1Cademy - Translating and Solving The sum of three-eighths and $$x$$ is one-half

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Translating and Solving 'The sum of three-eighths and $x$ is one-half'

To translate and solve the sentence "The sum of three-eighths and $x$ is one-half," apply the systematic sentence-to-equation translation process:

Locate the "equals" word: The word "is" translates to an equals sign ( $=$ ).
Translate the left side: "The sum of three-eighths and $x$ " uses the keyword sum to signal addition between $\frac{3}{8}$ and $x$ , giving $\frac{3}{8} + x$ .
Translate the right side: "One-half" is simply $\frac{1}{2}$ .

Combining these elements yields the equation $\frac{3}{8} + x = \frac{1}{2}$ .

To solve, isolate the variable by subtracting $\frac{3}{8}$ from both sides (Subtraction Property of Equality):

$\frac{3}{8} + x - \frac{3}{8} = \frac{1}{2} - \frac{3}{8}$

On the left side, $\frac{3}{8} - \frac{3}{8} = 0$ , leaving just $x$ . On the right side, the fractions $\frac{1}{2}$ and $\frac{3}{8}$ have different denominators, so rewrite $\frac{1}{2}$ with the common denominator $8$ : $\frac{1}{2} = \frac{4}{8}$ . Then subtract:

$x = \frac{4}{8} - \frac{3}{8} = \frac{1}{8}$

Finally, verify by substituting $\frac{1}{8}$ for $x$ in the original equation:

$\frac{3}{8} + \frac{1}{8} \stackrel{?}{=} \frac{1}{2}$

$\frac{4}{8} \stackrel{?}{=} \frac{1}{2}$

$\frac{1}{2} = \frac{1}{2} \checkmark$

Because both sides are equal, $x = \frac{1}{8}$ is confirmed as the correct solution. This example illustrates how the keyword "sum" in a sentence signals an addition equation involving fractions, which is solved by subtracting the known fraction from both sides. When that subtraction involves fractions with unlike denominators, a common denominator must be found before the numerators can be combined.

Updated 2026-04-21

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OpenStax Elementary Algebra

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