There are now x cans stacked on a shelf

2018 Handbook

2018 Test B Math · Question 56 of 57

Also in 2026 Practice Test B Math, question 48; 2025 Practice Test B Math, question 56.

Question

There are now $x$ cans stacked on a shelf that holds 36 cans when full. If 4 of these cans were removed, the shelf would be half full. What is the value of $x$ ?

Show the answer

Correct answer: D

The shelf, when full, holds 36 cans. When the shelf is half full, it holds 18 cans.

x − 4 = 18

x = 22

Question analysis

A shelf holds 36 cans when full, and removing 4 of the cans now on it would leave it half full. Students find how many cans are on the shelf now. It tests writing and solving a one-step equation from words, and the risk is undoing the removal in the wrong direction.

NYS standards: 6.EE.7 one-step equations
Grade level: Grade 6 content, by NYS standard
Compared with grade 7: Below grade 7 Grade 6 students solve equations like x − p = q, and this one needs only that and halving 36. It is a quick, routine word problem, easier than the two-step equations typical of grade 7.
Estimated difficulty: Routine: one familiar step, little to misread.
Thinking it demands: Depth of knowledge 2, skills and concepts · mainly apply
What makes it hard: Turning words into math minor
A very close wrong answer minor
Solution: 2 steps
Reading level: Question and choices: about grade 3, 32 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

Short and routine, this question still punishes a quick reading. Students must keep three amounts apart, the 36-can capacity, the 18 cans on a half-full shelf and the x cans there now, and then undo the removal by adding. Writing x − 4 = 18 before computing prevents every wrong choice.

Turning words into math minor Understanding the question
The 36 cans are the shelf's capacity, not its contents, and "if 4 of these cans were removed" describes a hypothetical, so the equation is x − 4 = 18. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)
A very close wrong answer minor Choosing the answer
18 sits just below the right 22, and both are near half of 36; only remembering that x counts the cans before the 4 were removed separates them. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

Strategy: Work out how many cans make a half-full shelf, then undo the removal of the 4 cans; write it as an equation in x if that helps.

Traps

Stops one step early: A choice is a number from partway through the solution.
Follows a key word: Picks the operation from a single word rather than the situation: "removed" so subtract, "of" so multiply, "more" so add, when the situation needs something else.

The wrong answers, classified

A Follows a key word (Hegarty, Mayer & Monk, 1995)
Reveals a surface strategy Expected pull: moderate
Halves 36 to get 18 and then subtracts the 4 removed cans, following the word removed, instead of adding them back to undo it.
B Misreads the question (Hegarty, Mayer & Monk, 1995)
Reveals a slip Expected pull: weak
Reads the 36 cans as what is on the shelf now, removes 4 to get 32, and halves that to 16.
C Stops one step early (King et al., 2004; Gierl et al., 2017)
Reveals partial knowledge Expected pull: strong
Finds the 18 cans on a half-full shelf and stops, forgetting that x counts the cans before the 4 were removed.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Undoing subtraction: $x - a = b \text{ means } x = b + a$; x is the number you want. a is the amount taken away, and b is what is left. To undo taking away a, add a back. Adding the same number to both sides keeps the equation balanced.

Sort out what you know
The shelf holds 36 cans when it is full. That is its size, not how many cans are on it now. Right now it has $x$ cans. If 4 cans were taken off, the shelf would be half full. You need $x$ .
Find half of a full shelf
A full shelf holds 36 cans, so half full is half of that.
$36 \div 2 = 18$
A half-full shelf has 18 cans.
Write an equation
Taking 4 cans away from $x$ leaves 18 cans.
$x - 4 = 18$
Watch out: 18 is the number after the cans are taken off. Picking 18 gives choice C.
Add the 4 cans back
Undo the subtraction by adding 4 to both sides.
$x = 18 + 4 = 22$
Watch out: subtracting the 4 again gives $18 - 4 = 14$ , choice A. The cans were taken from $x$ , so $x$ is more than 18.
Give the answer
There are 22 cans on the shelf now, before any are taken off. So the answer is D, 22.

Check your answer

Start with 22 cans and take off 4. That leaves $22 - 4 = 18$ cans. Half of 36 is 18, so the shelf is half full. The answer works.

The original DOE question

Question 113 as printed on page 164 of the DOE handbook. — As printed in the DOE’s Specialized High Schools Student Handbook 2018–2019, page 164, question 113. Open page 164 of the handbook (PDF)