Gael Woodward

Answered

2023-01-01

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?

Answer & Explanation

bagosiwp2

Expert

2023-01-02Added 16 answers

First:

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.

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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