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Arithematics

Arithematics

Order of operations, negative numbers and ordering

Bodmas Core + Extended 8 min read

On this page
  1. In this note
  2. Formula bank
  3. The order of operations
  4. Negative numbers
  5. Ordering numbers
  6. Ways they test it
  7. Mistakes to avoid
  8. Summary
  9. Practice
  10. Can I do this?
  11. Quiz

In this note

Syllabus 1.5 ordering, 1.6 the four operations
  • 1.6Use the order of operations and brackets, including putting brackets in to make a statement true.
  • 1.6Calculate with negative numbers (e.g. temperature changes).
  • 1.5Order fractions, decimals and percentages and use \(=, \ne, >, <, \ge, \le\).

Next: fractions. Converting forms: percentages, fractions and decimals.

Formula bank

Order of operationsLearn itBrackets → Indices → ÷ × → + −÷ and × have equal rank (left to right); so do + and −.Use it when any calculation with more than one operation.
Fraction lineLearn ita fraction line is a bracket round the top and the bottom\(\dfrac{12+8}{7-3}=(12+8)\div(7-3)\).Use it when typing fractions into a calculator.
Negatives: + and −Learn it+ (−a) = − a; − (−a) = + aUse it when adding or subtracting negatives.
Negatives: × and ÷Learn itsame signs → +; different signs → −\((-3)(-4)=12\); \(-20\div4=-5\).Use it when multiplying or dividing negatives.
Powers of negativesLearn it\((-3)^{2}=9\) but \(-3^{2}=-9\)the bracket decides what is squared.Use it when a negative with a power.

The order of operations

\(6+4\times2\) could be 20 or 14. Everyone (and every calculator) uses ONE agreed order so there is only one answer: 14.

General methodHow to work out any calculation

  1. Brackets first (a fraction line counts as brackets).
  2. Then powers and roots.
  3. Then × and ÷, working left to right.
  4. Then + and −, working left to right.

Example Work out \(20-12\div4\times2\).

÷ and × left to right, then subtract.

  1. \(12\div4=3\)
  2. \(3\times2=6\)
  3. \(20-6=14\)

Answer 14

Try it yourself

1
\(24\div4\times2\)
Show answer

Answer 12

2
\(5+3\times(6-2)^{2}\)
Show answer

Answer 53

3
\(\dfrac{15-3}{2\times3}\)
Show answer

Answer 2

General methodHow to put in one pair of brackets to make a statement true

  1. Work it out with no brackets: is the answer too big or too small?
  2. Try brackets round each pair of neighbouring numbers that would change the order.
  3. Check the new value equals the target.

Example Insert one pair of brackets: \(2+3\times4-1=19\).

Without brackets it is 13; make the + happen first.

  1. \((2+3)\times4-1=20-1=19\)

Answer \((2+3)\times4-1=19\)

Try it yourself

1
Insert one pair of brackets: \(8-6-2+3=7\).
Hint
A bracket after a minus changes the sign of what's inside.
Show answer

Answer \(8-(6-2)+3=7\)

Negative numbers

Number line-10-9-8-7-6-5-4-3-2-101234−2 − 8 = −10
Start at −2, move 8 left: −10.

General methodHow to calculate with negative numbers

  1. + and −: picture a number line. Adding moves right, subtracting moves left. Subtracting a negative is adding: 7 − (−2) = 9.
  2. × and ÷: do the numbers, then the sign: same signs → positive, different signs → negative.
  3. A rise or fall between two temperatures: bigger − smaller (5 − (−7) = 12).

Example Work out \(5-(-3)\times2\).

× first, then subtract a negative.

  1. \((-3)\times2=-6\)
  2. \(5-(-6)=5+6\)

Answer 11

Try it yourself

1
\(-5+12\)
Show answer

Answer 7

2
\((-5)\times(-2)-3\)
Show answer

Answer 7

3
The temperature rises from −7 °C to 5 °C. By how much?
Show answer

Answer 12 °C

Ordering numbers

General methodHow to order fractions, decimals and percentages

  1. Change every number to a decimal, to 3 or 4 decimal places.
  2. Compare digit by digit.
  3. Write the ORIGINAL numbers in the order asked.

With negatives, the number further from 0 is smaller: −6 < −4.

Example Order, smallest first: \(0.7,\ \tfrac23,\ 68\%,\ \tfrac{13}{20},\ 0.705\).

All to 3 decimal places, then back to the given forms.

  1. \(0.700,\ 0.667,\ 0.680,\ 0.650,\ 0.705\)

Answer \(\tfrac{13}{20},\ \tfrac23,\ 68\%,\ 0.7,\ 0.705\)

SymbolMeaning
\(\ne\)is not equal to
\(>\)greater than
\(<\)less than
\(\ge\)greater than or equal to
\(\le\)less than or equal to

Try it yourself

1
Smallest of 0.6, 5/8, 61%, 0.59?
Show answer

Answer 0.59

Ways they test it

Type 1 Work it out

"Work out \(20-12\div(8-6)\)." (1 mark)

Spot it several operations.

