Fractions: Recurring Decimals, Mixed Numbers and Order of Operations

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This page has four parts: (1) fractions that turn into recurring decimals, (2) subtracting mixed numbers, (3) multiplying a whole number by a mixed number and dividing a whole number by a fraction, and (4) using the laws of arithmetic to make fraction and decimal calculations easier. All of this can appear on the non-calculator paper, so the methods are written out by hand.

Key words

1. Fractions and recurring decimals

A fraction is a division: 3/8 means 3 ÷ 8. Doing the division gives the decimal. Sometimes the division stops (a terminating decimal); sometimes it never stops because the same remainders come round again and again (a recurring decimal).

Dot notation. Instead of writing "…", mathematicians put a dot over the digit that repeats. If a block of digits repeats, put a dot over the first and last digit of the block.

Learn these. You should recognise them on sight:

Calculators round the last digit. A calculator may show 2/3 as 0.6666666667. That 7 is only the calculator rounding — 2/3 = 0.666… recurring. Do not say "it terminates" just because the screen is full.

2. Subtracting mixed numbers

Always estimate first. Round each mixed number to the nearest whole number (or half). For 5 1/4 − 2 2/3: about 5 − 3 = 2, and a bit more because 2 2/3 was rounded up. So expect a little over 2.

Method A — change to improper fractions (safe and always works):

1. Write each mixed number as an improper fraction: 5 1/4 = 21/4 and 2 2/3 = 8/3.
2. Use a common denominator (the LCM of 4 and 3 is 12): 21/4 = 63/12 and 8/3 = 32/12.
3. Subtract the numerators: 63/12 − 32/12 = 31/12.
4. Change back to a mixed number and simplify: 31 ÷ 12 = 2 remainder 7, so 2 7/12. (7 and 12 have no common factor.) This matches the estimate.

Method B — wholes and fractions separately (quicker with small numbers, but watch the fractions):

1. Common denominator: 5 1/4 − 2 2/3 = 5 3/12 − 2 8/12.
2. 3/12 is smaller than 8/12, so you cannot take 8/12 away. Exchange one whole for 12/12: 5 3/12 = 4 15/12.
3. Wholes: 4 − 2 = 2. Fractions: 15/12 − 8/12 = 7/12.
4. Answer: 2 7/12.

Method C — count up on a number line (good for checking):

Simplest form. The answer must be a mixed number with the fraction part in its simplest form: 3 6/8 should be written 3 3/4, and 2 15/12 is not finished — it is 3 3/12 = 3 1/4.

3. Multiplying a whole number by a mixed number

Estimate: 4 × 2 3/5 is between 4 × 2 = 8 and 4 × 3 = 12. As 2 3/5 is a bit more than 2 1/2, the answer is a bit more than 4 × 2 1/2 = 10.

Method 1 — split the mixed number (this is the distributive law):

Method 2 — improper fraction: 2 3/5 = 13/5, so 4 × 13/5 = 52/5 = 52 ÷ 5 = 10 remainder 2 = 10 2/5. Both methods agree, and the answer is close to the estimate 10.

4. Dividing a whole number by a proper fraction

6 ÷ 3/4 asks: how many 3/4 fit into 6? Draw it: 6 wholes are 24 quarters, and 24 quarters make 8 groups of 3 quarters.

The quick method: multiply by the reciprocal. Dividing by 3/4 is the same as multiplying by 4/3 ("keep, change, flip": keep the 6, change ÷ to ×, flip 3/4 to 4/3).

1. 6 ÷ 3/4 = 6 × 4/3
2. = 24/3
3. = 8 ✓ (the same as the diagram)

Sense check: dividing by a fraction less than 1 gives an answer bigger than the number you started with. 3/4 fits into 1 more than once, so it fits into 6 more than 6 times.

5. Laws of arithmetic and order of operations with decimals and fractions

The order of operations is the same as for whole numbers: Brackets, Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right). The laws of arithmetic let you reorder or regroup to make a calculation easy without a calculator.

Brackets change the answer:

• 1/2 + 1/4 × 2/3 = 1/2 + 2/12 = 1/2 + 1/6 = 3/6 + 1/6 = 4/6 = 2/3 (multiply first)
• (1/2 + 1/4) × 2/3 = 3/4 × 2/3 = 6/12 = 1/2 (bracket first)
• 4.8 − 1.2 × 3 = 4.8 − 3.6 = 1.2 (not 3.6 × 3)
• (2.4 + 3.6) ÷ 0.1 = 6 ÷ 0.1 = 60

Subtraction and division are not commutative: 5 − 2 1/2 ≠ 2 1/2 − 5, and 6 ÷ 3/4 ≠ 3/4 ÷ 6. Only swap the order for + and ×.

