Question

John has 50 for soda and he must buy both diet and regular sodas. His total order must have at exactly two times as many cans of diet soda as cans of regular soda. What is the greatest number of cans of diet soda John can buy if regular soda is 0.50 per can and diet soda is $0.75 per can? 
 

• A. None of the other answers
• B. 51
• C. 25
• D. 75
• E. 50

Hint:

When dealing with "greatest number" or "maximum value" problems, set up an inequality. Solve for the variable in question. The maximum integer value that satisfies the inequality is usually the answer, but always double-check it against the original problem's conditions.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This is a word problem that can be modeled with a system of linear equations and inequalities. We need to maximize the number of diet sodas purchased under a budget constraint and a ratio constraint.
Step 2: Key Formula or Approach:
Let d be the number of cans of diet soda and r be the number of cans of regular soda.
We can set up the following equations based on the problem statement:
1. Budget Constraint: 0.75d + 0.50r ≤ 50
2. Ratio Constraint: d = 2r
We need to find the maximum possible integer value for d.
Step 3: Detailed Explanation:
We have two conditions:
0.75d + 0.50r ≤ 50
d = 2r
From the second equation, we can express r in terms of d: r = d/2.
Now, substitute this expression for r into the budget inequality:
0.75d + 0.50 (d/2) ≤ 50
Simplify the inequality:
0.75d + 0.25d ≤ 50
1.00d ≤ 50
d ≤ 50
This means the number of diet soda cans cannot exceed 50. The greatest possible number of cans of diet soda is 50.
Let's check if this solution is valid.
If d = 50, then r = d/2 = 50/2 = 25.
The total cost would be:
Cost = (0.75 × 50) + (0.50 × 25) = 37.50 + 12.50 = $50.00
This meets the budget exactly. Also, John buys both diet (50 cans) and regular (25 cans) sodas, satisfying all conditions.
Step 4: Final Answer:
The greatest number of cans of diet soda John can buy is 50.