Number and algebra · Lesson 2 of 4

Order of operations and mental strategies

A calculation is not a sentence. You do not read it left to right. There is a fixed order, and once you can see it, the no-calculator section becomes a set of shortcuts.

About 20 min · Interactive · 8 practice questions

Pick an expression or type your own, then press Next step. The stepper underlines the operation that must go next, explains why, and rewrites the expression one operation at a time. The left-to-right answer is shown so you can see where the common mistake lands.

Correct answer—
Left-to-right gives—
Steps done0

    Why there is an order at all

    The natural assumption is that a line of maths reads like a line of text: start at the left, do each thing as you reach it. Instead, the operations have ranks, and the higher rank goes first no matter where it sits in the line.

    Multiplication outranks addition because it is addition in bulk. 4 × 3 is a short way of writing 3 + 3 + 3 + 3. So in 8 + 4 × 3 the 4 × 3 is one compressed lump of adding, and you must unpack that lump before you can add 8 to it. Powers outrank multiplication for the same reason one level up: 3² is multiplication in bulk, 3 × 3.

    Brackets are different. They are not an operation; they are an instruction. (7 + 3) says "treat this group as one number". So whatever is inside brackets is worked out first, before anything outside can touch it.

    Division is multiplication's partner (dividing by 2 is multiplying by a half) and subtraction is addition's partner (taking 5 is adding −5). Partners share a rank. When two operations of the same rank sit side by side, you take them in the order written, left to right.

    That is the whole mechanism. People remember it with the word BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) or BIDMAS (Indices instead of Orders). The letters are a memory aid, not the rule: D before M in the word does not mean division goes before multiplication. They are equal, left to right.

    For the test, the consequence is simple. Before you calculate anything, scan the whole expression and find the highest-ranked operation. Do that one first. The figure below shows the ladder for one expression.

    Ladder diagram reducing 100 minus 2 times open bracket 7 plus 3 close bracket, one level at a time, to 80 100 − 2 × (7 + 3) 100 − 2 × 10 100 − 20 80 1. Brackets the group becomes 10 2. Multiplication 2 × 10 is bulk adding, so it goes before − 3. Subtraction only one operation left Left to right: 980
    Each rung does exactly one operation. The gold part is the operation chosen at that level; everything else waits. Reading left to right (100 − 2 = 98, then 98 × 10 = 980) gives an answer more than twelve times too big.

    The five rules

    The order is one rule, but there are four more things that trip people on test day: ties, hidden brackets, calculator entry, and checking. Each is a short recipe.

    1. The order

    1. Brackets: work out any group in brackets first, innermost first.
    2. Indices (powers, square roots): 3², √16.
    3. Multiplication and division, together, left to right.
    4. Addition and subtraction, together, left to right.

    5 + 2 × 3² = 5 + 2 × 9 = 5 + 18 = 23

    2. Ties go left to right

    1. If the next operations are × and ÷ (or + and −), do not pick the one that "looks" more important.
    2. Do the leftmost one, then the next, in reading order.

    18 ÷ 3 × 2 = 6 × 2 = 12 (not 18 ÷ 6 = 3)
    20 − 6 + 1 = 14 + 1 = 15 (not 20 − 7 = 13)

    3. Fraction bars and ÷ signs hide brackets

    1. A fraction bar groups everything above it and everything below it.
    2. When you rewrite it on one line, put brackets around the top and around the bottom.
    3. The same applies to anything under a square-root sign.

    (12 + 8) over (2 × 5) = (12 + 8) ÷ (2 × 5) = 20 ÷ 10 = 2

    4. Calculator entry

    1. A scientific calculator already follows the order of operations. Typing 8 + 4 × 3 = gives 20, not 36.
    2. So you only press the bracket keys where the question has brackets, including the hidden ones from a fraction bar.
    3. If the test calculator result disagrees with your head, the order is almost always the reason.

    For (34 + 26) ÷ 4 press ( 3 4 + 2 6 ) ÷ 4 = → 15. Without brackets: 34 + 6.5 = 40.5

    5. Check by estimation

    1. Round each number to one significant figure.
    2. Run the same order of operations on the rounded numbers in your head.
    3. If the exact answer is nowhere near the estimate, you have mis-ordered something or mistyped.

