Learn first: Natural numbers and operations
Fractions are where exam marks are lost most often: one wrong common denominator spoils a whole problem. A fraction expresses part of a whole, records the quotient of two numbers and shows ratios — so it appears constantly in algebra, geometry, percentages and probability. In this lesson we learn to carry out every operation reliably, from ordinary fractions to recurring decimals, understand the reasons behind the rules and work through fraction word problems.
a/b means: divide a whole into b equal parts and take a of them (b ≠ 0). b is the denominator (how many parts), a the numerator (how many are taken). a < b — a proper fraction (less than one), a ≥ b — an improper fraction.
Basic property: a/b = (a·k)/(b·k), k ≠ 0
Why? If each part is cut into k pieces, both the number of parts and the number taken grow k times, while the amount taken stays the same.
Adding: a/b + c/d = (ad + bc)/(bd) — in practice use the LCM of the denominators
Multiplying: a/b · c/d = ac/(bd); dividing: a/b : c/d = a/b · d/c
Why a common denominator? Only like “pieces” can be added: thirds and halves are first turned into sixths: 1/2 + 1/3 = 3/6 + 2/6 = 5/6, not 2/5.
Dividing equals multiplying by the reciprocal, because division undoes multiplication: (a/b · d/c)·(c/d) = a/b.
Convert mixed numbers to improper fractions before multiplying or dividing. Cancelling across before multiplying makes the arithmetic easier: 14/15 · 25/28 = (1·5)/(3·2) = 5/6.
Calculate 2/3 + 5/6 − 3/4.
A decimal is another way of writing a fraction with denominator 10, 100, 1000, …: 0.37 = 37/100.
Useful values: 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75, 1/5 = 0.2, 1/8 = 0.125.
The repeating block is put in brackets: 0.333… = 0.(3), 0.1666… = 0.1(6).
Purely recurring: 0.(ab) = ab/99; e.g. 0.(36) = 36/99 = 4/11
Mixed recurring: 0.a(b) = (ab − a)/90
General rule: (all digits after the point − the part before the block) / (as many 9s as digits in the block followed by as many 0s as digits before it). For example, 0.1(6) = (16 − 1)/90 = 15/90 = 1/6; 0.(9) = 9/9 = 1.
Calculate 7.2(1) − 4.4(2) + 31/90.
If an expression mixes ordinary and decimal fractions, bring everything to one form: values such as 0.25 = 1/4 and 0.(3) = 1/3 are easier as ordinary fractions. A multi-level fraction is simplified from the bottom level up.
Calculate 1/(1 + 1/(1 + 1/2)).
m/n of a is a·m/n; the whole from its part is b : (m/n)
Careful: the words “of the remainder” mean the fraction is taken of a new whole — what is left.
3/8 of a book has been read, which is 150 pages. How many pages does the book have?
Of 1200 som, 1/3 was spent, then 1/4 of the remainder. How much money is left?
It is sometimes very useful to write a fraction as a difference of two fractions. When the denominator is a product of two consecutive numbers:
1/(n(n + 1)) = 1/n − 1/(n + 1)
Check: 1/n − 1/(n + 1) = (n + 1 − n)/(n(n + 1)) = 1/(n(n + 1)). For example, 1/6 = 1/(2·3) = 1/2 − 1/3.
In a sum of such fractions neighbouring terms cancel and only the first and last parts remain — the sum “collapses like a telescope”. This turns a long calculation into one line and saves time in the exam.
Calculate 1/(1·2) + 1/(2·3) + … + 1/(9·10).