Clear Every Fraction in an Equation

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Solve x/2+x/3=10. Candidate answers: (A) 11 (B) 12 (C) 13 (D) 24.
• Independent check inputs: Solve x/2+x/3=15. Candidate answers: (A) 17 (B) 18 (C) 19 (D) 36.

Learn the idea

An equation’s solution count depends on its entire structure. Balanced simplification may isolate a variable, reduce to an identity, or expose a contradiction; each outcome has a different interpretation. Multiplying both sides by the least common denominator can turn fractional coefficients into whole-number coefficients. Each term must receive that multiplier, including constants on the opposite side.

Here is the method on the supplied teaching case. The least common denominator is 6. Multiply every term by 6: 3x+2x=60. Thus 5x=60 and x=12.

An equation multiplier applies to the whole side, not merely the fractions that look inconvenient. After solving the easier equation, substitute into the original fractional equation to verify equivalence.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: Solve x/2+x/3=45. Candidate answers: (A) 108 (B) 53 (C) 54 (D) 55. Show the mathematical reason, not only the selected answer. Expected result: 54. The least common denominator is 6. Multiply every term by 6: 3x+2x=270. Thus 5x=270 and x=54.

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