Question
Orange juice costs $2 per gallon, and grapefruit juice costs $3 per gallon. A recent shipment of equal amounts of the two types of juice costs a total of $250. How many gallons of orange juice are in the shipment?
- A50
- B62.5
- C100
- D125
Show the answer
Correct answer: A
Let x represent the number of gallons of orange juice. Since the shipment contains equal amounts of the two types of juice, x also represents the number of gallons of grapefruit juice. Write an equation to solve for x:
Question analysis
Orange juice costs $2 a gallon and grapefruit juice costs $3 a gallon, and a shipment with equal amounts of each costs $250. Students find how many gallons of orange juice are in the shipment. The word problem leads to a simple equation; the key phrase is equal amounts of juice, not equal amounts of money.
- Grade level
- Grade 7 content, by NYS standard
- Compared with grade 7
- At grade 7 Writing and solving an equation from a short word problem is standard grade 7 work. This one has a single unknown, since both juices come in the same number of gallons, and only two short steps.
- Estimated difficulty
- Straightforward: two steps or one careful reading.
- Thinking it demands
- Depth of knowledge 2, skills and concepts · mainly apply
- What makes it hard
- Turning words into math moderate
- Solution
- 3 steps
- Reading level
- Question and choices: about grade 6, 37 words.
- Students, last 30 days
- Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)
Why it is hard
The equation is short once the key phrase is read as equal gallons, since one gallon of each juice costs $5 together and $250 buys 50 such pairs. Read as equal spending, the same phrase leads to a listed answer that is not a whole number of gallons. The other trap is giving the gallons of both juices together when only orange juice is asked for.
Turning words into math moderate Understanding the question
“Equal amounts of the two types of juice” means the same number of gallons of each, x, so the total cost is 2x + 3x, not two equal halves of $250. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)
Strategy: Price one gallon of each juice together, then see how many such pairs $250 buys; that is the number of gallons of each kind.
Traps
- Misreads the question: Misses or misreads a condition or a word in the question: "at least", "each", "remaining", "not".
- Answers a different question: A choice is right for another quantity: the other unknown, the rest instead of the part, the part instead of the whole.
The wrong answers, classified
B Misreads the question (Hegarty, Mayer & Monk, 1995)
Reads “equal amounts” as equal spending, $125 on each juice, and divides $125 by $2 a gallon to get 62.5.
C Answers a different question (King et al., 2004; Hegarty, Mayer & Monk, 1995)
Finds the right count for each juice but gives the whole shipment, 50 + 50 = 100 gallons, instead of the orange juice alone.
D Numbers taken from the question (Hegarty, Mayer & Monk, 1995)
Divides $250 by 2, the orange juice price or the number of juices, combining numbers from the question without equal gallons of each.
Worked solution, step by step
Formulas used
- Total cost
- The price per gallon is what one gallon costs. Multiply it by the number of gallons to get what you pay in all. For example, 3 gallons at $2 a gallon cost $6.
Sort out the question
Orange juice costs $2 a gallon. Grapefruit juice costs $3 a gallon. The shipment has equal amounts of each, so it has the same number of gallons of each juice. It costs $250 in all. You need the number of gallons of orange juice.
Write the cost of each juice
Let x be the number of gallons of each juice. Use the cost formula for each one.
Add the costs and solve
The two costs add up to $250. Two x’s plus three x’s make five x’s.
There are 50 gallons of each juice.
Give the answer
Watch out: equal amounts means equal gallons, not equal dollars. Spending $125 on each juice gives 62.5 gallons, choice B. Also, 100 gallons is both juices together, choice C. So the answer is A, 50.
Check your answer
Find each cost. 50 gallons of orange juice cost dollars. 50 gallons of grapefruit juice cost dollars. Together, dollars, the total in the question.
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