On a certain day, orangeade was made by mixing a certain amount of orange juice with an equal amount of...
GMAT Word Problems : (WP) Questions
On a certain day, orangeade was made by mixing a certain amount of orange juice with an equal amount of water. On the next day, orangeade was made by mixing the same amount of orange juice with twice the amount of water. On both days, all the orangeade that was made was sold. If the revenue from selling the orangeade was the same for both days and if the orangeade was sold at $0.60 per glass on the first day, what was the price per glass on the second day?
- Translate the problem requirements: We need to understand that orangeade composition changes between days (Day 1: equal parts orange juice and water; Day 2: same orange juice but twice the water), yet total revenue stays the same. We're looking for the price per glass on Day 2.
- Set up the volume relationships: Use a concrete amount of orange juice to establish how many glasses are produced each day, recognizing that more water means more total volume and more glasses.
- Apply the equal revenue constraint: Since revenue = price × number of glasses, and revenue is the same both days, we can set up an equation relating the different prices and volumes.
- Solve for the unknown price: Use the revenue equation to find the price per glass on the second day.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what's happening on each day in plain English:
Day 1: We mix some amount of orange juice with an equal amount of water to make orangeade. If we use 1 cup of orange juice, we add 1 cup of water, giving us 2 cups total of orangeade.
Day 2: We use the same amount of orange juice (1 cup), but this time we add twice as much water as Day 1. So we add 2 cups of water, giving us 3 cups total of orangeade.
Key constraint: The total revenue (money earned) was the same both days, even though we made different amounts of orangeade.
What we know: Day 1 price was \(\$0.60\) per glass
What we need to find: Day 2 price per glass
Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships
2. Set up the volume relationships
Let's use a concrete example to make this easier to understand. Say we start with 1 unit of orange juice each day.
Day 1 composition:
• Orange juice: 1 unit
• Water: 1 unit (equal amount)
• Total orangeade: 2 units
Day 2 composition:
• Orange juice: 1 unit (same amount)
• Water: 2 units (twice the amount from Day 1)
• Total orangeade: 3 units
This means Day 2 produces 50% more orangeade than Day 1 (3 units vs 2 units).
If each "unit" represents the same volume per glass, then:
• Day 1: 2 glasses produced
• Day 2: 3 glasses produced
3. Apply the equal revenue constraint
Now here's the key insight: even though we made more glasses on Day 2, we earned the same total revenue both days.
Revenue formula: \(\mathrm{Revenue} = \mathrm{Price \, per \, glass} \times \mathrm{Number \, of \, glasses}\)
Since revenues are equal:
\(\mathrm{Day \, 1 \, Revenue} = \mathrm{Day \, 2 \, Revenue}\)
In plain English: The money from selling 2 glasses at \(\$0.60\) each equals the money from selling 3 glasses at the unknown price.
Setting up the equation:
Day 1: \(\$0.60 \times 2 \, \mathrm{glasses} = \$1.20\)
Day 2: \(\mathrm{Unknown \, price} \times 3 \, \mathrm{glasses} = \$1.20\)
Process Skill: APPLY CONSTRAINTS - Using the equal revenue condition to create our solving equation
4. Solve for the unknown price
Now we can find the Day 2 price:
\(\mathrm{Unknown \, price} \times 3 \, \mathrm{glasses} = \$1.20\)
\(\mathrm{Unknown \, price} = \$1.20 \div 3 \, \mathrm{glasses}\)
\(\mathrm{Unknown \, price} = \$0.40 \, \mathrm{per \, glass}\)
Logic check: This makes sense! We made more orangeade on Day 2 (it was more diluted), so we had to sell it at a lower price per glass to earn the same total revenue.
4. Final Answer
The price per glass on the second day was \(\$0.40\).
Verifying against the answer choices: This matches choice (D) \(\$0.40\).
Final verification:
• Day 1: \(2 \, \mathrm{glasses} \times \$0.60 = \$1.20 \, \mathrm{revenue}\)
• Day 2: \(3 \, \mathrm{glasses} \times \$0.40 = \$1.20 \, \mathrm{revenue}\) ✓
The answer is (D) \(\$0.40\).
Common Faltering Points
Errors while devising the approach
1. Misinterpreting "twice the amount of water"
Students often get confused about what "twice the amount" refers to. The question states that on Day 2, orangeade was made by mixing the same amount of orange juice with "twice the amount of water." Students may incorrectly think this means twice the amount of water compared to Day 2's orange juice (which would be 2 units of water for 1 unit of orange juice), when it actually means twice the amount of water compared to Day 1's water amount.
2. Missing the equal revenue constraint
Students may focus on calculating volumes and ratios but overlook the crucial constraint that "the revenue from selling the orangeade was the same for both days." Without recognizing this as the key relationship that allows us to solve for the unknown price, students cannot set up the correct equation.
3. Assuming equal pricing instead of using the constraint
Some students might assume that since the orange juice concentration is different, they should calculate price based on concentration ratios rather than using the equal revenue constraint. This leads them away from the correct approach of setting up Revenue₁ = Revenue₂.
Errors while executing the approach
1. Incorrect volume calculations
Even with the right approach, students may make errors in calculating the total volumes. For example, miscalculating Day 2's total as 2 units (1 orange + 1 water) instead of 3 units (1 orange + 2 water), or getting the ratio between Day 1 and Day 2 volumes wrong.
2. Setting up the revenue equation incorrectly
Students may set up the equation backwards or with wrong values, such as writing "\(\$0.60 \times 3 = \mathrm{Unknown \, price} \times 2\)" instead of "\(\$0.60 \times 2 = \mathrm{Unknown \, price} \times 3\)," leading to an incorrect final answer.
Errors while selecting the answer
No likely faltering points - the calculation directly yields \(\$0.40\), which clearly matches answer choice (D).
Alternate Solutions
Smart Numbers Approach
We can solve this problem by choosing a convenient amount of orange juice and working with concrete numbers throughout.
Step 1: Choose a smart number for orange juice
Let's say we start with 10 units of orange juice each day. This is a convenient number that will make our calculations clean.
Step 2: Calculate the composition and volume for each day
Day 1: Equal amounts of orange juice and water
• Orange juice: 10 units
• Water: 10 units (equal to orange juice)
• Total orangeade: 20 units
Day 2: Same orange juice, twice the amount of water
• Orange juice: 10 units
• Water: 20 units (twice the amount from Day 1)
• Total orangeade: 30 units
Step 3: Calculate revenue for Day 1
Day 1 revenue = \(20 \, \mathrm{units} \times \$0.60 \, \mathrm{per \, unit} = \$12.00\)
Step 4: Use the equal revenue constraint
Since revenue is the same both days:
Day 2 revenue = Day 1 revenue = \(\$12.00\)
Step 5: Find the price per glass on Day 2
Day 2: \(\$12.00 \div 30 \, \mathrm{units} = \$0.40 \, \mathrm{per \, unit}\)
Answer: (D) \(\$0.40\)
Note: The beauty of this approach is that any convenient number we choose for orange juice will yield the same final answer, demonstrating the robustness of the smart numbers method for this type of ratio problem.