A mobile service provider charges 11 dollars plus four dollars per minute for long-distance calls, whereas another provider charges 26 dollars plus one dollar 50 per minute. Write an equation that represents the number of minutes for which the two providers will charge the same amount and then solve it.
The total charge for each mobile service provider is equal to their fixed cost plus the cost of the long-distance calls. As we’re forming an equation, we’ll let the number of minutes spent on long-distance calls be equal to 𝑥. The first mobile service provider charges 11 dollars — that’s their fixed cost — plus four dollars per minute. Their total charge will therefore be 11 plus four multiplied by 𝑥: 11 plus four 𝑥. The second mobile service provider charges 26 dollars plus one dollar 50 per minute. Their total charge will therefore be equal to 26 dollars for the fixed cost plus 1.5 multiplied by 𝑥: 26 plus 1.5𝑥.
As the charges for the two mobile service providers are the same for this number of minutes, we can write this as an equation: 11 plus four 𝑥 is equal to 26 plus 1.5𝑥. Now, we want to solve our equation. The first step is to subtract 11 from each side. This gives four 𝑥 is equal to 15 plus 1.5𝑥. Now, there’re 𝑥 terms on both sides and we would like to group them all on one side of the equation. So the next step is to subtract 1.5𝑥 from both sides. This gives 2.5𝑥 is equal to 15.
To find the value of 𝑥, we need to divide both sides of the equation by 2.5. 𝑥 is equal to 15 over 2.5. Now, we’re currently being asked to divide by a decimal, which is quite a tricky thing to do. So in order to make this simpler, I’m going to multiply both the numerator and denominator of the fraction by two. This gives an equivalent fraction as I’ve done the same operation to both the numerator and the denominator. But now, it’s a much more straightforward division: 30 divided by five. This is equal to six.
So the equation that we were asked to form to represent the number of minutes for which the two providers will charge the same amount is 11 plus four 𝑥 is equal to 26 plus 1.5𝑥. The solution to this equation is 𝑥 is equal to six, meaning that for six minutes, the two providers will charge the same amount.