On the number line above, JK=3 1/2 , JM=9 3/4 , and LM=1 1/8 . What is the position of point L ?

On the number line above

2018 Handbook

2018 Test A Math · Question 54 of 57

Also in 2026 Practice Test A Math, question 47; 2025 Practice Test A Math, question 54.

Question

$Number line with points J, K, L and M marked from left to right. Only K has a value: 3/8.$

On the number line above, $JK=3\frac{1}{2}$ , $JM=9\frac{3}{4}$ , and $LM=1\frac{1}{8}$ . What is the position of point $L$ ?

A $5\frac{1}{8}$
B $5\frac{1}{4}$
C $5\frac{1}{2}$
D $6\frac{1}{4}$

Show the answer

Correct answer: C

First, use $JK=3\frac{1}{2}$ to find the location of $J$ :

$\frac{3}{8}-J=3\frac{1}{2}$

$J=\frac{3}{8}-3\frac{1}{2}=-3\frac{1}{8}$

Now, use $JM=9\frac{3}{4}$ to find the location of $M$ :

$M-\left(-3\frac{1}{8}\right)=9\frac{3}{4}$

$M+3\frac{1}{8}=9\frac{3}{4}$

$M=9\frac{3}{4}-3\frac{1}{8}=6\frac{5}{8}$

Finally, use $LM=1\frac{1}{8}$ to find the location of $L$ :

$6\frac{5}{8}-L=1\frac{1}{8}$

$L=6\frac{5}{8}-1\frac{1}{8}=5\frac{4}{8}=5\frac{1}{2}$

Question analysis

This number line question gives the lengths JK, JM and LM as mixed numbers and asks for the position of point L. Students must chain three distance steps, moving left from K to J, right to M and back to L, with fractions in eighths and a negative position. Losing track of where K sits is the main danger.

NYS standards: 7.NS.1c rational number subtraction
7.NS.1b rational number addition
Grade level: Grade 7 content, by NYS standard
Compared with grade 7: Above grade 7 Grade 7 students add and subtract rational numbers and find distances on a number line one at a time. This item chains three such steps with mixed numbers and a negative position, more than typical grade 7 work demands.
Estimated difficulty: Among the hardest on the SHSAT: a non-obvious approach, heavy reading, or a well-hidden trap.
Thinking it demands: Depth of knowledge 2, skills and concepts · mainly apply
What makes it hard: A very close wrong answer major
Error-prone arithmetic moderate
Reading a figure, table or graph moderate
Many steps minor
Solution: 4 steps
Reading level: Question and choices: about grade 9, 20 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

Every number is a mixed fraction, one position is negative, and the figure's even spacing hides how long KL really is. The distance route, KL = 9 3/4 − 3 1/2 − 1 1/8, avoids the negative entirely but ends one step short of a position unless K's 3/8 is added back. That last step is the one most students skip, which is why the question sits at the top of the scale.

A very close wrong answer major Choosing the answer
The choices 5 1/8, 5 1/4 and 5 1/2 lie within 3/8 of one another, so no estimate separates them; only exact eighths and adding K's 3/8 pick out the right one. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)
Error-prone arithmetic moderate Working it out
Every step mixes halves, quarters and eighths, and placing J means 3/8 − 3 1/2 = −3 1/8, a subtraction that crosses zero. (Brown & Burton, 1978; Junker & Sijtsma, 2001)
Reading a figure, table or graph moderate Understanding the question
Only K is labeled, and the points are drawn evenly spaced although KL is over four times LM, so the picture gives no help with sizes. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)
Many steps minor Working it out
Three linked moves, K to J, J to M and M back to L, each depend on the last, so an early slip carries through. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

Strategy: Start from K's labeled position, subtract JK to place J, add JM to place M, then subtract LM, keeping every fraction in eighths.

Traps

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.
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 KL = 9 3/4 − 3 1/2 − 1 1/8 = 5 1/8 and stops one step short, never adding K's position of 3/8.
B An arithmetic slip (Junker & Sijtsma, 2001)
Reveals a slip Expected pull: weak
Rewrites 9 3/4 as 9 4/8 instead of 9 6/8, so M lands at 6 3/8 and L at 6 3/8 − 1 1/8 = 5 1/4.
D Answers a different question (King et al., 2004; Hegarty, Mayer & Monk, 1995)
Reveals partial knowledge Expected pull: moderate
Finds KM = 9 3/4 − 3 1/2 = 6 1/4, the distance from K to M, and gives it without moving back 1 1/8 to L.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Distance on a number line: $d = b - a$; d is the distance between two points. b is the point on the right and a is the point on the left. Subtracting left from right always gives a positive distance. It works even when a is negative.
Equal fractions: $\frac{a}{b} = \frac{a \times k}{b \times k}$; a is the top number (numerator) and b is the bottom number (denominator). Multiplying both by the same number k gives an equal fraction. It is like cutting each piece into k smaller pieces. Use it to give fractions the same bottom before you add or subtract.

Sort out what you know
Only K has a value: $\frac{3}{8}$ . JK means the distance from J to K. You know JK = $3\frac{1}{2}$ , JM = $9\frac{3}{4}$ and LM = $1\frac{1}{8}$ .
You need the position of L, the number where L sits. The points are drawn evenly spaced, but the distances are not equal. So use the numbers, not the picture.
Write the fractions in eighths
The fractions use halves, fourths and eighths. Give them all a bottom of 8:
$3\frac{1}{2} = 3\frac{4}{8} \qquad 9\frac{3}{4} = 9\frac{6}{8}$
Watch out: $\frac{3}{4}$ is $\frac{6}{8}$ , not $\frac{4}{8}$ . That slip leads to choice B.
Find the distance from K to M
JM is JK and KM put together. So take JK away from JM:
$9\frac{6}{8} - 3\frac{4}{8} = 6\frac{2}{8}$
KM is $6\frac{2}{8}$ , or $6\frac{1}{4}$ . Watch out: this is a distance, not where L sits. Choice D stops here.
Find the distance from K to L
KM is KL and LM put together. So take LM away:
$6\frac{2}{8} - 1\frac{1}{8} = 5\frac{1}{8}$
KL is $5\frac{1}{8}$ . Watch out: this is still a distance. Choice A stops here.
Find the position of L
L is $5\frac{1}{8}$ to the right of K, and K is at $\frac{3}{8}$ . Add:
$\frac{3}{8} + 5\frac{1}{8} = 5\frac{4}{8} = 5\frac{1}{2}$
So the answer is C, $5\frac{1}{2}$ .

Check your answer

Place every point. J is $3\frac{4}{8}$ left of K, at $-3\frac{1}{8}$ . M is $9\frac{6}{8}$ right of J, at $6\frac{5}{8}$ . L is $1\frac{1}{8}$ left of M, at $5\frac{4}{8}$ , which is $5\frac{1}{2}$ . Same answer.

The original DOE question

Question 111 as printed on page 91 of the DOE handbook. — As printed in the DOE’s Specialized High Schools Student Handbook 2018–2019, page 91, question 111. Open page 91 of the handbook (PDF)