Example 3.1.6

Ahmed has saved \(\$2{,}500.00\) in his savings account and is going to start saving \(\$550.00\) per month. Julia has saved \(\$4{,}600.00\) in her savings account and is going to start saving \(\$250.00\) per month. If this situation continues, how many months later would Ahmed catch up with Julia in savings?

Ahmed saves \(\$550.00\) per month, so he can save \(550m\) dollars in \(m\) months. With the \(\$2{,}500.00\) he started with, after \(m\) months he has \(550m+2500\) dollars. Similarly, after \(m\) months, Julia has \(250m+4600\) dollars. To find when those two accounts will have the same amount of money, we write the equation

\begin{equation*} 550m+2500=250m+4600\text{.} \end{equation*}
\begin{align*} 550m+2500\amp=250m+4600\\ 550m+2500\subtractright{2500}\amp=250m+4600\subtractright{2500}\\ 550m\amp=250m+2100\\ 550m\subtractright{250m}\amp=250m+2100\subtractright{250m}\\ 300m\amp=2100\\ \divideunder{300m}{300}\amp=\divideunder{2100}{300}\\ m\amp=7 \end{align*}

Checking the solution \(7\text{:}\)

\begin{align*} 550m+2500\amp=250m+4600\\ 550(\substitute{7})+2500\amp\stackrel{?}{=}250(\substitute{7})+4600\\ 3850+2500\amp\stackrel{?}{=}1750+4600\\ 6350\amp\stackrel{\checkmark}{=}6350 \end{align*}

In summary, Ahmed will catch up with Julia's savings in \(7\) months.

in-context