9.5 Simultaneous equations concepts -- word problems

Simultaneous equation word problem

A roll of dimes and a roll of quarters lie on the table in front of you. There are three more quarters than dimes. But the quarters are worth three times the amount that the dimes are worth. How many of each do you have?

• Identify and label the variables.
  • There are actually two different, valid ways to approach this problem. You could make a variable that represents the number of dimes; or you could have a variable that represents the value of the dimes. Either way will lead you to the right answer. However, it is vital to know which one you’re doing! If you get confused half-way through the problem, you will end up with the wrong answer.

  Let’s try it this way:
  $d$ is the number of dimes
  $q$ is the number of quarters

• Translate the sentences in the problem into equations.
  • “There are three more quarters than dimes” $\to q=d+3$
  • “The quarters are worth three times the amount that the dimes are worth” $\to \text{25}q=3(\text{10}d)$
  • This second equation relies on the fact that if you have $q$ quarters, they are worth a total of $\text{25}q$ cents.
• Solve.
  • We can do this by elimination or substitution. Since the first equation is already solved for $q$ , I will substitute that into the second equation and then solve.

  $\text{25}\left(d+3\right)=3\left(\text{10}d\right)$

  $\text{25}d+\text{75}=\text{30}d$

  $\text{75}=\mathrm{5d}$

  $d=\text{15}$

  $q=\text{18}$

So, did it work? The surest check is to go all the way back to the original problem—not the equations, but the words. We have concluded that there are 15 dimes and 18 quarters.

“There are three more quarters than dimes.” $✓$

“The quarters are worth three times the amount that the dimes are worth.” $\to$ Well, the quarters are worth $\text{18}\cdot \text{25}=\$4\text{.}\text{50}$ . The dimes are worth $\text{15}\cdot \text{10}=\$1\text{.}\text{50}$ . $✓$