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Cambridge IGCSE0580

Ordering

Mathematics 0580 Chapter Notes

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Ordering
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In this chapter

  1. 1The comparison symbols
  2. 2Comparing different forms
  3. 3Writing the order

1. The comparison symbols

Six symbols compare two quantities. a < b says that a is less than b, and a > b says that a is greater than b. a mathrelrlap /= b says that a and b are not equal.

To compare two numbers, put the symbol that makes the statement true. Two quantities that have the same value take =. If the first is the smaller, it takes <. The symbols geqslant and leqslant also allow the two quantities to be equal.

a = b a mathrelrlap /= b

a < b a > b

a geqslant b a leqslant b

Worked example 12 marks

Write one of the symbols <, > or = in each statement to make it correct.

5/8 ... 62.5%

0.2² ... 0.4

2/3 ... 0.67

  1. 1

    5/8 = 0.625 = 62.5%, so the symbol is =

  2. 2

    0.2² = 0.2 × 0.2 = 0.04 and 0.04 < 0.4, so the symbol is <

  3. 3

    2/3 = 0.666… and 0.666… < 0.67, so the symbol is <

  4. 4

    answers: =, <, < [2]

2. Comparing different forms

Fractions, decimals and percentages cannot be compared until they are in the same form. Change every number to a decimal.

For a fraction, divide the numerator by the denominator. For a percentage, divide by 100, so 34% = 0.34. Keep every zero in its place: 0.036 is not 0.36.

Write enough decimal places to separate numbers that start alike. Show every conversion in your working.

a/b = a ÷ b x% = x/100

Examiner insight

“It is essential to find decimal equivalents of the square root and fraction to enough decimal places (in this case a minimum of 4) to decide on the order.” “An error was to regard 34% as 34 and so it was often seen as the largest item”

Worked example 12 marks

Write these numbers in order, starting with the smallest.

√[3]0.1 4/9 45% 0.443

  1. 1

    convert to decimals: √[3]0.1 = 0.4642…, 4/9 = 0.4444…, 45% = 0.45 [1]

  2. 2

    compare 0.443 < 0.4444 < 0.45 < 0.4642

  3. 3

    0.443 < 4/9 < 45% < √[3]0.1 [1]

3. Writing the order

Read the direction the question asks for. Starting with the smallest means the smallest goes on the left.

Write the original numbers in the answer. A decimal in the answer can score only if it is clear which item it refers to.

Check that you have put in order the same number of items as the question gave.

Examiner insight

“The fraction was often seen as the smallest and often when it was in the correct position the order was largest to smallest.” “Some who showed an order missed out a value; it would be good advice for candidates to check that the number of items they have put in order is the same as the number in the original list.”

Worked example 12 marks

Do not use a calculator. Write these numbers in order, starting with the largest.

2/5 35% 0.37

  1. 1

    convert to decimals: 2/5 = 0.4 and 35% = 0.35 [1]

  2. 2

    compare, largest first: 0.4 > 0.37 > 0.35

  3. 3

    2/5, 0.37, 35% [1]

Recap

  • Symbols: =, mathrelrlap /=, <, >, geqslant, leqslant.
  • Change every number to a decimal before comparing.
  • Fraction: numerator ÷ denominator. Percentage: ÷ 100.
  • Write enough decimal places to separate the numbers.
  • Starting with the smallest: the smallest goes on the left.
  • Check that no item is missing from your order.

End-of-chapter exercise

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

  1. Do not use a calculator. Write one of the symbols <, > or = in each statement. (a) 3/4 ... 0.7 (b) 0.25 ... 25% (c) 2/5 ... 0.043 marks
  2. Do not use a calculator. Write these numbers in order, starting with the smallest. 0.6 55% 1/2 0.0562 marks
  3. Write these numbers in order, starting with the largest. 5/7 0.71 72%2 marks
  4. Do not use a calculator. Write these numbers in order, starting with the smallest. 9/250 0.36 3.5% 0.042 marks
  5. Do not use a calculator. Write down all the integers n for which 3 leqslant n < 7.2 marks
  6. Write these numbers in order, starting with the smallest. 7/10 √(0.5) 71% 0.7052 marks
  7. In a test Asha scored 23/34 of the marks, Ben scored 68% and Chen scored 0.675 of the marks. State who had the highest score and who had the lowest.2 marks
  8. Do not use a calculator. Write one of the symbols <, > or = in each statement. (a) 3/8 ... 37.5% (b) 0.3² ... 0.9 (c) 1/3 ... 0.333 marks
  9. (a) Write these numbers in order, starting with the smallest. π/7 0.45 √(0.2) 22/49 44.8% (b) Lena rounds every number in the list to the same number of decimal places before she orders them. Find the smallest number of decimal places that gives five different values.4 marks
  10. Do not use a calculator. Four tanks have the same capacity. Tank A is 5/8 full. Tank B is 0.8² full. Tank C has 37% of its capacity empty. Tank D is 1/2 full, and then a further 0.13 of its capacity is poured in. (a) List the tanks in order, starting with the tank that holds the least water. Write = between any two tanks that hold the same amount. (b) A fifth tank is n/50 full, where n is an integer and 5/8 < n/50 leqslant 0.8². Find the value of n.5 marks

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