What are related calculations?

• Related calculations allow us to work out answers to difficult problems, using calculations that we already know the answer to
• If we know a single, simple calculation, we can often use it to find out the answer to many more difficult calculations using related calculations and inverse operations
  • Related calculations use multiples of ten 
  • Inverse operations reverse a calculation that has happened
• The  commutative property can also be used 
  • Adding and multiplying are  commutative
    • If a ×  b =  c, then b × a  =  c and if a +  b =  c, then b +  a  =  c
  • Subtracting and dividing are  not commutative

What are inverse operations?

• A mathematical operation is simply the thing that we do to a number to change it to another number
• Add, subtract, multiply and divide are all examples of operations
• Inverse operations are simply the thing that we can do to reverse this change
• Adding and subtracting are inverse operations
• Multiplying and dividing are inverse operations
• Inverse operations can be used to find out more tricky calculations quickly from things we already know
• For example,
  If we know that 3 × 5 = 15, then we also know that 15 ÷ 3 = 5 and 15 ÷ 5 = 3  
  If we know that 3 ² = 9, then we also know that √9 = 3

How can related calculations be used to simplify problems?

• If you are given a problem, such as 12 × 13 = 156,  other facts can be quickly deduced
  • 13 × 12 = 156 (commutative law)
    156 ÷ 13 = 12 (inverse operations)
    156 ÷ 12 = 13 (inverse operations)
• Using multiples of ten can also help to simplify other problems
  • 120 × 13 = (12 × 10) × 13 = 12 × 13 × 10 = 156 × 10 = 1560
  • 1.2 × 13 = (12 ÷ 10) × 13 = 12 × 13 ÷ 10 = 156 ÷ 10 = 15.6
  • 0.013 × 120 = (13 ÷ 1000) × (12 × 10) =13 × 12  ÷ 1000 × 10 = 156 ÷ 100 = 1.56
• Using a combination of multiples of ten and inverse operations can deduce the answers to many other related calculations
  • 15 600 ÷ 12 = (156 × 100) ÷ 12 = 156 ÷ 12 × 100 = 13 × 100 = 1300
• If the number you are  dividing by is a decimal, use a multiple of ten to change it to an integer before carrying out any calculations
• Always change both parts of the problem before using related calculations  
  • 1560 ÷ 1.2 = (1560 × 10) ÷ (1.2 × 10)  = 15600 ÷ 12 = 1300
• This may be easier to see by writing the problem as a fraction
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