Final answer:
To determine the cost of a bottle of soft drink A, set up equations based on the price difference with soft drink B, and the given amount spent compared to the quantity received. Solve the quadratic equation derived to get the price of soft drink A, which is found to be $3.20.
Step-by-step explanation:
The question is asking us to find the cost of a bottle of soft drink A given that it is $0.40 cheaper than soft drink B and that purchasing $48 of soft drink A yields 4 more bottles than purchasing soft drink B with the same amount of money. To solve this, we can set up two equations based on the given information.
Let x represent the cost of a bottle of soft drink A, and y represent the cost of a bottle of soft drink B. From the question, we know:
- y = x + $0.40 (since soft drink B is $0.40 more expensive than A)
- $48/x = $48/y + 4 (the number of bottles of A one can buy is 4 more than the number of bottles of B for the same amount of $48)
Substituting the first equation into the second, we have:
$48/x = $48/(x + $0.40) + 4
Multiplying through by x(x + $0.40) to clear denominators gives us:
$48(x + $0.40) = $48x + 4x(x + $0.40)
Expanding and simplifying:
$48x + $19.20 = $48x + 4x^2 + $1.60x
Collecting like terms and bringing all terms to one side gives us:
4x^2 + $1.60x - $19.20 = 0
To find the value of x, we can use the quadratic formula or factor the quadratic equation.
Solving, we get two potential values for x, but only one will make sense in the context of the problem (as the other might be negative, which is not possible for a price). Through calculation, we find that the cost of a bottle of soft drink A is $3.20.