Fractions & Decimals

Most fraction questions are comparisons, and long division is the slow way to answer them.

The four types:

1. Comparing two fractions. Cross-multiply: for a/b against c/d, compare a × d with c × b, and the larger product belongs to the larger fraction. With the same numerator the rule flips — 3/7 is smaller than 3/5, because the same thing shared among more people gives less.
2. Ordering a list. Take the LCM of the denominators once and rewrite every fraction over it; then only the numerators need comparing.
3. Converting between forms. 0.25 = 25/100 = 1/4; a recurring decimal becomes a fraction over as many nines as there are repeating digits, so 0.777… = 7/9 and 0.363636… = 36/99 = 4/11.
4. Complex fractions. A fraction stacked inside a fraction is simplified from the bottom upwards, one level at a time.

The single most common mistake is adding fractions by adding numerators and denominators separately: 2/3 + 3/4 is not 5/7. A denominator is a name, not a quantity — you can only add thirds to thirds, so both fractions must first be rewritten over the LCM: 8/12 + 9/12 = 17/12.

How to spot it in a second: 2/3 alone is already more than 5/7, so an answer smaller than either starting fraction cannot be a sum.

Speed tip: memorise 1/2 = 0.5, 1/3 = 0.333, 1/4 = 0.25, 1/6 = 0.166, 1/7 = 0.142, 1/8 = 0.125, 1/9 = 0.111 and 1/11 = 0.0909; most comparison questions are decided by the first two decimal places alone.

a/b vs c/d → compare a×d with c×b · same numerator → bigger denominator is smaller · order a list over the LCM of the denominators · 0.777… = 7/9, 0.3636… = 36/99 · adding needs a common denominator, never numerator+numerator over denominator+denominator

Arrange 3/5, 5/8 and 7/12 in ascending order. → The LCM of 5, 8 and 12 is 120, so the three become 72/120, 75/120 and 70/120. Comparing only the numerators gives 70 < 72 < 75, that is 7/12 < 3/5 < 5/8. Sanity check with decimals: 7/12 = 0.583, 3/5 = 0.600 and 5/8 = 0.625, which is the same order, and all three sit just above one half as they should.