1Cademy - Solving $$6 = \frac{1}{2}v + \frac{2}{5}v - \frac{3}{4}v$$ by Clearing Fractions

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Strategy for Solving Equations with Fraction or Decimal Coefficients

Example

Solving $6 = \frac{1}{2}v + \frac{2}{5}v - \frac{3}{4}v$ by Clearing Fractions

To solve $6 = \frac{1}{2}v + \frac{2}{5}v - \frac{3}{4}v$ , use the clearing fractions technique to eliminate all three fraction coefficients before combining like terms.

Step 1 — Find the LCD: The denominators are $2$ , $5$ , and $4$ . The least common denominator of these three numbers is $20$ .

Step 2 — Multiply both sides by $20$ and distribute: Apply the Multiplication Property of Equality by multiplying each side by $20$ , then use the Distributive Property so that $20$ reaches every term on the right:

$20 \cdot 6 = 20 \cdot \frac{1}{2}v + 20 \cdot \frac{2}{5}v - 20 \cdot \frac{3}{4}v$

Simplify each product: $20 \cdot \frac{1}{2} = 10$ , $20 \cdot \frac{2}{5} = 8$ , and $20 \cdot \frac{3}{4} = 15$ . The equation is now fraction-free:

$120 = 10v + 8v - 15v$

Step 3 — Combine like terms: On the right side, add the coefficients of $v$ : $10 + 8 - 15 = 3$ , so the equation becomes:

$120 = 3v$

Step 4 — Divide both sides by $3$ : Apply the Division Property of Equality:

$\frac{120}{3} = \frac{3v}{3}$

$40 = v$

Step 5 — Check by substitution: Replace $v$ with $40$ in the original equation:

$\frac{1}{2}(40) + \frac{2}{5}(40) - \frac{3}{4}(40) \stackrel{?}{=} 6$

$20 + 16 - 30 \stackrel{?}{=} 6$

$6 = 6 \checkmark$

Because both sides are equal, $v = 40$ is confirmed as the correct solution. This example shows that when an equation has multiple fraction coefficients on the same variable, clearing all fractions first using the LCD transforms the equation into one where combining like terms involves only whole numbers — here, $10v + 8v - 15v = 3v$ — which is far simpler than adding $\frac{1}{2} + \frac{2}{5} - \frac{3}{4}$ directly.

Updated 2026-04-21

Contributors are:

Claude Opus

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Who are from:

University of Michigan - Ann Arbor

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References

OpenStax Elementary Algebra

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