 The order of operations is the rule that tells us the sequence in which we should simplify an expression with multiple operations. Consider the following expression: 3 + 5 × 20 - 2 There are three operations. Which part do you calculate first? Order of Operations Steps:
Using these rules tells us how to evaluate the example above: 3 + 5 × 20 - 2 = 3 + 5 × 20 - 2 No brackets or exponents, so do the multiplication first = 3 + 100 - 2 = 3 + 100 - 2 End with the addition/subtraction rule = 101 Note that there is no rule which says addition comes before subtraction, or multiplication before division. So 6 - 2 + 5 = 6 - 2 + 5 = 4 + 5 = 9 and 18 ÷ 6 x 2 = 18 ÷ 6 x 2 = 3 x 2 = 6 More examples: In each step we'll underline what to do first: Example 1: 2 - 6 × (1 + 5) ÷ 3 + 4 = 2 - 6 × (1 + 5) ÷ 3 + 4 Brackets first = 2 - 6 × 6 ÷ 3 + 4 = 2 - 6 × 6 ÷ 3 + 4 then multiplication and division in order left to right = 2 - 12 + 4 = 2 - 12 + 4 and addition and subtraction left to right = - 6 Example 2: 3 - 15 ÷ (8 - 3) × 2 + 7 = 3 - 15 ÷ (8 - 3) × 2 + 7 = 3 - 15 ÷ 5 × 2 + 7 = 3 - 15 ÷ 5 × 2 + 7 multiplication and division in order left to right = 3 - 6 + 7 = 4 Example 3: 80 ÷ (6 + 7 × 2) - 1 = 80 ÷ (6 + 7 × 2) - 1 Brackets first, but ... = 80 ÷ (6 + 7 × 2) - 1 ... before the bracket, we must do the multiplication inside = 80 ÷ (6 + 14) - 1 Now we can do the bracket = 80 ÷ 20 - 1 = 80 ÷ 20 - 1 = 4 - 1 = 3 Example 4: 6 + (5 × 23 + 2) = 6 + (5 × 23 + 2) Again, the bracket is first, but ... = 6 + (5 × 23 + 2) before that we have to do the exponent = 6 + (5 × 8 + 2) and then do the multiplication = 6 + (40 + 2) now we can do the bracket = 6 + 42 = 48 Example 5: 9 - 32 ÷ 8 × 2 + 5 = 9 - 32 ÷ 8 × 2 + 5 do multiplication and division first, in order left to right = 9 - 8 + 5 = 6 Example 6: [(32 ÷ 8) + 3] × 2 = [(32 ÷ 8) + 3] × 2 Do the inner bracket first = [4 + 3] × 2 Now do the outer bracket = 7 x 2 = 14 Example 7: This type is frequently done wrong 6 ÷ 2(1 + 2) = 6 ÷ 2(1 + 2) Do the bracket first = 6 ÷ 2(3) there is a times sign understood to be between a number and a bracket = 6 ÷ 2 x 3 This is what the line really means = 6 ÷ 2 x 3 Do the line left to right in order = 9 Because the third line is often written with the bracket still there: 6 ÷ 2(3), some people mistakenly assume that that bracket must be done next. This is wrong. The convention is that when what was inside a bracket has been reduced to a single number, the bracket no longer needs to be written, so it's really 6 ÷ 2 x 3, which is done left to right. |