    298 − 4 × 52 ≈ 300 − 4 × 50 = 100. Exact: 298 − 208 = 90. Close, so trust it. Left to right would give 15 288: nowhere near.

    Seeing the difference with area

    A product is a rectangle: 4 × 3 is 4 columns of 3 squares. Adding 8 to that product puts 8 loose squares beside the rectangle. Putting brackets around 8 + 4 first makes a wider rectangle, 12 across and 3 high. The two pictures are not the same size, so the two expressions cannot have the same value.

    Area model comparing 8 plus 4 times 3 (eight loose squares and a 4 by 3 block, 20 squares) with open bracket 8 plus 4 close bracket times 3 (a 12 by 3 block, 36 squares) 8 + 4 × 3 (8 + 4) × 3 8 loose squares + 4 × 3 block = 12 = 8 + 12 = 20 squares 8 + 4 = 12 across 3 high = 12 × 3 = 36 squares
    Without brackets, only the 4 is multiplied by 3. With brackets, the whole 12 is. The brackets change which numbers the multiplication reaches.

    Mental strategies

    The final section of the test has no calculator. Nobody expects you to do long multiplication in your head. Instead, you change the sum into an easier sum with the same answer. Every strategy below is one such swap, and each is a fixed recipe.

    The common thread: the "hard" number is pushed to a round number (10, 100, 1000, or a multiple of 5) and the small difference is handled separately.

    Doubling and halving

    1. If one factor is even, halve it and double the other. The product does not change.
    2. Repeat until a factor hits 10, 100 or something you know.

    25 × 16 = 50 × 8 = 100 × 4 = 400
    15 × 12 = 30 × 6 = 180

    Compensation

    1. If a factor is just under 10, 100 or 1000, multiply by the round number instead.
    2. Take away the extra you added: (round − factor) lots of the other number.

    99 × 7 = 700 − 7 = 693
    6 × 998 = 6000 − 12 = 5988

    Splitting by place value

    1. Break one factor into tens and ones (and hundreds if needed).
    2. Multiply each part, then add the parts.

    23 × 4 = 20 × 4 + 3 × 4 = 80 + 12 = 92

    Multiplying by 5, 25 and 50

    1. × 5: multiply by 10, then halve.
    2. × 25: multiply by 100, then divide by 4 (halve twice).
    3. × 50: multiply by 100, then halve.

    46 × 5 = 460 ÷ 2 = 230
    24 × 25 = 2400 ÷ 4 = 600

    Dividing by 5, 25 and 50

    1. ÷ 5: double, then divide by 10.
    2. ÷ 25: multiply by 4, then divide by 100.
    3. ÷ 50: double, then divide by 100.

    135 ÷ 5 = 270 ÷ 10 = 27
    350 ÷ 25 = 1400 ÷ 100 = 14

    Round and adjust (add and subtract)

    1. If a number is within a few of a hundred or thousand, use the round number.
    2. Then undo the rounding: if you took away too much, add the difference back.

    346 − 198 = 346 − 200 + 2 = 148
    500 − 297 = 500 − 300 + 3 = 203

    Known facts

    1. Squares to 15² and small cubes are in the table below. Learn them.
    2. × 11 for a two-digit number: add the digits and put the sum between them (carry if over 9).
    3. Years: 2026 − 1987 is "13 to 2000, then 26 more" = 39.

    11 × 35 = 3 (3+5) 5 = 385
    11 × 47 = 4 (11) 7 → 517

    Percentage anchors

    1. 10% is ÷ 10. 5% is half of that. 1% is ÷ 100.
    2. Build any percentage from these: 15% = 10% + 5%; 12% = 10% + 1% + 1%.

    15% of $240: 10% = $24, 5% = $12, total $36
    GST on $85: 10% = $8.50

    Mental strategy coach

    Type a product, quotient, sum or difference

    Try 24 × 25, 99 × 7, 15 × 12, 23 × 4, 135 ÷ 5, 500 − 297 or 346 + 198. The coach picks a strategy by the rules above and shows every step, not just the answer. Whole numbers only.

