A motel clerk counts his $1 and $10 bills at the end of the day. He finds that he has a total of 48 bills having a combined monetary value of $138. How to find the number of bills of each denomination that he has?
Gael Woodward
Answered question
2023-01-01
Answer & Explanation
bagosiwp2
Beginner2023-01-02Added 16 answers
Let's call the number of $1 bills: s
Let's call the number of $10 bills: t
Using the data in the problem, we can then create the following two equations:
Given that we know the motel clerk has a total of 48 bills:
Equation 1: \(\displaystyle{s}+{t}={48}\)
We also know the 48 bills add up to $138, then:
Equation 2: \(\displaystyle\${1}{s}+\${10}{t}=\${138}\)
Step 1) The first equation for s must be solved:
\(\displaystyle{s}+{t}-{\color{red}{{t}}}={48}-{\color{red}{{t}}}\)
\(\displaystyle{s}+{0}={48}-{t}\)
\(\displaystyle{s}={48}-{t}\)
Step 2) In the second equation, swap out (48-t) for s, and then solve for t:
\(\displaystyle\${1}{s}+\${10}{t}=\${138}\) becomes:
\(\displaystyle\${1}{\left({48}-{t}\right)}+\${10}{t}=\${138}\)
\(\displaystyle{\left(\${1}\cdot{48}\right)}-{\left(\${1}\cdot{t}\right)}+\${10}{t}=\${138}\)
\(\displaystyle\${48}-\${1}{t}+\${10}{t}=\${138}\)
\(\displaystyle\${48}+{\left(-\${1}+\${10}\right)}{t}=\${138}\)
\(\displaystyle\${48}+\${9}{t}=\${138}\)
\(\displaystyle\${48}-{\color{red}{\${48}}}+\${9}{t}=\${138}-{\color{red}{\${48}}}\)
\(\displaystyle{0}+\${9}{t}=\${90}\)
\(\displaystyle\${9}{t}=\${90}\)
\(\displaystyle\frac{{\${9}{t}}}{{\color{red}{\${9}}}}=\frac{{\${90}}}{{\color{red}{\${9}}}}\)
\(\displaystyle\frac{{{\color{red}{\cancel{{{\color{black}{\${9}}}}}}}{t}}}{\cancel{{{\color{red}{\${9}}}}}}=\frac{{{\color{red}{\cancel{{{\color{black}{\$}}}}}}{\color{red}{\cancel{{{\color{black}{{90}}}}}}}{10}}}{{\color{red}{{\color{black}{\cancel{{{\color{red}{\$}}}}}}{\color{black}{\cancel{{{\color{red}{{9}}}}}}}}}}\)
\(\displaystyle{t}={10}\)
Step 3) Substitute 10 for t in the solution to the first equation at the end of Step 1 and calculate s:
\(\displaystyle{s}={48}-{t}\) becomes:
\(\displaystyle{s}={48}-{10}\)
\(\displaystyle{s}={38}\)
The motel clerk had 38 $1 bills and 10 $10 bills.
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