a/b = 3 + c/(d+x) The given equation relates the positive variables a, b, c, d, and x, where a...

1. TRANSLATE the problem information

• Given: \(\frac{\mathrm{a}}{\mathrm{b}} = 3 + \frac{\mathrm{c}}{\mathrm{d+x}}\) where all variables are positive and \(\mathrm{a \gt 3b}\)
• Find: Express x in terms of a, b, c, and d

2. INFER the solution strategy

• The variable x is buried inside a fraction within a larger equation
• Strategy: Work systematically to isolate x by first getting the fractional term by itself, then manipulating to extract x from the denominator

3. SIMPLIFY by isolating the fractional term

Subtract 3 from both sides:

\(\frac{\mathrm{a}}{\mathrm{b}} - 3 = \frac{\mathrm{c}}{\mathrm{d+x}}\)

4. SIMPLIFY the left side with a common denominator

Rewrite with denominator b:

\(\frac{\mathrm{a - 3b}}{\mathrm{b}} = \frac{\mathrm{c}}{\mathrm{d+x}}\)

5. SIMPLIFY by taking reciprocals to extract x from the denominator

Flip both sides:

\(\frac{\mathrm{b}}{\mathrm{a - 3b}} = \frac{\mathrm{d+x}}{\mathrm{c}}\)

6. SIMPLIFY by multiplying both sides by c

\(\frac{\mathrm{bc}}{\mathrm{a - 3b}} = \mathrm{d + x}\)

7. SIMPLIFY by subtracting d to isolate x

\(\mathrm{x} = \frac{\mathrm{bc}}{\mathrm{a - 3b}} - \mathrm{d}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when combining fractions in Step 4, writing \(\frac{\mathrm{a + 3b}}{\mathrm{b}}\) instead of \(\frac{\mathrm{a - 3b}}{\mathrm{b}}\).

When subtracting 3 from \(\frac{\mathrm{a}}{\mathrm{b}}\), they incorrectly think: \(\frac{\mathrm{a}}{\mathrm{b}} - 3 = \frac{\mathrm{a + 3b}}{\mathrm{b}}\) instead of \(\frac{\mathrm{a}}{\mathrm{b}} - 3 = \frac{\mathrm{a - 3b}}{\mathrm{b}}\). This fundamental algebraic mistake carries through the entire solution.

This leads them to select Choice A: \(\frac{\mathrm{bc}}{\mathrm{a+3b}} - \mathrm{d}\)

Second Most Common Error:

Incomplete SIMPLIFY execution: Students correctly work through all steps until the final isolation, but make a sign error when moving d to the other side.

From \(\frac{\mathrm{bc}}{\mathrm{a - 3b}} = \mathrm{d + x}\), they incorrectly write \(\mathrm{x} = \frac{\mathrm{bc}}{\mathrm{a - 3b}} + \mathrm{d}\) instead of \(\mathrm{x} = \frac{\mathrm{bc}}{\mathrm{a - 3b}} - \mathrm{d}\), forgetting that subtracting d from both sides changes the sign.

This leads them to select Choice B: \(\frac{\mathrm{bc}}{\mathrm{a-3b}} + \mathrm{d}\)

The Bottom Line:

This problem requires careful tracking of signs through multiple algebraic manipulations. Students who rush through fraction operations or don't double-check their algebra are prone to systematic errors that propagate through the solution.