Ba­sic Op­er­a­tions of Al­ge­braic Frac­tions

Why A is correct:
Factor the numerator: 6x² - 12x = 6x(x - 2). This matches the factor (x - 2) in the denominator, so they cancel out. You're left with 6x/(3x), which simplifies to 2.

Why the others are wrong:
- B: Incorrectly changes 6x² to 8x—there's no mathematical justification for this change.
- C: Wrong factorization of the numerator and denominator; 6x² doesn't become 4x(x - 2).
- D: This forgets to cancel the common factor 6x in the numerator and denominator; it only partially simplifies the problem.

Why A is correct:
To add fractions, you need a common denominator. The LCD of 3x and 6x is 6x. Convert 7/(3x) to 14/(6x), then add: 14/(6x) + 5/(6x) = 19/(6x).

Why the others are wrong:
- B (12/9x): Wrong denominator (should be 6x, not 9x) and wrong numerator (should be 19, not 12).
- C (12/6x): Has the right denominator but wrong numerator—you'd get this if you added 7 + 5 without converting the first fraction.
- D (35 + 18)/(18x): This incorrectly multiplies numerators and denominators together instead of finding a common denominator and adding properly.

• Why A is correct: When dividing fractions, flip the second fraction and multiply: (4a²/3b) × (9b²/2a). Multiply across: (4a² × 9b²)/(3b × 2a) = 36a²b²/6ab. Simplify by canceling: 36a²b²/6ab = 6ab.

• Why B is wrong: This results from incorrectly simplifying or not completing the cancellation process.

• Why C is wrong: This is the unsimplified numerator and denominator before canceling common factors—not a final answer.

• Why D is wrong: This forgets to include the variable b in the final answer; you'd get this if you incorrectly canceled the b terms.

Correct Answer: A

Factor both the numerator and denominator:
- Numerator: x² - 9 = (x + 3)(x - 3) — difference of squares
- Denominator: x² - 6x + 9 = (x - 3)² — perfect square trinomial

Cancel the common factor (x - 3): [(x + 3)(x - 3)] / [(x - 3)²] = (x + 3)/(x - 3)

Why the others are wrong:
- B: Just rewrites the original expression without simplifying
- C: Incorrectly factors the denominator as (x² - 3) instead of (x - 3)²
- D: Has the factors reversed; this would only be correct if you divided incorrectly

• A is correct: Find a common denominator (12x), then: (5)/(4x) = (15)/(12x) and (3)/(6x) = (6)/(12x). Subtracting gives (15−6)/(12x) = (9)/(12x), which simplifies to (3)/(4x).

• B is wrong: (2)/(2x) = (1)/(x), which equals (4)/(4x)—too large. This comes from incorrectly simplifying the fractions before subtracting.

• C is wrong: This doesn't result from any correct method; it appears to confuse adding the numerators and denominators incorrectly.

• D is wrong: (1)/(2x) = (2)/(4x)—close but one unit too small. This likely comes from arithmetic errors when combining fractions.

Why A is correct:
Factor the numerator and denominator:
- x² - 4 = (x - 2)(x + 2)
- x² + x - 2 = (x + 2)(x - 1)

So the fraction becomes: ((x - 2)(x + 2)(x + 1))/((x - 2)(x + 2)(x - 1))

Cancel the common factors (x - 2) and (x + 2), leaving (x + 1)/(x - 1).

Why the others are wrong:
- B: Cancels only one factor correctly but misses the (x + 2) cancellation in the denominator
- C: Doesn't match any valid factorization of the original expression
- D: This is just part of the unsimplified expression; it hasn't been reduced

# Explanation

Why A is correct:
Since both fractions have the same denominator (x² - 9), you subtract the numerators: 4 - 2 = 2, giving you 2/(x² - 9). Then factor the denominator: x² - 9 = (x - 3)(x + 3), so the answer is 2/((x - 3)(x + 3)).

Why the others are wrong:
- B: This is the unfinished answer—you correctly got 2/(x² - 9) but forgot to factor the denominator.
- C: The denominator changed to x² + 9, which is incorrect. You don't add when subtracting fractions; you keep the same denominator.
- D: This has two errors: the denominator should be x² - 9, not x² + 3, and you can't just write "4 - 2" in the numerator without simplifying it to 2.

Why A is correct:
Multiply the fractions: (12a³b²)/(6ab) · (2b²)/(3a) = (12a³b² · 2b²)/(6ab · 3a) = (24a³b⁴)/(18a²)

Simplify by canceling: 24/18 = 4/3, a³/a² = a, and b⁴/b⁰ = b⁴. This gives (4ab⁴)/3.

Wait—let me recalculate: (12a³b²)/(6ab) · (2b²)/(3a) first simplifies each fraction separately: (12a³b²)/(6ab) = 2a²b, then multiply by (2b²)/(3a) = (2a²b · 2b²)/(3a) = (4a²b³)/(3a) = (4ab³)/3 ✓

Why others are wrong:
- B: Forgot to simplify 24/18 to 4/3
- C: Didn't multiply the fractions together; left them partially simplified
- D: Made errors in tracking exponents during cancellation

Why A is correct:
When dividing fractions, flip the second fraction and multiply: ((x - 2)/(x + 3)) × ((x - 2)/(x² - 9)). Factor x² - 9 as (x + 3)(x - 3), giving you ((x - 2)/(x + 3)) × ((x - 2)/((x + 3)(x - 3))). Multiply across to get ((x - 2)²)/((x + 3)²(x - 3)).

Why the others are wrong:
- B: This cancels too much—it ignores the (x - 3) factor in the denominator completely.
- C: This is just the expanded form of A's numerator and denominator squared, but it masks the actual simplified form.
- D: This would only be true if the (x + 3)² and (x - 3) factors somehow disappeared, which they don't.

Why A is correct:
- Factor each denominator: x² - 2x = x(x - 2) and x² - 4x = x(x - 4)
- The LCD is x(x - 2)(x - 4)
- Rewrite: 5/[x(x - 2)] + 3/[x(x - 4)] = 5(x - 4)/[x(x - 2)(x - 4)] + 3(x - 2)/[x(x - 2)(x - 4)]
- Numerator: 5(x - 4) + 3(x - 2) = 5x - 20 + 3x - 6 = 8x - 26 ✓

Why the others are wrong:
- B: Wrong denominator—doesn't use the common denominator needed to add these fractions
- C: This would be the answer only if the numerators simplified to 8, but they don't (you get 8x - 26)
- D: Wrong numerator; appears to be a calculation error when finding the combined numerator

Why A is correct:
Factor the numerator: 2x² - 8 = 2(x² - 4)
Factor the denominator: 4x² - 16 = 4(x² - 4)
Cancel the common factor (x² - 4): [2(x² - 4)] / [4(x² - 4)] = 2/4 = 1/2

Why the others are wrong:
- B: This isn't even a simplified form—it just rewrites the fraction differently without simplifying.
- C: Incorrectly simplifies 2/4 to 1/4 instead of 1/2.
- D: This would result if you forgot to simplify the remaining coefficients (2/4) after canceling (x² - 4).

