Question

United Telephone charges a base rate of $\$ 10.00$ for service, plus an additional charge of $ \$ 0.25$ per minute. Atlantic Call charges a base rate of $ \$12.00$ for service, plus an additional charge of $ \$0.20$ per minute. For what number of minutes would the bills for each telephone company be the same? \[\]
A.40\[\]
B.20\[\]
C.60\[\]
D.30\[\]

Answer 1

Hint: We assume that the number of minutes is $x$ such that for each telephone company be the same. We express the bill for United Telephone $U$ and Atlantic Call $A$ as a function of $x$. We equate $U=A$ and then solve for $x$ to get the required result. \[\]

Complete step-by-step solution
Base rate of any service is the price paid to the service provider by just subscribing or installing the service. The customer has to pay additional charges per unit of time (minute, days, and months) to consume the service.\[\]
Let us assume the number of minutes to be $x$ such that for each telephone company be the same. We are given the question that United Telephone charges a base rate of $\$10.0$ for service, plus an additional charge of $\$ 0.25$ per minute. So the bill for United Telephone is
\[U=10+0.25x\]
We are also given the question that Atlantic Call charges a base rate of $\$12.00$ for service, plus an additional charge of $\$0.20$ per minute. So the bill for United Telephone is
 \[A=12+0.20x\]
We are further given in the question that bills for each telephone company be the same which means equal. So we have,
\[\begin{align}
  & U=A \\
 & \Rightarrow 10+0.25x=12+0.20x \\
\end{align}\]
Let us subtract 10 from both sides of the above equation in the above step to have,
\[\Rightarrow 0.25x=2+0.20x\]
Let us subtract $0.20x$ from both side of the above equation in the above step to have,
\[\begin{align}
  & \Rightarrow 0.25x-0.20x=2 \\
 & \Rightarrow x\left( 0.25-0.20 \right)=2 \\
 & \Rightarrow x\times 0.05= 2 \\
\end{align}\]
We divide both side of the above equation by 5 to have in the above step to have,
\[\Rightarrow x=\dfrac{2}{0.05}=\dfrac{2}{\dfrac{5}{100}}=\dfrac{2\times 100}{5}=40\]
So the correct option is A.

Note: We note that we need to know decimal addition, subtraction, and division to solve this problem. Most mistakes happen in this problem by multiplying $x$ with the base rate which we must be careful of. The equation we solved is a linear equation in one variable and it is solved by collecting variable terms (terms multiplied with $x$) at one side and numerical terms at the other side.