign rules as integers. (−3/4) × (−2/5) = +6/20 = 3/10 (negative × negative = positive). (−5/6) + 1/3: common denominator 6: −5/6 + 2/6 = −3/6 = −1/2. Keep track of signs at every step. A useful check: if both fractions are negative and you're adding them, the result should be more negative (larger magnitude). If you're adding a positive to a negative, the result's sign depends on which has the larger absolute value.

Mastery Practice

  1. Add and Subtract Fractions (Different Denominators) Fluency

    1. ½ + ⅓
    2. ¾ − ⅙
    3. 2⁄5 + 3⁄10
    4. 5⁄6 − 1⁄4
    5. 3⁄8 + 5⁄12
    6. 7⁄9 − 1⁄3
    7. 2⁄3 + 3⁄7
    8. 9⁄10 − 3⁄4
    9. 5⁄6 + 1⁄9
    10. 4⁄5 − 2⁄3

  2. Multiply Fractions (Including Mixed Numbers) Fluency

    1. ½ × ¾
    2. 2⁄3 × 3⁄5
    3. 4⁄7 × 7⁄8
    4. 2½ × ⅘
    5. 1⅓ × 1½
    6. 5⁄6 × 3⁄10
    7. 2¾ × ⅔
    8. 7⁄9 × 3⁄14

  3. Divide Fractions Fluency

    1. ¾ ÷ ½
    2. 2⁄5 ÷ 4⁄5
    3. 5⁄6 ÷ 5⁄3
    4. 1½ ÷ ¾
    5. 7⁄8 ÷ 7⁄4
    6. 2⅓ ÷ 1¾
    7. 3⁄10 ÷ 9⁄5
    8. 4 ÷ 2⁄3

  4. Mixed Operations with Fractions and Mixed Numbers Fluency

    1. ½ + ⅓ × 3⁄4
    2. 1½ − ⅔
    3. 3⁄4 × 2 + ⅛
    4. 2⅓ ÷ 7⁄9
    5. 3 − 1⅖
    6. ⅚ × ⅗ ÷ ½
    7. 2½ + 1¾ − ⅚
    8. 4⁄9 ÷ 2⁄3 + ⅙

  5. Order of Operations with Fractions Understanding

    1. (½ + ¼) × 2⁄3
    2. 3⁄4 ÷ (3⁄8 + 3⁄8)
    3. 5⁄6 − (1⁄3 × ½)
    4. 2 × (3⁄5 − 1⁄10)
    5. (4⁄5 + 2⁄5) ÷ 3⁄5
    6. ¼ + ½ × (2⁄3 ÷ ⅓)
    7. (5⁄8 − ⅜) × (1 + ¾)
    8. 7⁄10 ÷ (7⁄5 × ½)

  6. Real-World Fraction Problems Problem Solving

    1. A recipe calls for 2¼ cups of flour. Liam wants to make 1½ times the recipe. How much flour does he need?
    2. A bag of trail mix weighing 3½ kg is divided equally among 7 students. How much does each student receive?
    3. Georgia runs 3⁄4 of a kilometre on Monday, 1⅓ km on Tuesday, and 5⁄6 km on Wednesday. What is the total distance she ran over the three days?
    4. A plank of timber is 4½ metres long. How many pieces of 3⁄4 of a metre can be cut from it? How much is left over?
    5. A car travels at an average speed of 72½ km/h for 2⅔ hours. How far does it travel? (Use distance = speed × time)
    6. A water tank holds 120 litres. It is currently 5⁄8 full. After ¼ of the water is used, how many litres remain?

  7. Add and Subtract Mixed Numbers Fluency

    1. 1½ + 2¼
    2. 3⅔ − 1⅙
    3. 4⅕ + 2¾
    4. 5½ − 2⅔
    5. 3⅜ + 1½
    6. 6¼ − 3⅚
    7. 2&frac79; + 1⅓
    8. 4⅕ − 1½

  8. Rational Number Reasoning Understanding

    1. Between which two consecutive whole numbers does 17⁄5 lie? Explain.
    2. Place these numbers in ascending order: 1⅔,   7⁄4,   1.6,   11⁄6
    3. A student claims 5⁄3 ÷ 5⁄6 = 1⁄2. Show whether this is correct or not.
    4. Explain why dividing by a fraction less than 1 always gives an answer greater than the original number. Give an example.

  9. Multi-Step Fraction Problems Understanding

    1. A piece of rope is 4½ m long. It is cut into pieces of ¾ m each. How many pieces are cut? What length remains?
    2. If n × ⅔ = 10⁄9, find n.
    3. Evaluate: (1½ + ⅔) × (2 − ¾)
    4. A container holds 5¼ litres. Each cup holds ⅜ of a litre. How many full cups can be filled?

  10. Extended Real-World Problems Problem Solving

    1. A baker uses 1¾ cups of sugar for each batch of biscuits. She has 6½ cups of sugar available. How many full batches can she make, and how much sugar is left over?
    2. Three friends share a pizza. Zara eats 3⁄8 of the pizza, Marcus eats ¼, and Priya eats 5⁄16. What fraction is left over? Express your answer as both a fraction and a percentage.
    3. A factory produces 1⅔ tonnes of product per hour. How many tonnes are produced in a 7½ hour shift? Round your answer to two decimal places if necessary.
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