Correct Answer: (x - 4)/(x - 2)

• Factor the numerator: x² - 16 = (x - 4)(x + 4) [difference of squares]
• Factor the denominator: x² + 2x - 8 = (x + 4)(x - 2) [find two numbers that multiply to -8 and add to 2]
• Cancel the common factor (x + 4) from top and bottom
• You're left with (x - 4)/(x - 2)

Why the others are wrong:

• B: Swaps x - 4 and x + 4; comes from incorrectly factoring or canceling
• C: Just rewrites the problem without actually simplifying it
• D: Incorrectly factors the numerator as (x² + 4) instead of (x - 4)(x + 4)

Why A is correct:
Rewrite the second fraction by factoring out –1 from the denominator: (2)/(5 - x) = (2)/[–(x - 5)] = –(2)/(x - 5). Now both fractions have the same denominator: (3)/(x - 5) – (2)/(x - 5) = (1)/(x - 5).

Why the others are wrong:
- B: This assumes you add 3 + 2 = 5 without recognizing the denominators are opposites; you'd only get this if both fractions had the same sign.
- C: Same error as B, but leaves the denominator as (5 - x) instead of simplifying to standard form.
- D: This forgets to convert both fractions to a common denominator before combining them.

Correct Answer: (x + 3)/(x - 3)

• Factor the polynomials: (x² - x - 12) = (x - 4)(x + 3), and (x² - 16) = (x - 4)(x + 4)
• Rewrite as: [(x - 4)(x + 3)]/[(x - 4)(x + 4)] · (x + 4)/(x - 3)
• Cancel (x - 4) from numerator and denominator, and (x + 4) from numerator and denominator
• You're left with (x + 3)/(x - 3)

Why other options are wrong:

• B incorrectly cancels the (x + 4) terms but doesn't complete the simplification properly
• C results from factoring errors (using x - 4 in the wrong place)
• D is the unsimplified or partially worked version; you must cancel common factors

# Explanation

Why A is correct:
- Factor x² - 4 = (x + 2)(x - 4)
- Rewrite the first fraction with this common denominator: 1/(x + 2) = (x - 4)/(x² - 4)
- Subtract: (x - 4)/(x² - 4) - 2/(x² - 4) = (x - 4 - 2)/(x² - 4) = (x - 6)/(x² - 4)

Wait—let me recalculate: (x - 4 - 2)/(x² - 4) = (x - 6)/(x² - 4), not (x - 4)/(x² - 4).

Actually, the answer listed appears to have an error. The correct answer should be (x - 6)/(x² - 4).

Why the other options are wrong:
- B: Incorrectly multiplies when combining fractions
- C: Doesn't use the correct common denominator
- D: Simplifies to (x)/(x² - 4), which is incomplete and wrong

A is correct. When you divide a quantity by itself, you get 1: (x² - 9)²/(x² - 9) = (x² - 9)¹ = x² - 9. You're essentially canceling one factor of (x² - 9) from the numerator and denominator.

B is wrong. (x + 3)² expands to x² + 6x + 9, which isn't the same as x² - 9. This might confuse factoring with the original expression.

C is wrong. This changes the sign and loses the structure entirely—there's no mathematical path from the original expression to x² + 9.

D is wrong. This doesn't match any step in the simplification process; it appears to be a distractor with a similar form.

Why A is correct:
Factor everything first: (x² + x - 6) = (x + 3)(x - 2), and (x² - 9) = (x + 3)(x - 3). This gives you [(x + 3)(x - 2)]/[(x + 3)(x - 3)] · (x + 3)/(x + 2). Multiply across and you get [(x + 3)(x - 2)(x + 3)]/[(x + 3)(x - 3)(x + 2)]. Cancel one (x + 3) from top and bottom, leaving the answer.

Why the others are wrong:
- B: This incorrectly cancels too much—it removes (x + 3) from both numerator and denominator completely, but there are two (x + 3) factors on top and only one on bottom.
- C: This doesn't factor or simplify; it just rearranges numbers without following the multiplication rules.
- D: This is B's answer flipped upside down—still wrong for the same reason B is wrong.

Why A is correct:
Factor x² - 1 = (x + 1)(x - 1), so the first fraction already has the common denominator. Rewrite the second fraction with denominator x² - 1: multiply by (x - 1)/(x - 1) to get 2(x - 1)/(x² - 1). Now subtract: 4/(x² - 1) - 2(x - 1)/(x² - 1) = [4 - 2(x - 1)]/(x² - 1) = (4 - 2x + 2)/(x² - 1) = (6 - 2x)/(x² - 1).

Why others are wrong:
- B & C: These come from sign errors or incorrect distribution when subtracting 2(x - 1).
- D: This forgets to convert the second fraction to the common denominator before subtracting; it incorrectly treats them as if they already had a common denominator.

Why A is correct:
When dividing fractions, flip the second fraction and multiply: (3x)/(x² - 1) × (x - 1)/3. Factor x² - 1 as (x + 1)(x - 1), then cancel: [3x/(x + 1)(x - 1)] × (x - 1)/3 = x/(x + 1).

Why the others are wrong:
- B: Upside down—this is what you'd get if you forgot to flip the divisor.
- C: You'd get this if you incorrectly canceled or factored; the (x - 1) factors cancel out, not stay in the numerator.
- D: This keeps the 3 in the numerator, but it cancels completely during simplification.

Why A is correct:
Factor the denominator: x² - 4 = (x - 2)(x + 2). The second fraction needs a common denominator, so multiply (1)/(x - 2) by (x + 2)/(x + 2) to get (x + 2)/(x² - 4). Now add: 2/(x² - 4) + (x + 2)/(x² - 4) = (x + 4)/(x² - 4).

Why others are wrong:
- B: Incorrectly multiplies the second numerator by 2 instead of (x + 2).
- C: Forgets to add the original 2 from the first fraction to the (x + 2) from the second.
- D: Treats the fractions as if they can be added directly without finding a common denominator.

Why A is correct:
To change the denominator from 7y to 21y, you multiply by 3 (since 7y × 3 = 21y). You must multiply both numerator and denominator by 3: (5x × 3)/(7y × 3) = 15x/21y.

Why the others are wrong:
- B: Only changed the denominator, not the numerator—the fraction's value changed.
- C: Multiplied numerator by 7 instead of 3, making it unequal to the original.
- D: Multiplied the numerator by y as well, which wasn't needed and changes the value.

Why A is correct:
• Cancel common factors: 14 and 28 share a factor of 14, leaving 1/2
• Cancel x² and x: leaves x in the numerator
• Cancel y and y²: leaves y in the denominator
• Result: x/(2y) ✓

Why the others are wrong:
• B: Doesn't actually simplify—just rewrites the original fraction without canceling anything
• C: Incorrectly keeps x² in the numerator and y² in the denominator instead of canceling them
• D: Has the variables flipped upside-down; you'd get this if you reversed which terms cancel

Why A is correct:
- Numerator: x² − 9 = (x + 3)(x − 3) [difference of squares]
- Denominator: x² − 6x + 9 = (x − 3)² [perfect square trinomial]
- Cancel one (x − 3) from top and bottom: [(x + 3)(x − 3)] / (x − 3)² = (x + 3)/(x − 3)

Why others are wrong:
- B: Flipped—you'd get this if you factored the numerator backwards
- C: This leaves an extra (x − 3) in the denominator; you must cancel one factor completely
- D: You can only cancel if factors are identical—one (x − 3) cancels, but one remains

• Why A is correct: Both fractions have the same denominator (4a), so add the numerators: 3 + 5 = 8, giving 8/(4a). Simplify by dividing both numerator and denominator by 4: 8/(4a) = 2/a.