      Facts to know by heart

      These are the anchors the strategies lean on. In the no-calculator section, recognising 144 as 12² or 125 as 5³ saves a minute of working. Hide a column, say each answer, then click a striped cell to check.

      FactValueShortcut or link
      Squares
      11²1211 (1+1) 1 by the × 11 trick
      12²144A gross; 12 × 12
      13²16912² + 12 + 13
      14²196Just under 200: 15² − 15 − 14
      15²225Any number ending in 5: 1 × 2 = 2, then write 25
      1² to 10²1, 4, 9, 16, 25, 36, 49, 64, 81, 100Gaps grow by 2 each time: +3, +5, +7 …
      Cubes
      2³82 × 2 × 2
      3³279 × 3
      4³6416 × 4, also 8²
      5³12525 × 5; one eighth of 1000
      Doubles and halves
      Double 25 / half of 2550 / 12.5Quarter of 100
      Double 50 / half of 50100 / 25Half of 100
      Double 75 / half of 75150 / 37.5Three quarters of 100
      Double 125 / half of 125250 / 62.5Eighth of 1000
      Shortcuts
      × 5× 10, then halve36 × 5 = 360 ÷ 2 = 180
      × 25× 100, then ÷ 432 × 25 = 3200 ÷ 4 = 800
      × 50× 100, then halve18 × 50 = 1800 ÷ 2 = 900
      ÷ 5× 2, then ÷ 10215 ÷ 5 = 430 ÷ 10 = 43
      ÷ 25× 4, then ÷ 100450 ÷ 25 = 1800 ÷ 100 = 18

      Beat the calculator

      Ten no-calculator items, half order-of-operations and half mental arithmetic. The clock shows elapsed seconds only so you can see your own pace; there is no limit. After each answer the drill names the strategy that gets there fastest.

      Item 0 of 10 Elapsed 0 s Score 0
      Press Start to begin.

      Common traps

      • Reading left to right. 8 + 4 × 3 is 20, not 36. Scan the whole line for the highest-ranked operation before touching anything.
      • Ranking − below + or ÷ below ×. The letters of BODMAS suggest an order inside each pair. There is none. 18 ÷ 3 × 2 = 6 × 2 = 12, not 18 ÷ 6 = 3. 20 − 6 + 1 = 15, not 13.
      • Squaring the product. 2 × 3² = 2 × 9 = 18. The power belongs to the 3 only. (2 × 3)² = 36 needs the brackets.
      • A minus in front of brackets. 50 − (12 + 8) means take away the whole 20, so 30. Working it as 50 − 12 + 8 = 46 is wrong because the brackets were dropped.
      • Typing a fraction without brackets. (12 + 8)/(2 × 5) is 2. Typing 12 + 8 ÷ 2 × 5 gives 32, because the calculator correctly does 8 ÷ 2 × 5 first. Bracket the top and the bottom.
      • "Of" means multiply. "A quarter of 3 packs of 12" is 1/4 × 3 × 12 = 9. Treat "of" as × and it slots into the normal order.
      • Skipping the estimate. A mis-ordered calculation is usually wildly wrong (980 instead of 80). A one-second rounded estimate catches it; the exact arithmetic never will.

      ACER-style practice

      Eight questions in the formats the real test uses: multiple choice, type your answer, and true-or-false tables. Questions marked "no calculator" belong to the test's final section, where you cannot use one. Check each answer for a full explanation.

      0 of 8 checked

      What to remember

      • Order: brackets → indices → × and ÷ → + and −. Scan the whole line before you start.
      • × and ÷ are equal rank; + and − are equal rank. Ties go left to right: 18 ÷ 3 × 2 = 12.
      • A fraction bar is a pair of hidden brackets: (top) ÷ (bottom) on a calculator.
      • The power sticks to the number next to it: 2 × 3² = 18.
      • × 5 is × 10 then halve; × 25 is × 100 then ÷ 4; ÷ 5 is × 2 then ÷ 10.
      • Near a round number, compensate: 99 × 7 = 700 − 7; 346 − 198 = 346 − 200 + 2.
      • Estimate with rounded numbers to catch an order mistake before you commit.

      All questions on this page are original and are not reproduced from ACER material. LANTITE is administered by the Australian Council for Educational Research.