MathematicsExam code: 0580
Working with Fractions, Decimals and Percentages
Number
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
Exam tip
Show the subtraction, then cancel fully
Writing a recurring decimal as a fraction is set in 8 of the 21 papers, always on Paper 2. The subtraction of your two lines, or the equation it gives (90x = 35 for 0.3888…), earns the method mark even if the rest goes wrong, so write it down. When the simplest form is asked for, the last mark is only for the fully cancelled fraction: 35/90 earns every mark but the last, and 7/18 earns them all.
Worked example
A recurring decimal with a two-digit block
Write 0.272727… as a fraction in its lowest terms.
Solution:
- Let x = 0.272727…
- The repeating block is two digits long, so multiply by 100
- 100x = 27.272727…
- Subtract the first line from the second, so the endless tails cancel
- 100x − x = 27.272727… − 0.272727…, giving 99x = 27
- Divide by 99: x = 27/99
- Cancel by the common factor 9: 3/11
Worked example
A fraction plus a recurring decimal
Work out 2/3 + 0.1555… and give the result as a single fraction, fully cancelled.
Solution:
- Let x = 0.1555…; only the 5 repeats, starting at the second decimal place
- Make two lines with the same endless tail: 10x = 1.555… and 100x = 15.555…
- Subtract: 100x − 10x = 15.555… − 1.555…, giving 90x = 14, so x = 14/90
- Write 2/3 over the same denominator: 2/3 = 60/90
- Add: 60/90 + 14/90 = 74/90
- Cancel by 2: 37/45
- Check: 37 ÷ 45 = 0.8222…, the same as 0.6666… + 0.1555…