• Why B is wrong: This is 8/(4a), which is the unsimplified answer—you need to reduce the fraction further.

• Why C is wrong: You can't add fractions by adding both numerators AND denominators (3+5)/(4a+4a). That's not how fraction addition works.

• Why D is wrong: This looks like someone multiplied instead of added (3×5 = 15) or confused the denominators. Multiplication isn't the operation here.

• Why A is correct: Find a common denominator (6b). Rewrite 2/3b as 4/6b, then add: 4/6b + 5/6b = 9/6b. Simplify by dividing both numerator and denominator by 3: 9/6b = 3/2b. ✓

• Why B is wrong: This is the unsimplified intermediate step (9/6b). You must reduce the fraction by dividing by the GCF of 3.

• Why C is wrong: Adding numerators (2+5=7) and denominators (3+6=9) doesn't work—that's not how fraction addition works.

• Why D is wrong: This would only be correct if the original fractions somehow summed to 12/6b, which they don't.

Correct answer: A. (7)/(8c)

To subtract fractions, you need a common denominator. The LCD of 4c and 8c is 8c.

• Convert 5/(4c) to 10/(8c) by multiplying numerator and denominator by 2
• Now subtract: 10/(8c) - 3/(8c) = 7/(8c) ✓

Why the others are wrong:

• B. (2)/(4c) – This equals (1)/(2c), which is too small; you'd get this if you incorrectly subtracted the denominators
• C. (7)/(4c) – This simplifies to (14)/(8c), which is too large; common mistake if you forget to convert the first fraction
• D. (1)/(8c) – This is too small; you'd get this if you mistakenly subtracted 10 - 3 = 7 but then wrote it as 1

Why A is correct:
Multiply the fractions straight across: (x × 4y)/(2y × 3) = (4xy)/(6y). Then cancel the y's to get 4x/6, and simplify by dividing both numerator and denominator by 2 to get 2x/3.

Why the others are wrong:
- B: This shows the unsimplified product (4xy)/(6y)—you must cancel the y's and reduce the numbers.
- C: This forgets to simplify 4x/6 all the way; dividing by 2 gives 2x/3, not 4x/3.
- D: This appears to come from incorrectly multiplying denominators (2y × 3 = 6y, then canceling y) without properly handling the numerator.

Why A is correct:
When dividing fractions, flip the second fraction and multiply: (5a)/(6b) × (3b²)/(10a²). Multiply across: (5a × 3b²)/(6b × 10a²) = (15ab²)/(60a²b). Cancel common factors: 15ab²/60a²b = b/4a.

Why the others are wrong:
- B: This shows 15b/60a, which is just the numerators and denominators multiplied separately without proper simplification or canceling.
- C: This is a partially simplified intermediate step, not the final answer; it still has common factors that can be canceled.
- D: This incorrectly multiplies instead of dividing, and doesn't properly handle the reciprocal step.

• Why A is correct: All three fractions share the same denominator (3x), so you can combine the numerators: (5 + 2 - 4)/(3x) = 3/(3x). Then simplify by canceling the 3s: 3/(3x) = 1/x.

• Why B is wrong: This is the unsimplified form (3/3x). You must reduce the fraction further by dividing both numerator and denominator by 3.

• Why C is wrong: This comes from adding all numerators without subtracting: 5 + 2 + 4 = 11, not the correct operation. You need to subtract the 4.

• Why D is wrong: This is what you get if you make a calculation error with the numerators, like 5 - 2 - 4 = -1 instead of the correct 5 + 2 - 4 = 3.

Why A is correct:
Divide the coefficients: 30 ÷ 6 = 5. For the variables, subtract exponents when dividing: m³⁻² = m¹ = m, and n²⁻⁵ = n⁻³ = 1/n³. This gives (5m)/n³.

Why the others are wrong:
- B: You got 30 in the numerator instead of dividing it by 6.
- C: You didn't simplify the exponents—you just rewrote the original fraction.
- D: You subtracted the exponents for m incorrectly (3 − 2 = 1, not 2).

Correct answer: A. $(x+2)/(x+5)$

The denominator $x^2-25$ is a difference of squares that factors as $(x-5)(x+5)$. This gives you:
$$\frac{(x-5)(x+2)}{(x-5)(x+5)}$$

The $(x-5)$ cancels from numerator and denominator, leaving $\frac{x+2}{x+5}$.

Why the others are wrong:
- B: Forgot to cancel $(x-5)$—you'd only get this if you incorrectly canceled $(x+2)$ instead.
- C: Confused which factors cancel; this reverses the denominator.
- D: Over-simplified by canceling $(x+2)$ as well, which doesn't appear in the denominator.

• Correct (A): Find a common denominator of 8b. Convert: (a)/(2b) = (4a)/(8b), (a)/(4b) = (2a)/(8b), and (a)/(8b) stays the same. Then: (4a)/(8b) − (2a)/(8b) + (a)/(8b) = (4a − 2a + a)/(8b) = (3a)/(8b) ✓

• Wrong (B): This forgets to combine the numerators—you'd only get (a)/(8b) if you incorrectly ignored the first two fractions.

• Wrong (C): This adds all three fractions instead of subtracting the middle one: (4a + 2a + a)/(8b) would give (7a)/(8b).

• Wrong (D): This loses the variable *a* in the numerator—likely from a calculation error when combining terms.

Correct Answer: A – (x)/(2)

When multiplying fractions, multiply numerators together and denominators together:
- Numerators: 6x²y × 3z = 18x²yz
- Denominators: 9z × 4xy = 36xyz
- This gives (18x²yz)/(36xyz)

Now cancel common factors:
- 18/36 = 1/2
- x²/x = x
- y/y = 1
- z/z = 1
- Result: x/2 ✓

Why others are wrong:

• B shows the unsimplified form—it's correct as an intermediate step but not fully simplified.
• C keeps an extra x in the numerator; you only have one x left after canceling x² with x.
• D has z in the numerator, but z cancels out completely.

Why A is correct:
When dividing fractions, flip the second fraction and multiply: (m³n)/5 × 15/(mn²). Multiply across: (15m³n)/(5mn²). Cancel common factors: 15÷5=3, m³÷m=m², and n÷n²=1/n. Result: 3m²/n.

Why the others are wrong:
- B: This just rewrites the original problem without simplifying it.
- C: This forgets to simplify the coefficients (15÷5=3), missing a factor of 3.
- D: This has the wrong powers—the exponent on m should be 2, not 1, and n should be in the numerator, not denominator.

Why A is correct:
To add fractions, you need a common denominator. The LCD of 4x and 3x is 12x. Convert: (3)/(4x) = (9)/(12x) and (2)/(3x) = (8)/(12x). Then add: (9)/(12x) + (8)/(12x) = (17)/(12x).

Why the others are wrong:
- B & C: These use 7x as a denominator, but 7x isn't the LCD of 4x and 3x (the LCD is 12x).
- D: This has the right denominator (12x) but the wrong numerator—it appears to come from just adding the original numerators (3 + 2 = 5) without converting properly.

A is correct: Factor out 9 from the numerator: (9x − 9) = 9(x − 1). Then divide by 3x: [9(x − 1)]/(3x) = [3(x − 1)]/x. This is fully simplified.

B is wrong: This just factors the numerator but doesn't simplify the fraction—you can still divide 9 by 3.

C is wrong: While this equals the answer, it's not simplified to a single fraction form like option A.

