Also in 2025 Practice Test A Math, question 33.
Question
An unmarked straight stick will be laid end over end to measure a distance of exactly 72 feet. The same stick will be used in the same way to measure a distance of exactly 30 feet. What is the length of the longest possible stick that can be used for both measurements?
- A3 ft
- B4 ft
- C6 ft
- D8 ft
Show the answer
Correct answer: C
The length of the stick must be the greatest common factor of 72 and 30.
The factors of 30 are
1, 2, 3, 5, 6, 10, 15, and 30.
Of those, only 1, 2, 3, and 6 are also factors of 72. The greatest of these is 6.
Question analysis
This word problem asks for the longest stick that can be laid end over end to measure exactly 72 feet and exactly 30 feet. Students must recognize that the stick's length has to divide both distances evenly, so the task is finding a greatest common factor. The story wording hides the factor idea.
- NYS standards
- 6.NS.4 common factors and multiples
- Grade level
- Grade 6 content, by NYS standard
- Compared with grade 7
- Below grade 7 Finding a greatest common factor is a sixth-grade skill, and the arithmetic is light. The question is a little less routine than a bare factor problem only because students must see that the story calls for one.
- 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
- No obvious method moderate
- Hard words minor
- Solution
- 2 steps
- Reading level
- Question and choices: about grade 3, 54 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
Once the story is recognized as a greatest common factor problem, the arithmetic is light and the choices can simply be tested. The trap is checking only one of the two distances, because the longest choice works for 72 feet but not for 30. A student who lists the factors of 30 first, the smaller number, sees the answer quickly.
No obvious method moderate Working it out
Nothing in the stick story says "common factor"; students must see that laying the stick a whole number of times along 72 and 30 feet means it divides both. (Schoenfeld, 1985)
Hard words minor Understanding the question
The phrases "unmarked straight stick" and "laid end over end" are unusual wording for a math item; they describe whole-stick lengths, a reading hurdle rather than more math. (Perfetti, 2007; Freedle & Kostin, 1993)
Strategy: Translate 'measures both distances exactly' into 'divides both numbers evenly', list the factors of the smaller number, 30, and take the largest that also divides 72.
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
A Answers a different question (King et al., 2004; Hegarty, Mayer & Monk, 1995)
Finds a length that divides both 72 and 30, 3 feet, but not the longest one, missing that 6 also divides both.
B Misreads the question (Hegarty, Mayer & Monk, 1995)
Checks only that 4 divides 72, giving 18 sticks, and misses that 30 feet would take 7.5 sticks.
D Misreads the question (Hegarty, Mayer & Monk, 1995)
Takes the longest choice that divides 72 evenly, nine 8-foot lengths, without checking that 30 feet is not a whole number of 8-foot sticks.
Worked solution, step by step
Formulas used
- Number of sticks
- d is the distance to measure. s is the length of the stick. n is how many times you lay the stick down. For an exact measure, n must be a whole number. So s must divide d evenly.
Sort out the story
You lay a stick down again and again, end to end, like taking steps. The stick has no marks, so you can only count whole stick lengths.
It must fit exactly into 72 feet and exactly into 30 feet. You need the longest stick that does both.
Turn it into factors
Fitting exactly means no piece is left over. So the stick length must divide 72 evenly and divide 30 evenly.
A number that divides both is a common factor. You want the greatest common factor: the biggest number that divides both.
List the factors of 30
Start with the smaller number, 30, since its list is shorter. Its factors are 1, 2, 3, 5, 6, 10, 15 and 30.
Now test them on 72, starting from the biggest. 30, 15, 10 and 5 leave a remainder. But 6 works:
So 6 is the greatest common factor.
Check the other choices
Watch out: 8 feet fits 72 feet, 9 times. But 30 ÷ 8 = 3.75, not a whole number, so choice D fails. Choice B fails too, since 30 ÷ 4 = 7.5. A 3-foot stick works, but it is not the longest, so choice A is wrong.
So the answer is C, 6 ft.
Check your answer
Use the stick. A 6-foot stick goes down 12 times for 72 feet, since 12 × 6 = 72. It goes down 5 times for 30 feet, since 5 × 6 = 30. Both counts are whole numbers, so 6 feet works.
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