Points X , Y , and Z are on a straight line, and Y is between X and Z . Length YZ=3/5XY , and length XY=20 centimeters. What is the length of XZ ?

Points X, Y, and Z are on a straight line

2018 Handbook

2018 Test B Math · Question 21 of 57

Also in 2026 Practice Test B Math, question 19; 2025 Practice Test B Math, question 21.

Question

Points $X$ , $Y$ , and $Z$ are on a straight line, and $Y$ is between $X$ and $Z$ . Length $\overline{YZ}=\frac{3}{5}\overline{XY}$ , and length $\overline{XY}=20$ centimeters. What is the length of $\overline{XZ}$ ?

A12 cm
B24 cm
C30 cm
D32 cm

Show the answer

Correct answer: D

Since $\overline{XY}=20$ cm, use that to find $\overline{YZ}$ :

$\overline{YZ}=\frac{3}{5}\overline{XY}=\frac{3}{5}(20)=12\text{ cm}$

$\overline{XZ}=\overline{XY}+\overline{YZ}=20+12=32\text{ cm}$

Question analysis

This segment question places points X, Y and Z on a line with Y between the others, gives XY as 20 centimeters, and says YZ is three fifths of XY. Students find the length of the whole segment XZ. Keeping track of which segment the question asks for matters more than the arithmetic.

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 Finding a fraction of a length and adding segments takes only two short steps with friendly numbers. Grade 7 rational-number problems are usually longer or involve negatives, so this sits below grade 7.
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 6, 33 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

There is no figure, so students who sketch the three points see at once that XZ is made of the two pieces XY and YZ. The arithmetic, three fifths of 20, is friendly. Most errors come from answering with the piece that was just calculated instead of the whole segment the question asks for.

Turning words into math minor Understanding the question
The condition “Y is between X and Z” has to become XZ = XY + YZ, and the overbar notation for segment lengths may be unfamiliar to some students. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

Strategy: Find three fifths of 20 to get YZ, then add the two parts, since Y lies between X and Z.

Traps

Stops one step early: A choice is a number from partway through the solution.

The wrong answers, classified

A Stops one step early (King et al., 2004; Gierl et al., 2017)
Reveals partial knowledge Expected pull: strong
Finds YZ, three fifths of 20, and gives it as the answer without adding XY to get the whole segment.
B An arithmetic slip (Junker & Sijtsma, 2001)
Reveals a slip Expected pull: weak
Takes one fifth of 20 but forgets to multiply by 3, so adds 4 cm instead of 12 cm to XY.
C Little reason to choose it (Haladyna & Downing, 1993; Rodriguez, 2005)
Reveals a surface strategy Expected pull: weak
No standard route gives 30; a student would have to take YZ as half of XY rather than three fifths, adding 10 cm.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Fraction of a number: $\frac{a}{b} \text{ of } n = n \div b \times a$; n is the starting amount. b is the bottom of the fraction: split n into b equal pieces. a is the top: take a of those pieces. Here n is 20, b is 5 and a is 3.
Adding segment lengths: $\overline{XZ} = \overline{XY} + \overline{YZ}$; X, Y and Z are points on one line, with Y between X and Z. XY and YZ are the lengths of the two short pieces. XZ is the whole length. The two pieces join end to end to make the whole.

Picture the line
Points X, Y and Z are on one straight line, with Y between X and Z. So the order is X, then Y, then Z.
A segment is the straight piece between two points. The bar over two letters means the length of that segment.
XY is 20 cm. You need XZ, the whole length.
Find the length of YZ
YZ is three fifths of XY. To find three fifths of 20, first find one fifth. Then take 3 of those.
$\frac{3}{5} \text{ of } 20 = 20 \div 5 \times 3 = 4 \times 3 = 12$
So YZ is 12 cm.
Watch out: 12 cm is only one piece of the line. Stopping here gives choice A.
Add the two pieces
Y is between X and Z, so the two pieces join to make the whole.
$\overline{XZ} = \overline{XY} + \overline{YZ} = 20 + 12 = 32$
XZ is 32 cm.
So the answer is D, 32 cm.

Check your answer

Check the fraction: $\frac{12}{20} = \frac{3}{5}$ , so YZ really is three fifths of XY. Also, XZ must be longer than XY, since it holds XY and more. 32 is longer than 20.

The original DOE question

Question 78 as printed on page 157 of the DOE handbook. — As printed in the DOE’s Specialized High Schools Student Handbook 2018–2019, page 157, question 78. Open page 157 of the handbook (PDF) The handbook letters this question’s choices E–H; here they are A–D (E = A, F = B, G = C, H = D).