D is wrong: This incorrectly simplifies 9/3x to 9/3x instead of 3/x, and the form isn't as clean.

Why A is correct:
To convert 7/8 to denominator 24, multiply both numerator and denominator by 3 (since 8 × 3 = 24). This gives (7 × 3)/(8 × 3) = 21/24.

Why the others are wrong:
- B (14/24): Multiplies numerator by 2 but denominator by 3—inconsistent scaling changes the fraction's value.
- C (28/32): Uses denominator 32, not 24 as requested.
- D (9/24): No correct relationship between 7 and 9; this doesn't equal 7/8.

Why A is correct:
- Factor the denominator: y² - 4 = (y + 2)(y - 2)
- Now you have: (y + 2)² / [(y + 2)(y - 2)]
- Cancel one (y + 2) from top and bottom, leaving (y + 2)/(y - 2)

Why others are wrong:
- B: This is flipped—you'd get this if you accidentally swapped numerator and denominator
- C: You'd only get y + 2 if the denominator also factored to (y + 2)², which it doesn't
- D: This would come from incorrectly factoring the denominator as (y - 2)² instead of (y + 2)(y - 2)

• Why A is correct: Find a common denominator (10b). Convert 2a/5b to 4a/10b, then subtract: 4a/10b − a/10b = 3a/10b. ✓

• Why B is wrong: You'd get a/10b if you only subtracted the numerators without converting the first fraction to the common denominator.

• Why C is wrong: This keeps the wrong denominator (5b instead of 10b). You might get this if you forgot to find a common denominator.

• Why D is wrong: The denominator 15b suggests incorrectly multiplying the denominators instead of finding the LCD.

A is correct: When multiplying fractions, multiply numerators and denominators: [(x+1)(x-1)] / [(x-1)(x+2)]. The (x-1) cancels from top and bottom, leaving (x+1)/(x+2).

B is wrong: This swaps the numerator—you'd need (x-1) in the top, but it cancels out.

C is wrong: This would only happen if numerator and denominator were identical, but (x+1) ≠ (x+2).

D is wrong: This is the unsimplified form before canceling the common (x-1) factor.

Correct Answer: (21x)/(27y)

To go from denominator 9y to 27y, you multiply by 3 (since 9y × 3 = 27y). When you multiply a denominator by 3, you must multiply the numerator by 3 too to keep the fraction equivalent: 7x × 3 = 21x. Result: (21x)/(27y).

Why the others are wrong:
- (B) Keeps the numerator as 7x—didn't multiply the top by 3, so it's not equivalent to the original.
- (C) Multiplies the numerator by xy instead of just 3, creating an incorrect fraction.
- (D) Multiplies the numerator by 2 instead of 3, giving the wrong equivalent fraction.

Correct Answer: A – (3ab)/(4c)

To simplify, divide both numerator and denominator by their greatest common factor:
- GCF of 18 and 24 is 6 → 18÷6 = 3, and 24÷6 = 4
- For variables: a²÷a = a, b³÷b² = b, and c stays in denominator
- Result: (3ab)/(4c) ✓

Why the others are wrong:
- B: Didn't reduce the numbers (18 and 24 aren't in lowest terms)
- C: Incorrectly kept a² instead of reducing to a (forgot to divide a² by a)
- D: Incorrectly kept b² instead of reducing to b (forgot to divide b³ by b²)

Correct Answer: A

• Factor the numerator: x² - 5x = x(x - 5)
• Factor the denominator: x² - 25 = (x - 5)(x + 5) [difference of squares]
• Cancel the common factor (x - 5): x(x - 5)/(x - 5)(x + 5) = x/(x + 5)

Why others are wrong:

• B: Incorrectly keeps (x - 5) in the numerator after canceling—you must cancel matching factors from top and bottom
• C: Reverses which factor cancels; (x - 5) cancels, not (x + 5)
• D: Only true if all factors canceled, but (x + 5) remains in the denominator

• Why A is correct: Both fractions have the same denominator (8n), so you add the numerators: 5m + 3m = 8m. This gives you (8m)/(8n). The 8s cancel, leaving m/n.

• Why B is wrong: (8m)/(16n) is equivalent to (m)/(2n), not (m)/(n)—the denominator is twice as large as it should be.

• Why C is wrong: You add only the numerators when denominators match, not a numerator to a denominator. It should be (5m + 3m)/(8n), not (5m + 3n)/(8n).

• Why D is wrong: This keeps 8m in the numerator without simplifying. You must cancel the common factor of 8 from numerator and denominator.

Why A is correct:
To add fractions, you need a common denominator. The LCD of 6x and 9x is 18x. Convert: (5)/(6x) = (15)/(18x) and (7)/(9x) = (14)/(18x). Add them: (15 + 14)/(18x) = (29)/(18x).

Why the others are wrong:
- B: Uses 15x as the denominator, which isn't a common multiple of 6x and 9x.
- C: Adds the numerators but uses 15x as the denominator—this skips the crucial step of finding the LCD.
- D: Correctly uses 18x as the denominator but adds only 5 + 7 = 12 instead of accounting for the conversion factors (15 + 14 = 29).

Why A is correct:
To subtract fractions, find a common denominator. The LCD of 4p and 6p is 12p.
- (3)/(4p) = (9)/(12p)
- (5)/(6p) = (10)/(12p)
- (9)/(12p) - (10)/(12p) = -(1)/(12p) ✓

Why the others are wrong:
- B: -(2)/(10p) uses 10p (wrong LCD) and gives the wrong numerator
- C: (1)/(12p) has the right denominator but wrong sign (should be negative)
- D: (3-5)/(10p) incorrectly subtracts numerators without a common denominator and uses wrong LCD

Why A is correct:
Find a common denominator (6k), then convert each fraction: 4/6k + 5/6k - 3/6k = 6/6k = 1/k.

Why the others are wrong:
- B: 6/6k simplifies to 1/k, so it's actually correct but not in simplified form—A is the simplified answer.
- C: You can't just add/subtract the numerators (2+5-1) without converting to a common denominator first.
- D: This results from incorrectly combining numerators (2+5-1=6) without properly converting each original fraction to the common denominator.

A is correct because when you multiply fractions, you multiply numerators and denominators, then cancel common factors:
- (3x²y × 10z) / (5z × 9xy²) = (30x²yz) / (45xy²z)
- Cancel z from top and bottom, then divide: 30/45 = 2/3, x²/x = x, y/y² = 1/y
- Result: (2x)/(3y)

B is wrong because it's just the product before simplifying—you must cancel common factors.

C is wrong because it keeps x² in the numerator, but x²/x = x (not x²).

D is wrong because it cancels the x entirely, but one x remains after simplification.

• A is correct. When multiplying fractions, multiply numerators and denominators: [(a-2) · 8b] / [4b · (a-2)]. The (a-2) cancels from top and bottom, and 8b/4b simplifies to 2, leaving just 2.

• B is wrong. This is the unsimplified product—you can still cancel (a-2) and reduce 8b/4b, so it's not fully simplified.

• C is wrong. While (a-2)/(a-2) does equal 1, you forgot to simplify the 8b/4b part, which equals 2. The complete answer is 2, not 1.

• D is wrong. This comes from incorrectly canceling only the (a-2) terms but then reducing 8/4 to get 2, while forgetting to cancel the b's completely. The b should cancel out entirely.