Worked examples

Worked example 1 — (a) Write 7/9 as a decimal. (b) Draw a ring around the fraction that is equal to a recurring decimal: 2/5 5/6 7/20 3/8 [2 marks]

• (a) Ninths: the numerator repeats, so 7/9 = 0.777… (written 0.7 with a dot over the 7). Check: 7 ÷ 9 = 0 r 7, 70 ÷ 9 = 7 r 7, 70 ÷ 9 = 7 r 7 … the remainder 7 keeps coming back.
• (b) 2/5 = 0.4, 7/20 = 0.35, 3/8 = 0.375 all stop. 5/6 = 0.8333… so ring 5/6.

Worked example 2 — Work out 4 3/8 − 1 5/6. Give your answer as a mixed number in its simplest form. [2 marks]

1. Estimate: about 4 1/2 − 2 = 2 1/2.
2. Improper fractions: 4 3/8 = 35/8 and 1 5/6 = 11/6.
3. Common denominator 24: 35/8 = 105/24 and 11/6 = 44/24. (1 mark for a correct common denominator.)
4. 105/24 − 44/24 = 61/24 = 2 remainder 13 → 2 13/24. (13 is prime and does not divide 24, so this is the simplest form.) Close to the estimate ✓

Common mistake: subtracting the fractions the wrong way round: 4 − 1 = 3 and 5/6 − 3/8 = 11/24, giving 3 11/24. You cannot take 20/24 from 9/24 without exchanging a whole.

Worked example 3 — Work out 3 × 4 5/6. Give your answer as a mixed number in its simplest form. [2 marks]

Estimate: 3 × 5 = 15, a bit less. Split: 3 × 4 = 12 and 3 × 5/6 = 15/6 = 2 3/6 = 2 1/2. Add: 12 + 2 1/2 = 14 1/2 ✓. Common mistake: multiplying only the whole number (3 × 4 5/6 = 12 5/6) — the fraction part must be multiplied too.

Worked example 4 — A ribbon is 9 m long. It is cut into pieces that are each 3/5 m long. How many pieces are there? [2 marks]

1. "How many 3/5 fit into 9?" → 9 ÷ 3/5.
2. 9 × 5/3 = 45/3 = 15 pieces.
3. Check: 15 × 3/5 = 45/5 = 9 ✓. The answer is more than 9 because each piece is shorter than 1 m.

Worked example 5 — Without a calculator, work out 4.6 × 0.7 + 5.4 × 0.7. Show how you used a law of arithmetic. [2 marks]

Both terms have the factor 0.7, so use the distributive law: (4.6 + 5.4) × 0.7 = 10 × 0.7 = 7. Much quicker than working out 3.22 + 3.78.

Worked example 6 — Work out 2/3 + 1/3 × (5/2 − 1). [2 marks]

1. Brackets: 5/2 − 1 = 5/2 − 2/2 = 3/2.
2. Multiply: 1/3 × 3/2 = 3/6 = 1/2.
3. Add: 2/3 + 1/2 = 4/6 + 3/6 = 7/6 = 1 1/6.

Common mistake: adding first (2/3 + 1/3 = 1, then 1 × 3/2 = 3/2). Multiplication comes before addition.

Practice

All pages in this course

Start here

• Stage 8 Maths: The Tests and How to Prepare

Number

• Order of Operations and Calculating with Integers
• Factors, Multiples, Primes, HCF and LCM
• Types of Number and Index Laws
• Squares, Cubes and Roots (Positive and Negative)
• Multiplying and Dividing by 0.1 and 0.01, and Significant Figures
• Fractions: Recurring Decimals, Mixed Numbers and Order of Operations
• Percentage Increase, Decrease and Absolute Change
• Ordering and Calculating with Decimals
• Ratio and Proportion

Algebra

• Algebraic Expressions: Simplifying, Expanding and Factorising
• Writing Expressions and Formulae, and Changing the Subject
• Equations and Inequalities
• Sequences and the nth Term
• Functions and Straight-line Graphs (y = mx + c)
• Reading and Interpreting Real-life Graphs

Geometry and measure

• Quadrilaterals and Regular Polygons
• Circumference of a Circle, and Miles and Kilometres
• Area of Parallelograms and Trapezia
• 3D Shapes: Euler's Formula and Plans and Elevations
• Volume of Prisms and Surface Area
• Angles in Triangles and on Parallel Lines
• Constructions with Ruler and Compasses
• Bearings and Midpoints
• Translation, Reflection, Rotation and Enlargement

Statistics and probability

• Data Collection and Sampling
• Representing Data
• Averages, Range and Comparing Distributions
• Probability: Complementary and Combined Events

Practice tests

• Stage 8 Maths — Practice Test 1
• Stage 8 Maths — Practice Test 2
• Stage 8 Maths — Practice Test 3