 Gael Woodward

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.

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