Working with Fractions, Decimals and Percentages

Terminating and recurring

• Every rational number has a decimal form that behaves in one of just two ways
• A terminating decimal stops after a finite number of places
  • 1/8 is 0.125
• A recurring decimal repeats a block of digits without end
  • 3/11 is 0.272727…
• A dot above a digit marks the repeat; where a block repeats, dots sit on its first and last digits
  • 0.888… is written with a single dot over the 8
  • 0.272727… carries dots over the 2 and the 7
• Recurring decimals are rational, not irrational, because every one of them can be written as a fraction

Converting a recurring decimal to a fraction

• Set the decimal equal to x, written out far enough to show the pattern
• Multiply by 10 as many times as it takes for two of the lines to carry exactly the same endless tail
• Where the block starts immediately after the point, one multiplication is enough, by 10 raised to the length of the block: one digit needs 10, two digits need 100, three need 1000
• Where the repeat starts later, two multiplied lines are needed instead: one that carries the point to the start of the block, and one a full block further on
  • For 0.1666… the lines 10x = 1.666… and 100x = 16.666… share a tail, and subtracting gives 90x = 15, so x = 1/6
• Subtract the two matching lines, so the endless tails cancel
• Divide to make x the subject, then cancel to lowest terms