In a scale drawing of a triangular banner

2018 Handbook

2018 Test A Math · Question 26 of 57

Also in 2026 Practice Test A Math, question 23; 2025 Practice Test A Math, question 26.

Question

In a scale drawing of a triangular banner, one side measures 16 centimeters and the other two sides each measure 12 centimeters. On the actual banner, these two sides each measure 36 feet. What is the length of the remaining side of the actual banner?

A16 ft
B32 ft
C40 ft
D48 ft

Show the answer

Correct answer: D

Let $x$ be the remaining side of the actual banner.

Set up a proportion:

$\frac{x}{16}=\frac{36}{12}$

$12x=576$

$x=48$ ft

Question analysis

A scale drawing word problem about a triangular banner: the drawing's sides are 16 cm, 12 cm and 12 cm, and the two equal sides are 36 feet on the real banner. Students find the actual length of the remaining side. The key is using a scale factor, not adding the difference between lengths.

NYS standards: 7.G.1 scale drawings
7.RP.2b constant of proportionality
Grade level: Grade 7 content, by NYS standard
Compared with grade 7: At grade 7 Scale drawings are a named grade 7 topic, and this is a direct one-factor problem. It matches typical grade 7 work; the only pull away is reasoning by differences instead of by multiplication.
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 minor
Solution: 2 steps
Reading level: Question and choices: about grade 4, 46 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

A scale drawing multiplies every length by the same factor, here 3 feet per centimeter, and the question checks whether students use that factor or a difference. The 16 cm side is 4 cm longer than the others, and adding 4 to 36 feels natural but treats the scale as an addition. Students who find the factor from the matching sides first have one multiplication left.

Turning words into math minor Understanding the question
"These two sides each measure 36 feet" refers back to the 12 cm sides, so the scale pairs 12 cm with 36 ft, centimeters on the drawing against feet on the banner. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

Strategy: Find the scale from the matching sides, feet per centimeter, then multiply the 16 cm side by that same factor.

Traps

Misunderstands the concept: A wrong belief about a concept: that percents can be added, that area grows like length, that averages can be averaged, that a bigger denominator means a bigger fraction.
Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.

The wrong answers, classified

A Units or scale (King et al., 2004)
Reveals a surface strategy Expected pull: weak
Copies the 16 cm from the drawing as 16 feet, without applying the scale that turns each centimeter into 3 feet.
B Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)
Reveals a misconception Expected pull: weak
Reasons with differences instead of a scale factor and also subtracts, 36 − 4 = 32, though the 16 cm side is the longest.
C Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)
Reveals a misconception Expected pull: strong
Adds the 4 cm difference between 16 and 12 to the 36 feet, keeping a constant difference instead of a constant ratio.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Scale factor: $k = \frac{\text{actual length}}{\text{drawing length}}$; k is the scale factor: how many real units match 1 unit on the drawing. Divide a real length by its matching length on the drawing to find it. Every length on the drawing gets multiplied by the same k. That keeps the real shape the same as the drawing.

Sort out the question
A scale drawing is a smaller copy with the same shape. The drawing is a triangle with sides of 16 cm, 12 cm and 12 cm. On the real banner, the two 12 cm sides are 36 feet each. You need the real length of the 16 cm side.
Find the scale factor
Use the matching sides: 12 cm on the drawing is 36 feet on the banner.
$k = \frac{\text{actual length}}{\text{drawing length}}$
$k = \frac{36}{12} = 3$
So every 1 cm on the drawing stands for 3 feet on the banner.
Scale up the third side
Multiply the 16 cm side by the same scale factor.
$16 \times 3 = 48$
The third side is 48 feet. Watch out: a scale multiplies lengths. It does not add to them. 16 is 4 more than 12, but adding 4 to 36 gives 40, choice C. So the answer is D, 48 ft.

Check your answer

The 16 cm side is the longest side on the drawing. So it must be the longest on the banner too, longer than 36 feet. 48 fits. Also, $48 \div 16 = 3$ , the same scale as $36 \div 12 = 3$ .

The original DOE question

Question 83 as printed on page 86 of the DOE handbook. — As printed in the DOE’s Specialized High Schools Student Handbook 2018–2019, page 86, question 83. Open page 86 of the handbook (PDF)