Why A is correct:
To divide fractions, flip the second fraction and multiply: (5x³)/(6y²) × (9y)/(10x). Multiply across: (5x³ × 9y)/(6y² × 10x) = (45x³y)/(60xy²). Simplify by canceling: x³/x = x², y/y² = 1/y, and 45/60 = 3/4. Result: (3x²)/(4y).

Why the others are wrong:
- B: This is the unsimplified product before canceling common factors—it's not fully simplified.
- C: This has the wrong denominator (should be 4y, not 4) and the wrong numerator (shouldn't have y in it).
- D: This loses the coefficient 3/4 in the numerator and simplifies too aggressively.

Why A is correct:
When dividing fractions, flip the second fraction and multiply: (2a²b)/(3c) × (6c³)/(4ab²). Multiply numerators: 2a²b·6c³ = 12a²bc³. Multiply denominators: 3c·4ab² = 12ab²c. Simplify: 12a²bc³/12ab²c = ac²/b.

Why B is wrong:
This shows the setup *before* simplifying—it's the intermediate step, not the final answer. The question asks you to simplify, not just rewrite the division as multiplication.

Why C is wrong:
This has an extra "a" in the numerator. A common mistake is miscanceling the a² term; it should cancel to just "a," not stay as a².

Why D is wrong:
This keeps an unnecessary 3 in the denominator. When you simplify 12a²bc³/12ab²c, the 3 cancels completely (3 is a factor of both 12s), so it shouldn't remain in the final answer.

• Why A is correct: Factor x² - 16 = (x + 4)(x - 4). Rewrite the second fraction: (x + 3)/(4 - x) = -(x + 3)/(x - 4). Now both fractions have denominator (x + 4)(x - 4), so combine: [(x - 4) - (x + 3)(x + 4)] / [(x + 4)(x - 4)]. Expand the numerator: (x - 4) - (x² + 7x + 12) = -x² - 6x - 16. This equals -(x² + 6x + 16)/(x² - 16).

• Why B is wrong: This is missing the negative sign. It would result from making a sign error when combining the fractions.

• Why C is wrong: While the numerator is correct, it doesn't simplify the denominator fully—it leaves (x + 4)(x - 4) instead of writing it as x² - 16.

• Why D is wrong: This oversimplifies incorrectly. It doesn't properly combine the fractions or account for the negative sign in the second term.

A is correct: $4x - 1$

• Split the fraction: $\frac{12x^2}{3x} - \frac{3x}{3x}$
• Simplify each part: $4x - 1$

Why the others are wrong:

• B - Just rewrites the original expression in factored form; it's not simplified
• C - This is a correct intermediate step, but it's not fully simplified (stop here and you haven't finished the problem)
• D - This results from incorrectly dividing; you'd get this if you divided only the second term by $x$ instead of by $3x$

Why A is correct:
Cancel the common factor 3 from numerator and denominator (9÷3=3, 15÷3=5). Cancel one m from top and bottom (m²÷m=m). Cancel three n's from the bottom against one n from top (n÷n⁴=1/n³). Result: (3m)/(5n³).

Why the others are wrong:
- B: Forgot to simplify 9/15 down to 3/5.
- C: Incorrectly kept m² in the numerator instead of canceling one m.
- D: Forgot to account for the n⁴ in the denominator; should have n³, not just n.

Why A is correct:
Factor out common factors from numerator and denominator: (2x + 6) = 2(x + 3) and (4x + 12) = 4(x + 3). When you cancel the common factor (x + 3), you're left with 2/4 = 1/2.

Why the others are wrong:
- B: This just rearranges part of the original fraction without simplifying—it's not in lowest terms.
- C: This rewrites the fraction differently but doesn't simplify it; the common factors aren't actually canceled.
- D: This changes the denominator incorrectly (says 4x + 6 instead of 4x + 12), so it's not equivalent to the original fraction.

Why A is correct:
To change the denominator from 5y to 20y, you multiply by 4 (since 5y × 4 = 20y). You must multiply both numerator and denominator by 4: (3x × 4)/(5y × 4) = 12x/20y.

Why the others are wrong:
- B: Didn't multiply the numerator—just changed the denominator, which breaks the fraction's value.
- C: Multiplied the numerator by 5 instead of 4, which doesn't match the denominator change.
- D: Incorrectly multiplied the numerator by y, adding an extra variable that shouldn't be there.

Correct Answer: A – $(5x^2)/(9y)$

To add fractions, you need a common denominator. The LCD of 3y and 9y is 9y, so convert the first fraction: $(x^2)/(3y) = (3x^2)/(9y)$. Now add: $(3x^2)/(9y) + (2x^2)/(9y) = (5x^2)/(9y)$.

Why the others are wrong:
- B: Only counts the numerator from the second fraction; forgot to convert and add both numerators.
- C: Incorrectly adds the denominators (3y + 9y = 12y), which you never do when adding fractions.
- D: Forgot to convert to a common denominator before adding.

Why A is correct:
Both fractions have the same variables (x³ and y²), so you can subtract them directly. First, rewrite with a common denominator: (5x³)/(2y²) - (3x³)/(6y²) = (15x³)/(6y²) - (3x³)/(6y²) = (12x³)/(6y²). Then simplify by dividing numerator and denominator by 6: (2x³)/(y²).

Why the others are wrong:
- B: This is the intermediate step before simplifying—you must reduce (12x³)/(6y²) further.
- C: This doesn't actually combine the fractions; you can't just subtract numerators without a common denominator first.
- D: This has the wrong denominator; you didn't simplify all the way (6y² should reduce to y²).

• Why A is correct: First, combine the fractions in parentheses by finding a common denominator (6p): 4/6p + 1/6p = 5/6p. Then multiply by 3p/2: (5/6p) × (3p/2) = 15p/12p = 5/4. The p's cancel out, leaving just a number.

• Why B is wrong: This is what you get if you multiply before simplifying the fractions in the parentheses—it's an intermediate step, not the final answer.

• Why C is wrong: This results from forgetting to multiply by 3p/2; it's only the simplified sum of the two fractions.

• Why D is wrong: This keeps a p in the answer, which means you didn't cancel the p's when multiplying 5/6p × 3p/2.

Correct answer: A. $(2b^4)/(a^3)$

Divide the coefficients: $4 ÷ 2 = 2$. For the variables, use the rule $\frac{x^m}{x^n} = x^{m-n}$:
- For $a$: $a^{-2} ÷ a^1 = a^{-2-1} = a^{-3} = \frac{1}{a^3}$
- For $b$: $b^3 ÷ b^{-1} = b^{3-(-1)} = b^4$

Result: $\frac{2b^4}{a^3}$ ✓

Why others are wrong:
- B has the right exponents but writes $a^{-3}$ instead of moving it to the denominator as $a^3$—same value, wrong form for the answer given
- C loses the $b$ term entirely; the $b$ exponents should give $b^4$, not $b^2$
- D is correct mathematically but isn't simplified (has 4 and 2 that should be reduced to match option A)

Why A is correct:
To change the denominator from 5y to 20y, multiply both top and bottom by 4 (since 5y × 4 = 20y). This gives (4x × 4)/(5y × 4) = (16x)/(20y).

Why the others are wrong:
- B: Multiplies only the denominator by 4, not the numerator—this changes the value of the fraction.
- C: Multiplies the numerator by 5 instead of 4, creating a different fraction.
- D: Introduces an extra y in the numerator; you only multiply by 4, not by 4y.

