Elizabeth has $$ 10, $ 5, and $ 1$ bills worth $$ 101$ .
She has five more five dollar bills than ten dollar bills and $4$ times more $1$ dollar bills than $10$ dollar bills.
How many of each kind of bills does Elizabeth have?

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Solution

Find each kind of bill Elizabeth:
It is given that, Elizabeth has $$ 10, $ 5, and $ 1$ bills worth $$ 101$ .
She has $5$ more $$ 5$ bills than $$ 10$ bills and $4$ times more $$ 1$ bills than $$ 10$ bills.
Let Elizabeth has $x$ number of $$ 10$ bills, which worth is $$ 10 x$ , and $x + 5$ number of $$ 5$ bills, which worth is $$ 5 (x + 5)$ , and $4 x$ number of $$ 1$ bills, which worth is $4 x$ .
The total worth of bills is $$ 101$ :
$\begin{array}{rcl} 10 x + 5 (x + 5) + 4 x & = & 101 \\ \Rightarrow & 10 x + 5 x + 25 + 4 x = 101 \\ \Rightarrow & 19 x + 25 = 101 \\ \Rightarrow & 19 x = 76 \\ \Rightarrow & x = 4 \end{array}$
$4$ number of $$ 10$ bills, $9$ number of $$ 5$ bills and $4$ number of $$ 1$ bills.
Therefore, Elizabeth has $4$ number of $$ 10$ bills, $9$ number of $$ 5$ bills, and $4$ number of $$ 1$ bills.

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