Simplify: $\frac{3x - y}{xy} - \frac{2x + 3y}{2xy} + \frac{1}{2}$

Step 1: Analyze the question We need to combine three fractions into one. The first step is to find a common denominator.

Step 2: Concept The denominators are xy, 2xy and 2. The LCM is 2xy, so we rewrite every fraction with denominator 2xy.

Step 3: Solution

• First fraction: (3x − y)/(xy) = 2(3x − y)/(2xy) = (6x − 2y)/(2xy)
• Second fraction: (2x + 3y)/(2xy), already over 2xy
• Third fraction: 1/2 = xy/(2xy)

Now combine them:

[(6x − 2y) − (2x + 3y) + xy] / (2xy)

Be careful with the minus sign in front of the second bracket:

= (6x − 2y − 2x − 3y + xy) / (2xy)

= (4x − 5y + xy) / (2xy)

Step 4: Answer

Step 5: Tip When subtracting a fraction whose numerator has more than one term, put that numerator in brackets first. Sign errors, such as forgetting to change +3y to −3y, are the most common way to lose marks on this type of question. Options A, B and C come from exactly those kinds of slips.