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Cambridge IGCSE0580

The four operations

Mathematics 0580 Chapter Notes

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The four operations
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In this chapter

  1. 1Whole-number calculations
  2. 2Negative numbers and temperature
  3. 3Multiplying and dividing decimals
  4. 4Order of operations and brackets
  5. 5Fractions and mixed numbers
  6. 6Calculator work and rounding
  7. 7Money problems in stages

1. Whole-number calculations

Use the four operations on integers: add, subtract, multiply and divide. Keep each digit in its correct place value, especially when you divide.

When a question asks for one number, such as the nearest multiple, give that one number. A list of multiples is not an answer.

Examiner insight

“Common errors were responses of 1311 and 1313 while a minus sign was occasionally seen before the correct answer.” “Many did not perform the division correctly, with 47 being the most common incorrect answer from an error with the place value of 4.”

Worked example 13 marks

(a) Subtract 9 from 1321.

(b) Divide 814 by 2.

(c) Write down the multiple of 6 that is nearest to 20.

  1. 1

    (a) 1321 - 9 = 1312 [1]

  2. 2

    (b) 814 ÷ 2 = 407 [1]

  3. 3

    (c) multiples of 6: 12, 18, 24. The nearest to 20 is 18 [1]

2. Negative numbers and temperature

Subtracting a negative number is the same as adding the positive number.

In a temperature question, a rise is an addition and a fall is a subtraction. To find the difference between two temperatures, subtract the lower from the higher.

a - (-b) = a + b

difference = higher - lower

Examiner insight

“This question was very well answered although a small number of candidates added the temperatures leading to answers of 13 or −13.” “Errors were often made in subtracting 5 from –7 with –7 – (−5) or −7 + 5 seen.”

Worked example 11 mark

In a town, the lowest temperature one night is −12.6 °C. The highest temperature the next day is 25.7 °C. Calculate the difference between these two temperatures.

  1. 1

    25.7 - (-12.6) = 25.7 + 12.6 = 38.3 °C [1]

3. Multiplying and dividing decimals

To multiply decimals, multiply as whole numbers, then place the decimal point. Do not read a power as a multiplication by 2.

To divide by a decimal, write the division as a fraction and multiply the numerator and the denominator by the same power of 10. The divisor becomes a whole number.

0.7² = 0.7 × 0.7 = 49/100

2 ÷ 0.04 = (2 × 100)/(0.04 × 100) = 200/4

Examiner insight

“Some completed the multiplication correctly then inserted the decimal point in the incorrect place giving answers such as 4.9 or 0.049.” “Some found 2 ÷ 4 = 0.5 then adjusted the wrong way so instead of multiplying by 100 they divided by 100 to reach 0.005.”

Worked example 12 marks

Work out.

(a) 0.7²

(b) 2 ÷ 0.04

  1. 1

    (a) 0.7 × 0.7: 7 × 7 = 49, two decimal places, so 0.49 [1]

  2. 2

    (b) 2 ÷ 0.04 = 200/4 = 50 [1]

4. Order of operations and brackets

Work out brackets first, then multiplication and division, then addition and subtraction. At the same level, work from left to right.

One pair of brackets can go round more than two terms. Do not put a minus sign that comes before a number inside the brackets by mistake.

6 × (5 - 3 + 5) = 6 × 7 = 42

Examiner insight

“There were some who had two pairs of brackets, ignoring the clear instruction.” “A common error was to include the minus sign before 80 inside the brackets.”

Worked example 12 marks

Put one pair of brackets into each calculation to make it correct.

(a) 6 × 5 - 3 + 5 = 42

(b) 4 - 80 + 20 ÷ 10 = -6

  1. 1

    (a) 6 × (5 - 3 + 5) = 6 × 7 = 42 [1]

  2. 2

    (b) 4 - (80 + 20) ÷ 10 = 4 - 100 ÷ 10 = 4 - 10 = -6 [1]

5. Fractions and mixed numbers

Change a mixed number to an improper fraction before you multiply or divide. To divide, invert the second fraction and multiply.

