1Cademy - Translating and Solving Three-fourths of $$p$$ is $$18$$

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Translating and Solving 'Three-fourths of $p$ is $18$ '

To translate and solve the sentence "Three-fourths of $p$ is $18$ ," apply the systematic sentence-to-equation translation process:

Locate the "equals" word: The word "is" translates to an equals sign ( $=$ ).
Translate the left side: "Three-fourths of $p$ " uses the word of to signal multiplication between the fraction $\frac{3}{4}$ and the variable $p$ , giving $\frac{3}{4}p$ .
Translate the right side: The right side is simply $18$ .

Combining these elements yields the equation $\frac{3}{4}p = 18$ .

To solve, isolate the variable by multiplying both sides by $\frac{4}{3}$ , the reciprocal of the fractional coefficient $\frac{3}{4}$ (Multiplication Property of Equality):

$\frac{4}{3} \cdot \frac{3}{4}p = \frac{4}{3} \cdot 18$

On the left side, $\frac{4}{3} \cdot \frac{3}{4} = 1$ by the inverse property of multiplication, so $1 \cdot p = p$ . On the right side, $\frac{4}{3} \cdot 18 = \frac{72}{3} = 24$ . The solution is:

$p = 24$

Finally, verify by substituting $24$ for $p$ in the original equation:

$\frac{3}{4} \cdot 24 \stackrel{?}{=} 18$

$18 = 18 \checkmark$

Because both sides are equal, $p = 24$ is confirmed as the correct solution. This example highlights two important translation cues: the word "is" signals an equals sign, and the word "of" — when it appears between a fraction and a variable — signals multiplication. Recognizing "of" as multiplication is essential for correctly setting up equations that involve fractional parts of unknown quantities.

Updated 2026-04-21

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OpenStax Elementary Algebra

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