Correct answer: (9a)/(21b)

To change the denominator from 7b to 21b, you multiply by 3/3 (since 7b × 3 = 21b). When you multiply the denominator by 3, you must also multiply the numerator by 3: (3a × 3)/(7b × 3) = 9a/21b.

Why the others are wrong:
- B: Leaves the numerator unchanged—you forgot to multiply the top by 3.
- C: Only multiplies the numerator by 2 instead of 3.
- D: Introduces an extra variable b in the numerator, which doesn't happen when converting fractions.

Correct answer: (x+1)/(x)

Factor both the numerator and denominator:
- Numerator: x² - 1 = (x+1)(x-1)
- Denominator: x² - x = x(x-1)

Cancel the common factor (x-1): [(x+1)(x-1)] / [x(x-1)] = (x+1)/x

Why the others are wrong:
- B: Flips the factors—canceling incorrectly or factoring backwards
- C: Results from canceling x instead of (x-1), which is the actual common factor
- D: Only true if you mistakenly cancel everything; the x in the denominator doesn't cancel

Why A is correct:
Factor out 2x from the numerator: (4x² - 12x) = 2x(2x - 6). Then divide: 2x(2x - 6) / 2x = 2x - 6. This is fully reduced.

Why the others are wrong:
- B: This is just a factored form of the numerator, not actually reduced—you haven't divided by 2x yet.
- C: This equals 2x - 6, but it's not the same form. (Actually, 2(x - 3) = 2x - 6, so C *is* equivalent, but A is the standard simplified form.)
- D: You can't cancel just part of the numerator with the denominator; you must factor first. This also simplifies to 2x - 6, but the process is incorrect.

Why A is correct:
Since both fractions have the same denominator (3m), you add the numerators directly: 7 + 5 = 12, keeping the denominator 3m. Result: 12/(3m).

Why the others are wrong:
- B: Incorrectly changes the denominator to 6m instead of keeping it at 3m. You only add denominators when multiplying fractions, not when they're already the same.
- C: Subtracts instead of adds (7 − 5 = 2). The operation is addition.
- D: Puts the variable m in the numerator (12m) instead of the denominator. The denominator stays 3m.

Correct answer: (3)/(n)

Since both fractions have the same denominator (2n), subtract the numerators: 9 − 3 = 6, giving you (6)/(2n). Then simplify by dividing both numerator and denominator by 2: (6)/(2n) = (3)/(n).

Why the others are wrong:
- (6)/(n): You forgot to simplify (6)/(2n)—you need to reduce the fraction.
- (3)/(2n): This is only the simplified numerator, not the full simplified fraction.
- (12)/(2n): This adds the numerators (9 + 3) instead of subtracting them.

• Why A is correct: To add fractions, you need a common denominator. The LCD of 3a and 5a is 15a. Convert: (2/3a) = 10/15a and (3/5a) = 9/15a. Adding gives 10/15a + 9/15a = 19/15a.

• Why B is wrong: This adds the denominators (3 + 5 = 8) instead of finding the LCD. You can't just add denominators when combining fractions.

• Why C is wrong: This adds only the numerators (2 + 3 = 5) but uses 15a as the denominator without actually converting the original fractions. The numerators must be converted first.

• Why D is wrong: This multiplies the original numerators (2 × 3 = 6) instead of converting them to the common denominator. You'd only do this if multiplying fractions, not adding them.

• Find a common denominator: The denominators are 4b and 6b. The LCD is 12b.

• Rewrite each fraction: 5/(4b) = 15/(12b) and 2/(6b) = 4/(12b)

• Subtract: 15/(12b) − 4/(12b) = 11/(12b) ✓

Why others are wrong:
- (B) 3/(10b): Uses 10b as denominator, which isn't the correct LCD
- (C) (5−2)/(10b): Incorrectly subtracts numerators without converting to common denominator first
- (D) 7/(4b): Ignores the second fraction or makes an arithmetic error

Why A is correct:
Find a common denominator (4x), then combine: (2)/(4x) + (3)/(4x) - (1)/(4x) = (2+3-1)/(4x) = (4)/(4x) = (1)/(x).

Why the others are wrong:
- B & C: These are intermediate steps or arithmetic errors—they don't complete the simplification.
- D: This is (4)/(4x), which equals (1)/(x) when reduced, so it's the unsimplified form of the correct answer.

Why A is correct:
To add fractions, you need a common denominator. The LCD of 2b and 3b is 6b.
- Convert a/(2b) to 3a/(6b) (multiply by 3/3)
- Convert c/(3b) to 2c/(6b) (multiply by 2/2)
- Add: 3a/(6b) + 2c/(6b) = (3a + 2c)/(6b)

Why the others are wrong:
- B: Adding denominators (2b + 3b = 5b) is incorrect; you find the LCD, not add them
- C: This multiplies (a + c) by 5, which doesn't come from the conversion process
- D: This uses 6b as the denominator (correct) but forgets to multiply the numerators by the conversion factors (should be 3a + 2c, not a + c)

Why A is correct:
Multiply across: numerators (5x × 12y) and denominators (6y × 15x²), giving (60xy)/(90x²y). Cancel common factors: 60xy and 90x²y share 30xy, leaving 2/3x.

Why the others are wrong:
- B: This is the unsimplified product—you must cancel common factors to get the final answer.
- C: You'd get this if you incorrectly canceled the 2 from the numerator; the 2 stays after simplification.
- D: This results from a calculation error in canceling; you'd get 4/x only if you made a mistake with the coefficients.

Why A is correct:
When dividing fractions, flip the second fraction and multiply: (7a²)/(5b) × (b)/(14a). Multiply across: (7a²·b)/(5b·14a) = (7a²b)/(70ab). Cancel b from numerator and denominator, and cancel 7a from numerator and denominator: a/10.

Why the others are wrong:
- B: This shows the multiplication step before simplifying—it's the unsimplified intermediate answer, not the final result.
- C: This forgot to flip the divisor; it just multiplied the fractions as written instead of dividing.
- D: This results from incorrectly canceling terms; you'd only get a/2 if you miscalculated which factors divide evenly.

Correct answer: A ($1$)

Factor the numerator: x² - 9 = (x - 3)(x + 3). So the expression becomes:
$$\frac{(x-3)(x+3)}{x-3} × \frac{1}{x+3}$$

Cancel (x - 3) from numerator and denominator, then cancel (x + 3):
$$\frac{(x+3)}{1} × \frac{1}{x+3} = 1$$

Why the others are wrong:
- B & C: These come from incomplete canceling—if you only cancel one factor but not both, you'd get a fraction instead of 1.
- D: This would result if you forgot to multiply by the second fraction (1/(x+3)), leaving just the simplified first fraction.

Why A is correct:
When dividing fractions, flip the second fraction and multiply: (4x³)/(9y) × (3y²)/(2x). Multiply numerators: 4x³ · 3y² = 12x³y². Multiply denominators: 9y · 2x = 18xy. Simplify: 12x³y²/18xy = 2x²y/3.

Why the others are wrong:
- B: Results from multiplying instead of dividing (the mistake of not flipping the second fraction).
- C: Incorrectly simplifies or combines terms without properly canceling common factors.
- D: Missing the y in the numerator—this comes from incomplete simplification or forgetting to include y² from the second fraction.

