How is 1/x-1/2x written as a single fraction? (Assume x≠ 0 .)

How is 1/x − 1/(2x) written

2023 Handbook

2023 Test B Math · Question 38 of 57

Question

How is $\frac{1}{x}-\frac{1}{2x}$ written as a single fraction? (Assume $x\neq 0$ .)

A $\frac{1}{2x}$
B $\frac{1}{3x}$
C $-\frac{1}{x}$
D $-\frac{1}{2x}$

Show the answer

Correct answer: A

$\frac{1}{x}-\frac{1}{2x}=\frac{2}{2x}-\frac{1}{2x}=\frac{1}{2x}$

Question analysis

This algebra question asks students to write 1/x minus 1/(2x) as a single fraction, assuming x is not zero. It tests finding a common denominator when the denominators contain a variable. Getting both fractions over the same denominator before subtracting, and keeping track of the sign, are what matter.

NYS standards: 7.EE.1 linear expressions with rational coefficients
7.EE.2 rewriting expressions to show relationships
Grade level: Grade 7 content, by NYS standard
Compared with grade 7: At grade 7 Grade 7 students subtract fractions with unlike denominators and work with linear expressions, but seldom see a variable in a denominator. The steps match those for numerical fractions, so it is a short task once the common denominator is found.
Estimated difficulty: Straightforward: two steps or one careful reading.
Thinking it demands: Depth of knowledge 1, recall and reproduction · mainly apply
What makes it hard: Variables instead of numbers moderate
Error-prone arithmetic minor
Solution: 2 steps
Reading level: Question and choices: about grade 11, 14 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

This is subtraction of unlike fractions with a letter in the bottom, and every wrong choice is a familiar fraction error: combining the denominators straight across by adding or subtracting, or subtracting the numerators the wrong way round. Substituting x = 1 makes it 1 − 1/2 = 1/2, which rules out the negative choices and 1/3 at once.

Variables instead of numbers moderate Understanding the question
The denominators are x and 2x, so the common denominator is built with a letter, and no number checks the result unless students substitute a value such as x = 1. (Koedinger & Nathan, 2004; Enright, Morley & Sheehan, 2002)
Error-prone arithmetic minor Working it out
Only the first fraction changes: 1/x becomes 2/2x, and then the subtraction 2 − 1 must be done in the order written, first numerator minus second. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

Strategy: Rewrite 1/x with the denominator 2x by multiplying its top and bottom by 2, then subtract the numerators and keep the common denominator.

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.
Negatives and direction: Arithmetic with negative numbers, the order of a subtraction, or direction on a number line. (Moving a term across the equals sign without changing its sign is a faulty procedure.)

The wrong answers, classified

B A faulty procedure (Brown & Burton, 1978; Tatsuoka, 1983)
Reveals a misconception Expected pull: weak
Adds the denominators, x + 2x = 3x, and keeps a numerator of 1, as if two unit fractions combined by adding their bottoms.
C A faulty procedure (Brown & Burton, 1978; Tatsuoka, 1983)
Reveals a misconception Expected pull: moderate
Subtracts the denominators straight across, x − 2x = −x, and keeps 1 on top, so the result 1/(−x) is written −1/x.
D Negatives and direction (Brown & Burton, 1978)
Reveals a slip Expected pull: moderate
Rewrites 1/x as 2/2x correctly but subtracts the numerators in reverse order, 1 − 2 = −1, giving a negative result.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Equivalent fractions: $\frac{a}{b} = \frac{a \times k}{b \times k}$; a is the top, or numerator. b is the bottom, or denominator. k is any number except 0. Multiplying the top and bottom by the same number gives an equal fraction, because k over k is 1.
Subtracting fractions with the same bottom: $\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}$; a and b are the tops. c is the shared bottom. When the bottoms match, the pieces are the same size. So subtract the tops and keep the bottom.

Sort out the question
You need to write $\frac{1}{x} - \frac{1}{2x}$ as one fraction. The problem says x is not 0. That matters because you can’t divide by 0.
Find a common bottom
To subtract fractions, their bottoms must match. 2x is 2 times x. So multiply the top and bottom of $\frac{1}{x}$ by 2.
$\frac{1}{x} = \frac{1 \times 2}{x \times 2} = \frac{2}{2x}$
Now both fractions have the bottom 2x.
Subtract the tops
Subtract the tops in order. Keep the bottom.
$\frac{2}{2x} - \frac{1}{2x} = \frac{2 - 1}{2x} = \frac{1}{2x}$
Watch out: Never add or subtract the bottoms. Adding them gives $\frac{1}{3x}$ , choice B. Subtracting them gives $-\frac{1}{x}$ , choice C. And keep the order: 2 − 1, not 1 − 2. The wrong order gives choice D.
So the answer is A, $\frac{1}{2x}$ .

Check your answer

Try x = 1. The problem becomes $1 - \frac{1}{2} = \frac{1}{2}$ . Choice A gives $\frac{1}{2 \times 1} = \frac{1}{2}$ . It matches. Choices B, C and D give $\frac{1}{3}$ , −1 and $-\frac{1}{2}$ .

The original DOE question

Question 95 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 2024 Admissions, page 202, question 95. Open page 202 of the handbook (PDF)