2.1 From words to equations
Most aptitude problems in algebra are not difficult once they are written down correctly; the difficulty lies in the translation. The method is always the same. One chooses a symbol for the quantity asked for (or for a quantity from which it follows easily), expresses every statement in the problem as a relation involving that symbol, and solves. A final check against the original wording guards against the commonest error, which is to answer a slightly different question from the one set.
A frequent pattern describes a whole through its parts. If fractions \(f_1, f_2, \dots, f_k\) of an unknown total \(x\) are accounted for and the remainder is a known amount \(R\), then
\[x\left(1 - f_1 - f_2 - \cdots - f_k\right) = R \quad\Longrightarrow\quad \boxed{x = \frac{R}{1 - \sum_i f_i}}.\]
The fractions are best added over their least common denominator. For instance, \(\tfrac14 + \tfrac16 + \tfrac18 + \tfrac1{12} = \tfrac{6+4+3+2}{24} = \tfrac{15}{24}\), so a remainder \(R\) corresponds to \(\tfrac{9}{24}\) of the whole. Percentages are fractions with denominator 100, and phrases such as “40% empty” should be converted at once into the equivalent positive statement, “60% full”.
Problems about ages are linear because every person ages at the same rate. It is usually clearest to place all the events on a single time line and to measure every age from one reference event; the differences between the ages of two people never change.
