Operations with Integers and Fractions

Properties and order of operations

When an expression has many operations, everyone must do them in the same order:

1. Brackets
2. Orders (powers, square roots)
3. Division and Multiplication, left to right
4. Addition and Subtraction, left to right

Some countries call it BODMAS, BIDMAS, BEDMAS or PEMDAS. It is the same rule.

Useful properties

• Commutative: a + b = b + a; a × b = b × a
• Associative: (a + b) + c = a + (b + c)
• Distributive: a × (b + c) = a × b + a × c
• Identity: a + 0 = a; a × 1 = a

Square numbers and mental math

Learn the squares 1² to 15² by heart: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. Then √144 = 12 is instant.

Mental math tricks

• Split: 7 × 48 = 7 × 40 + 7 × 8 = 280 + 56 = 336 (distributive).
• Round and fix: 19 × 6 = 20 × 6 − 6 = 114.
• Double and halve: 25 × 16 = 50 × 8 = 100 × 4 = 400.
• Divide in steps: 420 ÷ 12 = 420 ÷ 6 ÷ 2 = 70 ÷ 2 = 35.

Adding, subtracting, multiplying and dividing integers

An integer is a whole number that can be positive, negative or zero.

• Add: use chips. +1 and −1 make a zero pair. (−6) + 4: four pairs cancel, −2 is left.
• Subtract: add the opposite. 3 − (−5) = 3 + 5 = 8.
• Multiply and divide: work out the size, then the sign. Same signs → positive; different signs → negative. (−7) × 3 = −21; (−20) ÷ (−4) = 5.

Adding, subtracting, multiplying and dividing fractions

• Add or subtract: find a common denominator, then add the tops. 2/3 − 1/4 = 8/12 − 3/12 = 5/12.
• Multiply: top × top over bottom × bottom. 2/3 × 3/5 = 6/15 = 2/5. You may cancel first.
• Divide: keep the first fraction, change ÷ to ×, flip the second. 5/6 ÷ 2/3 = 5/6 × 3/2 = 15/12 = 5/4.
• Change mixed numbers to improper fractions first: 1½ = 3/2.

Why flip? Dividing asks "how many of these fit?". How many 1/8 pieces fit in 3/4? 6, which is 3/4 × 8.

Proportional situations

Two quantities are proportional when their ratio stays the same. If 4 pens cost ₹60, then 1 pen costs ₹15 (the unit rate), so 7 pens cost ₹105.

Or set up a proportion: 4/60 = 7/x, so x = 60 × 7 ÷ 4 = 105.

Try it

Take a glass of juice made from 1 part syrup and 4 parts water. How much syrup for 1 litre of drink? (1/5 of 1000 mL = 200 mL.) Then scale the recipe in the 3D free play.

Key formulas and definitions

• Order: Brackets → Powers → × ÷ (left to right) → + − (left to right)
• a − b = a + (−b)
• (+)(+) = +, (−)(−) = +, (+)(−) = −
• a/b + c/d = (ad + bc)/bd
• a/b × c/d = ac/bd; a/b ÷ c/d = a/b × d/c
• Proportion: a/b = c/d ⇔ ad = bc

Worked examples

Common mistakes

• Working left to right without the order rule: 3 + 4 × 2 is 11, not 14.
• Adding denominators: 1/2 + 1/3 is not 2/5. Make a common denominator first.
• Thinking two negatives always give a positive. That is true for × and ÷, but (−3) + (−4) = −7.
• Flipping the first fraction when dividing. Keep the first; flip only the second.