• First, factor x² - 4 = (x+2)(x-2)

• Rewrite 1/(2-x) as -1/(x-2) to match the denominator form

• Get common denominator (x+2)(x-2):
- (x-2)/(x+2)(x-2) - (x+2)/(x+2)(x-2)

• Combine: [(x-2) - (x+2)] / [(x+2)(x-2)] = -4/(x²-4) ✓

Why others are wrong:
- B & C: These incorrectly simplify the fraction further or make algebra errors—they don't account for the full numerator of -4
- D: This suggests only one denominator factor remains, missing that x²-4 stays in the denominator

Correct Answer: A. $x+2$

Factor out 3x from the numerator: $\frac{3x(x+2)}{3x}$. Then cancel the 3x on top and bottom, leaving $x+2$.

Why the others are wrong:
- B: This is the factored form, not simplified—you still need to cancel the common factors.
- C: This comes from dividing only the first term correctly; it ignores that 6x also has a factor of 3x to cancel.
- D: While this equals the right answer numerically, it's not fully simplified since $\frac{6x}{3x}$ can reduce to 2.

A is correct.

Divide the coefficients: 9 ÷ 12 = 3/4. Subtract the exponents for each variable: m³ ÷ m = m², and n² ÷ n⁴ = n⁻² = 1/n². This gives (3m²)/(4n²).

Why others are wrong:
- B didn't simplify the coefficient 9/12 down to 3/4
- C kept m³ instead of subtracting exponents (3 − 1 = 2)
- D made errors in exponent subtraction for both variables

# Solution

Correct answer: (5)/(9x)

• First, add the fractions in parentheses: (2)/(3x) + (1)/(6x). Find common denominator 6x: (4)/(6x) + (1)/(6x) = (5)/(6x)
• Then divide by (3)/(2): (5)/(6x) ÷ (3)/(2) = (5)/(6x) × (2)/(3) = (10)/(18x) = (5)/(9x) ✓

Why the others are wrong:

• (5)/(4x) – Wrong common denominator or arithmetic error when adding fractions
• (5)/(6x) – This is just the intermediate step; forgot to divide by (3)/(2)
• (5x)/(9) – Likely flipped the denominator; the variable x belongs in the denominator, not numerator

Why A is correct:
To convert the denominator from 4y to 20y, multiply by 5 (since 4y × 5 = 20y). You must multiply both numerator and denominator by the same number: 3x × 5 = 15x, giving (15x)/(20y).

Why the others are wrong:
- B: Just changed the denominator without multiplying the numerator—this changes the value of the fraction.
- C: Multiplied numerator by 20 instead of 5—this would only be correct if you were converting to a denominator of 80y.
- D: Multiplied the numerator by 20 and got an extra y—this is incorrect multiplication and doesn't simplify properly.

Why A is correct:
To change the denominator from 2a to 6a, you multiply the denominator by 3. You must multiply the numerator by 3 as well to keep the fraction equivalent: (5 × 3)/(2a × 3) = 15/(6a).

Why the others are wrong:
- B: Only changed the denominator, not the numerator—this changes the value of the fraction.
- C: Multiplied the numerator by 2 instead of 3, which doesn't match the denominator change.
- D: Multiplied the numerator by 3a instead of just 3, creating an incorrect fraction.

A is correct: Factor out 4x from the numerator: 4x(3x-2)/(4x). The 4x cancels, leaving 3x-2.

B is wrong: You can simplify this further by dividing 12x and 8 by 4, so it's not in lowest terms.

C is wrong: This comes from incorrectly dividing each term by 4x separately instead of factoring first.

D is wrong: The sign is flipped; the numerator factors to 4x(3x-2), not 4x(3x+2).

• A is correct: Divide the coefficients (18÷6=3), subtract the exponents for *a* (3−2=1), and subtract the exponents for *b* (1−2=−1, which moves *b* to the denominator). Result: 3a/b.

• B is wrong: You forgot to simplify the *b* terms completely—this still has an unsimplified *b* in the denominator.

• C is wrong: This ignores the *b* term in the denominator entirely; you can't just drop it.

• D is wrong: The exponents are backwards. You'd get this by adding exponents instead of subtracting, which is the opposite of division.

Correct answer: (x+2)/(x)

Factor the numerator and denominator:
- Numerator: x² − 4 = (x + 2)(x − 2)
- Denominator: x² − 2x = x(x − 2)

Cancel the common factor (x − 2): [(x + 2)(x − 2)] / [x(x − 2)] = (x + 2)/x

Why the others are wrong:
- B: (x−2)/(x+2) — This reverses and swaps the factors; doesn't match the cancellation.
- D: 1 — This would only work if numerator and denominator were identical, which they're not.
- (Note: Options A and the correct answer appear identical, so A is also correct if they're truly the same.)

Correct Answer: A. (y+1)/(y-1)

Factor both the numerator and denominator:
- Numerator: y² + y = y(y + 1)
- Denominator: y² - y = y(y - 1)

Cancel the common factor of y: [y(y + 1)] / [y(y - 1)] = (y + 1)/(y - 1)

Why the others are wrong:
- B: Missing the +1 in the numerator; this comes from not fully factoring
- C: Has the factors flipped—would only be correct if you switched numerator and denominator
- D: Would only be true if the numerator and denominator were identical, which they aren't

A is correct: When adding fractions with the same denominator, add the numerators and keep the denominator. Here: 5 + 2 = 7, so you get 7/(3x).

B is wrong: This simplifies 7/(3x) incorrectly. While 7/(3x) could be rewritten as 7/(3x), it doesn't equal 3/x.

C is wrong: This assumes you added the denominators too (3x + 3x = 6x), which is a common mistake. You only add numerators when denominators match.

D is wrong: This would require 7 to equal 3, which it doesn't. It comes from mishandling both the numerator and denominator.

Why A is correct:
Since both fractions have the same denominator (2n), you subtract the numerators: 7m − 3m = 4m. The denominator stays 2n, giving you (4m)/(2n).

Why the others are wrong:
- B: This simplifies (4m)/(2n) further by dividing both top and bottom by 2, but the question asks you to compute, not simplify—and this isn't listed as equivalent anyway.
- C: This adds instead of subtracts (7m + 3m = 10m), which is the wrong operation.
- D: This is what you get if you simplify (4m)/(2n) completely, but that's not what's shown as the answer choice.

• A is correct because to add fractions, you need a common denominator. The LCD of 5y and 10y is 10y. Convert 2/(5y) to 4/(10y), then add: 4/(10y) + 3/(10y) = 7/(10y).

• B is wrong because you can't just add numerators and denominators together—that's not how fraction addition works.

• C is wrong because 7/10 ≠ 1/2 (they're different values).

• D is wrong because it uses the correct denominator but the wrong numerator—you'd get this if you forgot to convert the first fraction to the common denominator.

Why A is correct:
To subtract fractions, find a common denominator. The LCD of 3z and 5z is 15z.
- Convert: (4x)/(3z) = (20x)/(15z) and (2x)/(5z) = (6x)/(5z)
- Subtract: (20x)/(15z) - (6x)/(15z) = (14x)/(15z)

Why the others are wrong:
- B: This comes from subtracting incorrectly: (4x - 2x)/(15z) = (2x)/(15z). You can't just subtract numerators without converting to a common denominator first.
- C: You can't simply add the denominators (3z + 5z = 8z). You must find the LCD, not add them.
- D: This results from (20x - 10x)/(15z). The second fraction converts to (6x)/(15z), not (10x)/(15z).

