Changing Recurring Decimals into Fractions Skipton Girls’ High School

Silent Starter- Divide these numbers Answers 35 48 63 77 89 175 ÷ 5 = 432 ÷ 9 = 756 ÷ 12 = 1155 ÷ 15 = *1513 ÷ 17 = Use the Bus Stop Method!!

Changing Fractions to Decimals When changing improper (top heavy) fractions e.g 21 5 to mixed numbers we know to divide by the denominator Therefore.. Remember to divide by the denominator when changing fraction to decimals.

Example 3 8 . 3 7 5 . 3 6 4 8 3

What about this? 4 9 . 4 4 4….. . 4 4 4 9 4

Changing Recurring Decimal to Fractions Objective- (L7) To be able to change decimals that recurring e.g. 0.55555555, 0.4141414141… and 0.564564564… to a fraction.

Example 1 Step 1 x = 0.5555555555.... Make the decimal equal to x Step 2 10x = 5.555555555.... How many digits are repeated ? So what do I need to multiply by ? 1 digit is repeated so I need to multiply by 10 Step 3 10 x = 5.55555555… x = 0.55555555… - What do I need to subtract to be left with just a whole number? 9x = 5.00000000 I need to subtract x which is 0.55555… So 9x = 5 Step 4 How do I get x on it’s own? x = 5 9 I divide by 9

Example 2 Step 1 x = 0. 41 41 41 41.... Make the decimal equal to x Step 2 100 x = 41. 41 41 41.... How many digits are repeated ? So what do I need to multiply by ? 2 digits are repeated so I need to multiply by 100 Step 3 100 x = 41. 41 41 41 x = 0. 41 41 41 - What do I need to subtract to be left with just a whole number? 99x = 41. 00 00 00 I need to subtract x which is 0.41 41 41… So 99 x = 41 Step 4 How do I get x on it’s own? x = 41 99 I divide by 99

x = 41 99 Can we cancel down or simplify the fraction? Step 5 Always check and see if you can simplify the fraction by dividing the numerator by 3 or 9. Can we divide 41 by either 3 or 9? No – so the fraction is already in it’s simplest form

Step 1 Example 2 x = 0. 564 564 564.... Make the decimal equal to x Step 2 1000 x = 564. 564 564.... How many digits are repeated ? So what do I need to multiply by ? 3 digits are repeated so I need to multiply by 1000 1000 x = 564. 564 564.... x = 0. 564 564 - Step 3 What do I need to subtract to be left with just a whole number? 999x = 564. 000 000 I need to subtract x which is 0.41 41 41… Step 4 So 999 x = 564 How do I get x on it’s own? x = 564 999 Divide by 999

x = 564 999 x = 188 333 Can we cancel down or simplify the fraction? Step 5 Can we cancel down or simplify the fraction? x = 564 999 Always check and see if you can simplify the fraction by dividing numerator by 3 or 9 1 8 8 2 2 3 5 6 4 Can we divide 564 exactly by 3 or 9? 3 9 9 9 3 Yes it divides exactly by 3, 188 times 3 goes exactly into 999, 333 times x = 188 333

Your Turn! Make the Q equal to x 0.6767676767 0.2121212121 0.666666666 0.3939393939 0.44444444 0.327327327 0.555555555 0.865865865 0.888888888 0.672672672 0.780378037803

Answers 1) 67/99 2) 2/3 3) 4/9 4) 5/9 5) 8/9 6) 21/99 or 7/33 7) 39/99 or 13/33 8) 327/999 or 109/333 9) 865/999 10) 672/999 or 224/333 867/1111

What about this? Can we use the technique you’ve just learned on this recurring decimal? What else could be we do which is similar?

To finish... Which of these fractions give recurring decimals? 3/10 2/3 3/8 3/7