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Arithematics

Arithematics

Using a calculator, and mixed arithmetic

Practice Core + Extended 7 min read

On this page
  1. In this note
  2. Formula bank
  3. Typing it in correctly
  4. Reading the display
  5. Estimating to check
  6. Mixed practice: the whole chapter
  7. Ways they test it
  8. Mistakes to avoid
  9. Summary
  10. Practice
  11. Can I do this?
  12. Quiz

In this note

Syllabus 1.6, 1.9 and 1.14: calculators and mixed number work
  • 1.14Use a calculator efficiently: brackets, fraction key, negative key, powers, roots.
  • 1.14Read the display in context: money, time, rounding up or down.
  • 1.9Estimate to check that a calculator answer is sensible.

This page pulls the chapter together: order of operations, fractions, decimals.

Formula bank

Fractions on the calculatorLearn it( top ) ÷ ( bottom )or use the fraction template.Use it when any fraction with sums in it.
NegativesLearn ituse the (−) key; bracket before squaring: ( (−) 3 ) x²(−) 3 x² gives −9.Use it when a negative number with a power.
Decimal hoursLearn itminutes = decimal part × 607.2 h = 7 h 12 min.Use it when a time answer.
EstimateLearn itround every number to 1 significant figure60 × 0.3 ÷ 2 = 9.Use it when checking, or "estimate".

Typing it in correctly

General methodHow to type any calculation

  1. Put brackets round every top and every bottom of a fraction.
  2. Use the (−) key for negatives, and bracket a negative before squaring it.
  3. Close a root's bracket after the whole thing inside it.
  4. Standard form: the ×10ˣ (EXP) key.
  5. Write the full display before rounding.

Example Calculate \(\dfrac{12.4+\sqrt{5.29}}{3.1^{2}-2.6}\) to 3 s.f.

Brackets round top and bottom; full display; round.

  1. \((12.4+\sqrt{5.29})\div(3.1^{2}-2.6)\)
  2. \(=14.7\div7.01=2.097\,004\ldots\)

Answer 2.10

Try it yourself

1
\(\dfrac{\sqrt{256}+4.8}{0.25}\)
Show answer

Answer 83.2

2
\(\dfrac{2.7^{3}+1.9}{4.2-\sqrt{3.61}}\) to 2 d.p.
Show answer

Answer 9.38

Reading the display

General methodHow to turn a display into an answer

  1. Money: 2 decimal places ($2.50).
  2. Time in hours: the decimal part × 60 gives minutes.
  3. Things that must all fit (buses, tins, paying a bill): round UP.
  4. How many whole items you can make: round DOWN.
DisplayContextAnswer
2.5money ($)$2.50
7.2hours7 h 12 min
13.785 7…7 friends must cover $96.50$13.79 (round up)
6.4buses needed7
6.8whole boxes filled6

Try it yourself

1
A display shows 4.75 hours. How many minutes past 4 hours?
Show answer

Answer 45

2
230 people travel in 52-seat buses. How many buses?
Show answer

Answer 5

Estimating to check

General methodHow to estimate any calculation

  1. Round every number to 1 significant figure.
  2. Work it out in your head.
  3. Compare with the calculator: very different means a typing mistake.

Example Estimate \(\dfrac{58.6\times0.312}{2.07}\).

1 s.f.: 60, 0.3, 2.

  1. \(\tfrac{60\times0.3}{2}=\tfrac{18}2\)

Answer 9 (calculator: 8.83)

Try it yourself

1
Estimate \(\dfrac{39.2\times0.48}{1.97}\).
Show answer

Answer 10

Mixed practice: the whole chapter

No calculator. Each one is a different skill from this chapter.

Mixed practice (no calculator)

1
\(7+3\times(-2)\)
Show answer

Answer 1

2
\(\tfrac56-\tfrac14\)
Show answer

Answer \(\tfrac7{12}\)

3
\(0.04\times0.3\)
Show answer

Answer 0.012

4
\(300\div0.06\)
Show answer

Answer 5000

5
\(\tfrac29\times\tfrac34\)
Show answer

Answer \(\tfrac16\)

6
Halfway between \(\tfrac3{10}\) and \(\tfrac25\)
Show answer

Answer \(\tfrac7{20}\)

7
Difference between 9 °C and −5 °C
Show answer

Answer 14

8
\(\tfrac38\) of 64
Show answer

Answer 24

Ways they test it

Type 1 Calculate (0580 P2)

"Calculate \(\dfrac{12.4+\sqrt{5.29}}{3.1^{2}-2.6}\) to 3 s.f." (1 mark)

Spot it "calculate" with roots, powers, fractions.

