Answer
$t=341p$, where $t$ is the total monetary value in cents and $p$ is the number of pennies.
Work Step by Step
Let's assign a couple of variables:
Number of pennies: $p$
Number of nickels (coins): $n$
Number of dimes (coins): $d$
Number of quarters (coins): $q$
First, let's set equations to the given data:
There are twice as many nickels as pennies, so: $n=2p$
There are four times dimes than nickels, so: $d=4n$
And there are as many quarters as nickels and dimes combined, so: $q=n+d$
Now, let's make changes to the 2nd and 3rd equations so that their number of coins can be in terms of the number of pennies:
Substituting $n=2p$ into $d=4n$ gives $d=4(2p) = 8p$
Substituting $n=2p$ and $d=8p$ into $q=n+d$ gives $q=2p+8p= 10p$
If we add all variables immediately, we'll get an equation with the total number of coins. We don't want that. Instead, we want to find the total monetary value. Therefore, we'll convert them to their monetary value: $q_{v}=25q$, $d_{v}=10d$, $n_{v}=5n$, and the pennies will be left as $p$ because they are worth 1 cent.
Substituting, we get:
${n_{v} \over 5} =2p$ -> $n_{v}=10p$
${d_{v} \over 10} =8p$ -> $d_{v}=80p$
${q_{v} \over 25} =10p$ -> $q_{v}=250p$
Now we add them all up to find the total monetary value $t$:
$t=q_{v}+d_{v}+n_{v}+p$
Once again, substituting will leave us with:
$t=250p+80p+10p+p$
$t=341p$
We're going to confirm our answer by assuming that there are 3 pennies in the purse. That means there will be
3 pennies, 6 nickels, 24 dimes, and 30 quarters. In monetary value, they are worth $3$ cents, $30$ cents, $240$ cents, and $750$ cents, respectively. Adding them up will give a total of $1023$ cents.
Using the simple equation we want to prove, $t=341 \cdot 3$ also equals $1023$, confirming our equation.