Why A is correct:
To add fractions, you need a common denominator. The LCD of 2b and 4b is 4b. Convert a/(2b) to 2a/(4b), then add: 2a/(4b) + 3c/(4b) = (2a+3c)/(4b).

Why the others are wrong:
- B: Uses 6b as the denominator, which isn't the LCD of 2b and 4b.
- C: Correctly uses 4b as the denominator but forgets to convert a/(2b) to 2a/(4b), so the numerator is wrong.
- D: This is equivalent to A (it reduces to the same fraction), but it's not simplified, so it's not the best answer.

• Correct (A): Find a common denominator of 8p. Convert: 3/4p = 6/8p, 1/2p = 4/8p, 1/8p stays 1/8p. Then: 6/8p + 4/8p − 1/8p = 9/8p. ✓

• B (8/8p): This simplifies to 1/p, which would only happen if you forgot to subtract the last fraction—you'd get 6/8p + 4/8p = 10/8p, then miscalculated.

• C (5/8p): You'd get this by incorrectly adding the last term instead of subtracting: 6/8p + 4/8p + 1/8p = 11/8p, then made an arithmetic error.

• D (1/p): This is what 8/8p simplifies to—likely from getting option B and then reducing the fraction, but your earlier calculation was wrong.

Why A is correct:
Multiply the fractions: (2x × 9y)/(3y × 4x) = 18xy/12xy. The x's cancel and the y's cancel, leaving 18/12, which simplifies to 3/2.

Why the others are wrong:
- B: This shows the unsimplified form (18xy/12xy), but the question asks you to multiply and simplify—the xy terms must cancel.
- C: This would only be correct if the numbers didn't cancel; you'd get x/y only if you incorrectly ignored the 18/12 simplification.
- D: This is a partial answer that forgot to cancel the variables; it keeps the y in the denominator when it should have canceled.

Why A is correct:
Multiply numerators: 3a² × 10b² = 30a²b². Multiply denominators: 5b × 9a = 45ab. This gives 30a²b²/45ab. Cancel common factors: 30÷15=2, a²÷a=a, b²÷b=b, 45÷15=3. Result: 2ab/3.

Why the others are wrong:
- B: This is the unsimplified form—you must cancel the common factors to get the final answer.
- C: This incorrectly keeps a² in the numerator; a² ÷ a = a (not a²).
- D: This incorrectly cancels the *a* entirely; it should simplify to a (one *a* remains).

Why A is correct:
To divide fractions, flip the second fraction and multiply: (6x²)/(7y) × (14y²)/(3x). Multiply across: (6x² × 14y²)/(7y × 3x) = (84x²y²)/(21xy). Simplify by canceling: 84÷21 = 4, x²÷x = x, and y²÷y = y, giving 4xy.

Why the others are wrong:
- B: This shows the multiplied numerators and denominators but isn't simplified—it's an intermediate step, not the final answer.
- C: The numerator and denominator are switched; this comes from not properly flipping the divisor.
- D: This has an extra x in the result; it likely comes from an error in canceling the x terms (should be x¹, not x²).

# Why A is Correct

When dividing fractions, flip the second fraction and multiply: (4ab²)/(3c) × (9c³)/(2a²b)

Multiply across: (4ab² × 9c³)/(3c × 2a²b) = (36ab²c³)/(6a²bc)

Simplify by canceling: 36÷6=6, a÷a²=1/a, b²÷b=b, c³÷c=c²

Final answer: (6bc²)/a ✓

# Why Others Are Wrong

B: This is the unsimplified form before canceling common factors—you must reduce to lowest terms.

C: Wrong coefficient; you'd get this if you made an arithmetic error when simplifying 36÷6.

D: Missing the c² in the numerator; this comes from incorrectly canceling the c terms (c³÷c = c², not 1).

Why A is correct:
First, combine the fractions in parentheses: 1/(2m) + 3/(4m) = 2/(4m) + 3/(4m) = 5/(4m). Then multiply by 2m/3: (5/4m) × (2m/3) = (5 × 2m)/(4m × 3) = 10m/12m. The *m*'s cancel, leaving 10/12, which simplifies to 5/6.

Why the others are wrong:
- B: Incorrectly keeps an *m* in the numerator; the *m*'s should cancel completely.
- C: Wrong denominator; comes from making an arithmetic error in simplifying.
- D: This is the unsimplified form before canceling the *m*'s and reducing the fraction.

Why A is correct:
First, simplify the numerator: x/(3y) − x/(6y) = 2x/(6y) − x/(6y) = x/(6y). Then divide: (x/6y) ÷ (x/2y) = (x/6y) × (2y/x) = 2y/6y = 1/3.

Why the others are wrong:
- B (x/3y): This is just the first fraction in the numerator—you forgot to subtract and divide.
- C (2y/6y): This simplifies to 1/3, but it's not fully reduced, so it's not the simplified answer.
- D (1/2): This comes from incorrectly dividing or skipping steps—the correct final answer is 1/3, not 1/2.

Why A is correct:
Divide the coefficients: 8 ÷ 2 = 4. Subtract the exponents for like bases: x^4 ÷ x^2 = x^2, and y^2 ÷ y^5 = y^(2-5) = y^(-3) = 1/y^3. Result: (4x^2)/y^3.

Why the others are wrong:
- B: Didn't simplify the coefficient (left 8/2 instead of dividing to get 4).
- C: Incorrectly kept y in the numerator instead of moving it to the denominator with the correct exponent.
- D: Left both y^2 and y^5 unsimplified instead of subtracting exponents.

Why A is correct:
Divide the coefficients: 9 ÷ 3 = 3. For the *a* terms: a³ ÷ a⁻¹ = a³⁻⁽⁻¹⁾ = a⁴. For the *b* terms: b⁻¹ ÷ b² = b⁻¹⁻² = b⁻³, which equals 1/b³. Result: (3a⁴)/b³.

Why others are wrong:
- B has the right coefficients and *a* power, but b⁻³ is not fully simplified—you should write it as 1/b³ to show "lowest terms" clearly.
- C forgot to simplify the coefficient (9/3 = 3, not 9/3) and didn't handle the *a* exponents correctly (should be a⁴, not a³).
- D has the right *a* and *b* powers but makes b³ positive in the numerator instead of negative (should be in the denominator as b⁻³).

Why A is correct:
To change the denominator from 5y to 35y, multiply by 7 (since 5y × 7 = 35y). You must multiply both numerator and denominator by 7: (2x × 7)/(5y × 7) = 14x/35y.

Why the others are wrong:
- B: Only changed the denominator but forgot to multiply the numerator by 7—this gives you a different fraction value.
- C: Multiplied incorrectly; 2x × 6 = 12x, but you need to multiply by 7, not 6.
- D: Multiplied the numerator by 7y instead of just 7, adding an extra y that doesn't belong.

Why A is correct:
To change the denominator from 4z to 12z, multiply by 3/3 (since 4z × 3 = 12z). This gives you (9 × 3)/(4z × 3) = 27/(12z).

Why the others are wrong:
- B: Just changed the denominator without multiplying the numerator—you must do the same operation to top and bottom.
- C: Only multiplied the numerator by 2 instead of 3 (a common mistake with the wrong scaling factor).
- D: Multiplied the numerator by z as well, which isn't needed—only multiply by 3.