Question

A construction firm is trying to decide which of two models of a crane to purchase. Model A costs $90,000 and requires $4,000 per year to maintain. Model B has an initial cost of $70,000 and a maintenance cost of $10,000 per year. For how many years must model A be used before it becomes more economical than B?

Answer 1

Let's say hmm let's use the figures in thousands, so
model A costs 90 + 4 per year, so... ok.. after "t" years, then the cost for model A is then 90 + 4*t, or A = 90 + 4t.

now, model B is 70 + 10 per year, so,... after "t" years, model B is then
B = 70 + 10 * t, or B = 70 + 10t.

if we were to assume A < B, namely the cost of A is less than B, what would the years be? namely, what's "t"?

$\bf A\ \textless \ B\implies 90+4t \ \textless \ 70 + 10t\implies 90-70 \ \textless \ 10t-4t \\\\\\ 20 \ \textless \ 6t\implies \cfrac{20}{6}\ \textless \ t\implies \cfrac{10}{3}\ \textless \ t\implies 3\frac{1}{3}\ \textless \ t \\\\\\ \textit{3 years and }\frac{1}{3}\textit{ or \underline{3 years and 4 months}}$

after that period, then A becomes cheaper than B.