  1. Brackets, powers, × ÷, + − in order.

Type 2 Insert brackets

"Insert one pair of brackets: \(12-4\times2+1=17\)."

Spot it "insert one pair of brackets".

  1. Work out without brackets.
  2. Try each position; check.

For example \(12-4\times2+1=17\)

  1. \((12-4)\times2+1=17\)

Answer \((12-4)\times2+1\)

Type 3 Negatives in context

"The temperature is 3 °C and falls by 11 °C."

Spot it temperatures, bank balances, floors.

  1. Write it as a calculation with signs.
  2. Use the number line.

Type 4 Powers of negatives

"Work out \((-2)^{3}-(-3)^{2}\times2\)." (2 marks)

Spot it a negative in brackets with a power.

  1. Odd power of a negative → negative; even → positive.
  2. Then the order of operations.

For example \((-2)^{3}-(-3)^{2}\times2\)

  1. \(-8-9\times2=-8-18\)

Answer −26

Type 5 Order a mixed list

"Write \(\tfrac34\), 0.74, 77%, \(0.\dot7\), 0.758 in order." (2 marks)

Spot it mixed forms.

  1. All to decimals (\(0.\dot7=0.777\ldots\)).
  2. Answer in the given forms.

Mistakes to avoid

  • ✗ \(6+4\times2=20\)✓ 14× before +.
  • ✗ \(20-12\div4\times2=18.5\)✓ 14÷ and × go left to right.
  • ✗ \(-3^{2}=9\)✓ −9The power is done before the minus sign.
  • ✗ \(7-(-2)=5\)✓ 9Subtracting a negative adds.
  • ✗ Ordered list written as decimals✓ the given formsThe answer must match the question.

Summary

  • Brackets → Indices → ÷ × (left to right) → + − (left to right).
  • A fraction line is a bracket.
  • \((-3)^{2}=9\) but \(-3^{2}=-9\).
  • Same signs × or ÷ → positive; − (negative) = +.
  • Order mixed numbers by changing all to decimals.

If you see …

  • ÷ and × togetherleft to right.
  • a minus before a bracketit changes every sign inside.
  • mixed fractions, decimals, %decimals first.

Practice

Practice: easy, medium, hard

Easy

1
\(10-2\times3\)
Show working
  1. \(2\times3=6\)
  2. \(10-6\)

Answer 4

2
\(4+6\div2-1\)
Show working
  1. \(6\div2=3\)
  2. \(4+3-1\)

Answer 6

3
The temperature is 3 °C. It falls by 11 °C. New temperature?
Show working
  1. \(3-11=-8\)

Answer −8 °C

Medium

1
\(2\times(3+4)^{2}\)
Guide me

What is it asking? the value

What do we know? bracket, power, ×

Plan 7² = 49, then × 2

Show working
  1. \(7^{2}=49\)
  2. \(2\times49\)

Answer 98

2
\(9-(4-7)\)
Guide me

What is it asking? the value

What do we know? a bracket with a negative result

Plan 4 − 7 = −3, then 9 − (−3)

Show working
  1. \(4-7=-3\)
  2. \(9+3\)

Answer 12

3
Write in order, smallest first: \(\tfrac56\), 0.82, 83%, \(\tfrac45\).
Guide me

What is it asking? the order

What do we know? four forms near 0.8

Plan decimals: 0.833, 0.82, 0.83, 0.8

Show working
  1. \(0.800<0.820<0.830<0.833\)

Answer \(\tfrac45,\ 0.82,\ 83\%,\ \tfrac56\)

Hard

1
Work out \((-2)^{3}-(-3)^{2}\times2\).
Guide me

What is it asking? the value

What do we know? powers of negatives

Plan powers, then ×, then −

Show working
  1. \((-2)^{3}=-8\), \((-3)^{2}=9\)
  2. \(-8-18=-26\)

Answer −26

PartAnswer / workingMarksGuidance
−8 and 9 seenM1
−26A1
Total: 2 marks
2
Insert one pair of brackets: \(5+3\times6-2=17\).
Guide me

What is it asking? the bracket position

What do we know? without brackets it is 21

Plan make 6 − 2 happen first

Show working
  1. \(5+3\times4=17\)

Answer \(5+3\times(6-2)=17\)

PartAnswer / workingMarksGuidance
\(5+3\times(6-2)\)B1
Total: 1 mark
3
Order, smallest first: \(\tfrac34\), 0.74, 77%, \(0.\dot7\), 0.758.
Guide me

What is it asking? the order

What do we know? five values near 0.75

Plan 3 decimal places; \(0.\dot7=0.777\ldots\)

Show working
  1. 0.740, 0.750, 0.758, 0.770, 0.777

Answer \(0.74,\ \tfrac34,\ 0.758,\ 77\%,\ 0.\dot7\)

PartAnswer / workingMarksGuidance
decimals seenM1
correct order in given formsA1
Total: 2 marks

Can I do this?

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Quiz

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