Lupe, who works at a fast food restaurant received $6.10 in tips one afternoon, all in quarters, dimes, and nickels. There were five less dimes than quarters and seven less nickels than dimes. How many of each kind of coin was there?

Question

mathstudent55 · Answer

Let's choose variables.
Let n = number of nickels
Let d = number of dimes
Let q = number of quarters

Since a nickel is worth $0.05, a dime $0.10 and a quarter $0.25, the total value of the coins is

0.05n + 0.10d + 0.25q = 6.10

mathstudent55 · Answer

Now we can get two other equations using the information we are given.

"There were five less dimes than quarters" means
d = q - 5

"seven less nickels than dimes" means
n = d - 7

mathstudent55 · Answer

Now we have a system of equations in three variables that we can solve.

anonymous · Answer

Okay, I get confused after this. I'm confused on how to solve because I have two variables.

mathstudent55 · Answer

$\begin{cases} 0.05n + 0.10d + 0.25q = 6.10 \ d = q - 5 \ n = d - 7 \end{cases}$

Rewrite the second and third equations so all variables line up below the first equation:

$\begin{cases} 5n + 10d + 25q = 610 ~~~~~~~Eq.1\ ~~~~~~~~~~~~~~d - ~~~q = - 5 ~~~~~~~~Eq. 2\ ~~n ~~~~~-d ~~~~~~~~~~= - 7 ~~~~~~~~Eq. 3\end{cases}$


$~5n + 35d = 485$             25 * Eq. 2 + Eq. 1
$35n - 35d = -245$            35 * Eq. 3

$40n = 240$                Add last two equations above.

$n = 6$

Substitute n = 6 in Eq. 3 to get

$6 - d = -7$

$-d = -13$

$d = 13$

Substitute d = 13 in Eq. 2 to get

$13 - q = -5$

$-q = -18$

$q = 18$

Answer: 6 nickels, 13 dimes, 18 quarters

Check: 
6 * $0.05 + 13 * $0.10 + 18 * $0.25 = $0.30 + $1.30 + $4.50 = $6.10

anonymous · Answer

Thank you so much!

mathstudent55 · Answer

You're welcome.