Linear Equations & Word Problems

What you need to know

• A linear equation in one unknown has exactly one solution, unless the $x$ terms cancel. Then you get something always true, as when $2(x+1)=2x+2$ becomes $0=0$ (every number works), or never true, as when $2x+1=2x+3$ becomes $0=2$ (no number works). Example: $3x+7=22$ has the one solution $x=5$.
• Distance $=$ rate $\times$ time, and work done $=$ rate $\times$ time. Finishing a job alone in $t$ hours means a rate of $\frac1t$ of the job per hour. When workers or pipes work together, their rates add. Example: pipes that fill a tank in $4$ and $12$ hours together fill $\frac14+\frac1{12}=\frac13$ of it per hour, so they take $3$ hours.
• Average speed is total distance divided by total time, never the average of the speeds. Example: drive $30$ miles at $30$ mph ($1$ hour) and back at $15$ mph ($2$ hours). That is $60$ miles in $3$ hours, or $20$ mph, not $22.5$.
• Mixtures. The amount of the pure substance stays the same when you mix. If $V_1$ liters at concentration $c_1$ and $V_2$ liters at $c_2$ make a mix at concentration $c$, then $c_1V_1+c_2V_2=c(V_1+V_2)$. Example: $2$ liters of $10\%$ acid and $3$ liters of $20\%$ acid hold $0.2+0.6=0.8$ liters of acid in $5$ liters, so the mix is $16\%$ acid.
• Ages. The difference between two people's ages never changes. Example: Ann is $8$ years older than Ben, so she is twice his age when Ben is $8$ and she is $16$.

How AMC 10 tests it

• Problems 1–8: a short word problem (ages, coins, a trip in two parts) that becomes one equation like $ax+b=c$.
• Problems 5–12: work or pipe problems where the key idea is to add rates, not times.
• Round-trip speed problems where the average of the two speeds is one of the wrong answer choices.
• Problems asking for something other than the natural unknown: $x+y$, the other person's age, a percent.

Standard approaches

1. Give a letter to the unknown you are asked for, and turn each sentence into one equation.
2. Turn everything into a rate per unit of time. For trips, call the distance $1$ or a handy common multiple.
3. Pick a friendly total to avoid fractions: the least common multiple of the times, or $100$ for percents. Example: if Al does a job in $4$ hours and Bo in $6$, call the job $12$ units. Their rates are $3$ and $2$ units per hour, so together the job takes $\frac{12}5$ hours.
4. If the equation gets messy, test the answer choices against the story.

Worked example

A pump fills an empty tank in $6$ hours, and a drain empties a full tank in $12$ hours. The tank starts empty with the pump running and the drain closed. After $t$ hours the drain is opened, and the tank becomes exactly full $5$ hours after that. What is $t$?

(A) $3$ (B) $\frac{10}{3}$ (C) $\frac{7}{2}$ (D) $4$ (E) $\frac{9}{2}$

Call a full tank $1$. The pump adds $\frac16$ of a tank per hour and the drain removes $\frac1{12}$. With both open, the tank gains $\frac16-\frac1{12}=\frac2{12}-\frac1{12}=\frac1{12}$ of a tank per hour.

During the first $t$ hours only the pump runs, so it adds $\frac t6$. During the next $5$ hours both are open, adding $5\cdot\frac1{12}=\frac5{12}$. Together these make one full tank:
$$\frac t6+\frac5{12}=1\quad\Longrightarrow\quad \frac t6=\frac7{12}\quad\Longrightarrow\quad t=\frac{42}{12}=\frac72 .$$
Check: in $\frac72$ hours the pump fills $\frac7{12}$, and $\frac7{12}+\frac5{12}=1$.

The answer is $\boxed{\textbf{(C)}\ \frac72}$.

Pitfalls

• Averaging the two speeds instead of dividing total distance by total time.
• Adding times instead of rates. Example: if one person takes $3$ hours alone and another takes $4$, together they take $\frac{12}7$ hours, not $7$, which is longer than either one alone.
• Solving for $x$ but answering a different quantity from the one asked.
• Forgetting a unit conversion, such as minutes to hours. Example: $20$ minutes is $\frac13$ hour, not $0.2$ hour.

Traps that recur

• Assuming that once Michael catches the truck he stays ahead, or missing the two meetings where the gap only touches zero at the end of a stop or a drive. (2008 AMC 10B #25)
• Forgetting to subtract the lunch break from every day's working time, or converting 2:12 PM and 7:12 PM to hours incorrectly (372 and 672 minutes after 8:00). (2012 AMC 10A #19)
• Comparing the second mean to the original mean instead of to the enlarged list's mean (it should be m + 2 - 1 = m + 1 with n + 2 numbers). (2002 AMC 10B #25)
• Writing c + m = 8p and m/4 + c/6 = 8, then stalling with two equations in three unknowns instead of using that c and m are positive. (2000 AMC 10 #22)