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Cambridge IGCSE0607

Ordering

International Mathematics 0607 Chapter Notes

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Ordering
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In this chapter

  1. 1Ordering numbers, fractions and percentages
  2. 2Ordering quantities with units
  3. 3The symbols =, ≠, >, <, ⩾ and ⩽

1. Ordering numbers, fractions and percentages

To order quantities by magnitude means to put them in order of size. Read the question for the direction: starting with the smallest is smallest first.

Fractions, decimals and percentages cannot be compared as they are written. Write every value in the same form, usually as decimals, then compare place by place.

On a calculator paper, enter the fraction as a division (4 ÷ 5) and read off the decimal. On a non-calculator paper, convert by hand.

4/5 = 0.8

82% = 0.82

Examiner insight

“Some candidates put the fraction before the 0.78 .” “However, in a minority of cases 0.4 came before 25%.”

Worked example 11 mark

Write 4/5, 82% and 0.78 in order of size, starting with the smallest.

  1. 1

    Write each as a decimal: 4/5 = 0.80, 82% = 0.82, and 0.78 stays.

  2. 2

    Compare: 0.78 < 0.80 < 0.82.

  3. 3

    0.78, 4/5, 82% [1]

2. Ordering quantities with units

Lengths, masses and times given in different units must be changed to the same unit before they are compared. The number alone is misleading: 8000 cm is a bigger length than 9 m.

Choose the most convenient unit, then work through the list once. Write the answer in the original units unless the question says otherwise.

Check the direction of the order at the end.

1 m = 100 cm

Examiner insight

“Although many candidates did find the correct order, others wrote the terms in descending order instead.”

Worked example 12 marks

Write these lengths in order of size starting with the smallest.

8000 cm 9 m 7.5 m 600 cm

  1. 1

    Change to metres: 8000 cm = 80 m and 600 cm = 6 m.

  2. 2

    Compare: 6 < 7.5 < 9 < 80. [1]

  3. 3

    600 cm < 7.5 m < 9 m < 8000 cm [1]

3. The symbols =, ≠, >, <, ⩾ and ⩽

The symbols compare two quantities. The small end of < and > points at the smaller quantity.

= means equal to and ≠ means not equal to. > means greater than and < means less than.

geqslant means greater than or equal to and leqslant means less than or equal to. The line under the symbol means the two quantities may be equal, so “at least 6” is x geqslant 6.

a = b

a ≠ b

a > b

a < b

a geqslant b

a leqslant b

Examiner insight

“It was very rare to see the inequality sign rather than the equals sign used for ‘at least’.”

Worked example 13 marks

(a) Write < or > between 0.35 and 0.4.

(b) Write x is at least 6 using a symbol.

(c) List the integers n for which 2 < n leqslant 5.

  1. 1

    (a) 0.4 = 0.40 and 0.35 < 0.40, so 0.35 < 0.4 [1]

  2. 2

    (b) at least 6 includes 6, so x geqslant 6 [1]

  3. 3

    (c) 2 is not allowed and 5 is allowed: 3, 4, 5 [1]

Recap

  • Ordering means putting quantities in order of size. Check whether the question says smallest or largest first.
  • Write fractions, decimals and percentages in the same form before comparing.
  • Change quantities to the same unit before comparing.
  • = equal to, ≠ not equal to, > greater than, < less than.
  • geqslant greater than or equal to, leqslant less than or equal to.
  • Check the direction of the order at the end.

End-of-chapter exercise

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

  1. Write down the meaning of the symbol ≠.1 mark
  2. Do not use a calculator. Write < or > between 0.09 and 0.1.1 mark
  3. Do not use a calculator. Write 3/5, 0.55 and 58% in order of size, starting with the smallest.2 marks
  4. Write these lengths in order of size, starting with the smallest. 450 cm 4.2 m 0.04 km 4600 mm2 marks
  5. A lift can carry at most 8 people. There are n people in the lift. Write this statement using a symbol.1 mark
  6. Do not use a calculator. List the integers n for which -3 leqslant n < 2.2 marks
  7. Do not use a calculator. Write 5/8, 7/12 and 3/5 in order of size, starting with the largest.3 marks
  8. Four runners have these times. 1 min 5 s 62 s 1.05 minutes 1 min 2.5 s Write the times in order, starting with the fastest.3 marks
  9. Do not use a calculator. A machine accepts a bolt only if its length L cm satisfies 2.45 leqslant L < 2.5. Five bolts have these lengths. 29/20 cm 25 mm 2.449 cm 212/25 cm 0.0247 m (a) Write the five lengths in order of size, starting with the smallest. (b) Write down the lengths of the bolts the machine accepts.4 marks
  10. Five long jumps are measured. 63/8 m 638 cm 6.4 m 6375 mm 0.00639 km (a) Two of the jumps are equal. Write a statement about these two jumps using the symbol =. (b) Write the five jumps in order of size, starting with the longest. Use the symbols > and = between them. (c) A jump of J metres reaches the final when J geqslant 6.38. Find how many of the five jumps reach the final.5 marks

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