  1. Brackets.
  2. Full display, then round.

Type 2 Several short non-calculator parts

"(a) 0.04 × 0.3 (b) 300 ÷ 0.06 (c) 5/6 − 1/4 (d) (−4)² − 3 × 5"

Spot it a list of small calculations.

  1. Each with its own method.
  2. Exact answers.

Type 3 Round for the context

"Seven friends share a bill of $96.50, each paying enough to cover it."

Spot it people, buses, tins, money.

  1. Divide.
  2. Round the way the story needs.

For example $96.50 ÷ 7

  1. \(13.785\,7\ldots\)
  2. \(7\times13.78=96.46\) is too little

Answer $13.79

Type 4 Estimate and comment

"Estimate by rounding to 1 s.f., then use your calculator and comment." (3 marks)

Spot it "estimate".

  1. 1 s.f. for each number; working shown.
  2. Calculator value; say whether the estimate is high or low.

Type 5 Time from decimal hours

"A plumber works 3.35 hours. Write this in hours and minutes."

Spot it decimal hours.

  1. 0.35 × 60 = 21 minutes.

Mistakes to avoid

  • ✗ 7.2 h = 7 h 20 min✓ 7 h 12 min0.2 × 60 = 12.
  • ✗ $2.5✓ $2.50Money to 2 d.p.
  • ✗ (−) 3 x² = 9✓ −9 (bracket it)The power is done first.
  • ✗ 2.1 to 3 s.f.✓ 2.10Show the third figure.
  • ✗ 6.4 buses → 6✓ 7Everyone needs a seat.

Summary

  • Type what you MEAN: brackets round tops and bottoms.
  • Read the display in context: money, time, round up or down.
  • Estimate with 1 s.f. to catch typing mistakes.

If you see …

  • decimal hours× 60 for minutes.
  • "how many … needed"round up.
  • "how many whole … can be made"round down.

Practice

Practice: easy, medium, hard

Easy

1
\((-4)^{2}-3\times5\)
Show working
  1. \(16-15\)

Answer 1

2
A plumber works 3.35 hours. How many minutes past 3 hours?
Show working
  1. \(0.35\times60\)

Answer 21

3
Estimate \(\dfrac{58.6\times0.312}{2.07}\).
Show working
  1. \(\tfrac{60\times0.3}2\)

Answer 9

Medium

1
Seven friends share $96.50; each pays the same and together they cover the bill. How much each?
Guide me

What is it asking? each person's share

What do we know? $96.50 ÷ 7 is not whole cents

Plan divide, then round UP

Show working
  1. \(13.785\,7\ldots\)
  2. round up

Answer $13.79

2
Without a calculator, \(\tfrac34\times36+0.2\times15-2^{2}\).
Guide me

What is it asking? the value

What do we know? three terms

Plan each term, then add and subtract

Show working
  1. \(27+3-4\)

Answer 26

3
A plumber works 3.35 hours at $28 an hour. Her pay?
Guide me

What is it asking? the pay

What do we know? 3.35 × 28

Plan multiply

Show working
  1. \(3.35\times28=93.80\)

Answer $93.80

Hard

1
Calculate \(\dfrac{2.7^{3}+1.9}{4.2-\sqrt{3.61}}\), giving your answer to 2 d.p.
Guide me

What is it asking? the value

What do we know? powers, a root, a fraction

Plan top, bottom, divide

Show working
  1. top \(=21.583\)
  2. bottom \(=2.3\)
  3. \(9.3839\ldots\)

Answer 9.38

PartAnswer / workingMarksGuidance
9.383…M1
9.38A1
Total: 2 marks
2
Calculate \(\dfrac{12.4+\sqrt{5.29}}{3.1^{2}-2.6}\) to 3 s.f.
Guide me

What is it asking? the value

What do we know? a fraction with a root and a square

Plan brackets round top and bottom

Show working
  1. \(14.7\div7.01=2.097\ldots\)

Answer 2.10

PartAnswer / workingMarksGuidance
2.10B12.1 is not 3 s.f.
Total: 1 mark
3
Estimate \(\dfrac{39.2\times0.48}{1.97}\) by rounding to 1 s.f.
Guide me

What is it asking? an estimate

What do we know? 39.2, 0.48, 1.97

Plan 1 s.f.: 40, 0.5, 2

Show working
  1. \(\tfrac{40\times0.5}2=10\)

Answer 10

PartAnswer / workingMarksGuidance
40, 0.5, 2 seenM1
10A1
Total: 2 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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