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Lena is building a fence

2024 Handbook

2024 Test B Math · Question 12 of 57

Question

Lena is building a fence. She will need to dig holes to help support the posts that hold up the fence. The holes need to have a depth of 3133\frac{1}{3} feet below the ground. Each post is 10 feet long. What is the height of the part of the post that is above the ground?

  • A6236\frac{2}{3} feet
  • B7137\frac{1}{3} feet
  • C10 feet
  • D131313\frac{1}{3} feet
Show the answer

Correct answer: A

The posts are each 10 feet long, and 3133\frac{1}{3} feet of each post is below the ground, so the height of the part of the posts that is above the ground is the difference between the two amounts:

10−313=10-3\frac{1}{3}=

101−103=\frac{10}{1}-\frac{10}{3}=

303−103=203=623 feet\frac{30}{3}-\frac{10}{3}=\frac{20}{3}=6\frac{2}{3}\text{ feet}

Question analysis

A fence-building word problem: each post is 10 feet long, and 3 1/3 feet of it goes into a hole in the ground. Students find how much of the post stands above ground by subtracting a mixed number from a whole number. Regrouping the whole number into thirds is the step that trips students up.

NYS standards
  • 7.NS.3 four operations with rational numbers
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
Below grade 7 Subtracting a mixed number from a whole number is fraction work from before grade 7, set here in a one-step story. It asks much less than grade 7 problems, which add negative numbers or several steps.
Estimated difficulty
Routine: one familiar step, little to misread.
What makes it hard
Solution
1 step
Reading level
Question and choices: about grade 2, 61 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 situation is plain, a post partly buried in a hole, and the subtraction 10 − 3 1/3 is the whole task. The only real trap is the regrouping, since there are no thirds to take the 1/3 from until one foot is split into three. Checking that the two parts of the post add back to 10 feet catches the slip.

  1. Error-prone arithmetic minor Working it out

    Taking 3 1/3 from the whole number 10 requires regrouping 10 as 9 3/3, the step where students subtract the whole numbers and simply carry the 1/3 along. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

Strategy: Subtract the buried part from the whole post, rewriting 10 as 9 and 3/3, or as 30/3, so that the thirds can be taken away.

Traps

  • A faulty procedure: A systematic but wrong procedure: adding fractions straight across, applying an exponent to each term of a sum, dropping a sign when moving a term.
  • Wrong operation: Adds instead of multiplying, divides the wrong way round, turns a ratio or rate upside down, or undoes a step in the wrong direction (subtracts what should be added back).
  • 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

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Parts of a whole
above=whole−below\text{above} = \text{whole} - \text{below}
The whole is the full length of the post. Below is the part in the hole, and above is the part you can see. The two parts add up to the whole, so subtract to find one part.
One whole in thirds
1=331 = \frac{3}{3}
One whole can be cut into 3 equal thirds. So 1 is the same as 3 over 3. Use this when you need to take thirds away from a whole number.

  1. Sort out the question

    Each post is 10 feet long. Part of it goes into a hole 3133\frac{1}{3} feet deep. The rest sticks up above the ground. You need the height of the part above the ground.

  2. Decide to subtract

    The part below and the part above make up the whole post. So take the part below away from the whole.

    above=whole−below\text{above} = \text{whole} - \text{below}

    above=10−313\text{above} = 10 - 3\frac{1}{3}

    Watch out: do not add. 10+313=131310 + 3\frac{1}{3} = 13\frac{1}{3}, choice D, makes the post longer than it is.

  3. Turn one whole into thirds

    The 10 has no thirds to take 1/3 from. So take 1 whole from the 10 and cut it into 3 thirds.

    1=331 = \frac{3}{3}

    10=9+33=93310 = 9 + \frac{3}{3} = 9\frac{3}{3}

    Now there are thirds to subtract from.

  4. Subtract wholes and thirds

    Subtract the whole numbers, then the thirds.

    933−313=6239\frac{3}{3} - 3\frac{1}{3} = 6\frac{2}{3}

    That is 9−3=69 - 3 = 6 and 33−13=23\frac{3}{3} - \frac{1}{3} = \frac{2}{3}. Watch out: the one third must be taken away too. Doing 10 − 3 = 7 and then keeping the one third gives 7137\frac{1}{3}, choice B.

    So the answer is A, 6236\frac{2}{3} feet.

Check your answer

Add the two parts back together: 623+313=933=106\frac{2}{3} + 3\frac{1}{3} = 9\frac{3}{3} = 10. That is the whole post. Also, the answer must be less than 10 feet, since part of the post is under the ground.

More questions like this

The original DOE question

Question 69 as printed on page 202 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2025 Admissions, page 202, question 69. Open page 202 of the handbook (PDF)