To add or subtract, use a common denominator. Give the answer in the form the question asks for, in its simplest form. When a question asks for a mixed number, an improper fraction or a decimal loses the final mark.

21/3 = 7/3

a/b ÷ c/d = a/b × d/c

Examiner insight

“However, many lost the final mark by not reading that the question asked for a mixed number answer.” “The main errors in the process were to invert the first improper fraction or not invert the second fraction when multiplying.”

Worked example 13 marks

A book has 420 pages. Meg reads 14 pages each day. Meg has read 1/3 of the number of pages so far. Find how many days it will take Meg to finish reading the rest of the book.

  1. 1

    pages left = 2/3 × 420 [1]

  2. 2

    = 280 pages [1]

  3. 3

    days = 280 ÷ 14 = 20 [1]

6. Calculator work and rounding

On a calculator, keep the order of operations: work out the power, then multiply, then subtract.

Write down a more accurate decimal first. Then round it once to the accuracy asked for.

10 - (7/8)³ × 2 = 10 - 1.3398…

8.660… = 8.7 (1 d.p.)

Examiner insight

“Some candidates incorrectly rounded 8.660 to 8.6 or rounded to 8.66.” “A few candidates making this error did not show a more accurate decimal before rounding to 1 decimal place.”

Worked example 12 marks

Calculate 10 - (7/8)³ × 2. Give your answer correct to 1 decimal place.

  1. 1

    (7/8)³ × 2 = 1.3398… [1]

  2. 2

    10 - 1.3398… = 8.660… = 8.7 (1 d.p.) [1]

7. Money problems in stages

In a practical question, work in stages. Find each cost, add the costs, then subtract the total from the amount paid to find the change.

change = amount paid - total cost

Examiner insight

“Many candidates gave the correct answer, usually by working in stages.”

Worked example 13 marks

A café charges char36 5.75 for a cake and char36 4.40 for a milkshake. Sue buys 3 cakes and 2 milkshakes. Work out how much change she receives from a char36 50 note.

  1. 1

    3 × 5.75 = 17.25 and 2 × 4.40 = 8.80 [1]

  2. 2

    total = 17.25 + 8.80 = 26.05 [1]

  3. 3

    change = 50 - 26.05 = char36 23.95 [1]

Recap

  • Brackets first, then multiply and divide, then add and subtract. Left to right at the same level.
  • Subtracting a negative number is adding the positive: a - (-b) = a + b.
  • Temperature difference = higher − lower.
  • Decimal division: multiply both numbers by the same power of 10 first.
  • Mixed number to improper fraction first. To divide, invert the second fraction and multiply.
  • Give the answer in the form asked: mixed number, fraction or decimal.
  • Write down a more accurate decimal, then round once to the accuracy asked for.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. Work out 4000 - 387.1 mark
  2. The temperature in a freezer is −18 °C. It rises by 7 °C and then falls by 12 °C. Find the final temperature.2 marks
  3. Do not use a calculator. Work out 20 - 3 × 4 + 6 ÷ 2.2 marks
  4. Put one pair of brackets into 5 × 2 + 3 - 1 = 20 to make it correct.1 mark
  5. Do not use a calculator. Work out 4.2 ÷ 0.07.1 mark
  6. Do not use a calculator. Work out 21/3 × 11/2. Give your answer as a mixed number.3 marks
  7. Calculate (7.2 - 3.9 × 1.6)/(0.5² + 1.1). Give your answer correct to 2 decimal places.3 marks
  8. Do not use a calculator. Work out 7/8 - 3/4 ÷ 9/10. Give your answer as a fraction in its simplest form.4 marks
  9. Do not use a calculator. A lift in a mine is at a height of -71/2 m, measured from ground level. It goes down 21/4 m each second for 6 seconds. It then rises by 2/5 of its distance below ground level at that moment. Find the final height of the lift. Give your answer as a mixed number.5 marks
  10. A roll of wire is 50 m long and costs 84 dollars. A worker cuts 24 pieces, each 13/7 m long, from the roll. The wire left over is cut into 4 equal lengths. Calculate the cost, in dollars, of the wire in one of these 4 lengths.5 marks

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