1Cademy - Translating and Solving $$n$$ divided by $$8$$ is $$-32$$

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Translating and Solving ' $n$ divided by $8$ is $-32$ '

To translate and solve the sentence " $n$ divided by $8$ is $-32$ ," apply the systematic sentence-to-equation translation process:

Locate the "equals" word: The word "is" translates to an equals sign ( $=$ ).
Translate the left side: " $n$ divided by $8$ " uses the keyword divided by to signal division, giving $\frac{n}{8}$ .
Translate the right side: The right side is simply $-32$ .

Combining these elements yields the equation $\frac{n}{8} = -32$ .

To solve the equation, isolate the variable by multiplying both sides by $8$ (the Multiplication Property of Equality):

$8 \cdot \frac{n}{8} = 8 \cdot (-32)$

On the left side, $8 \cdot \frac{n}{8} = \frac{8n}{8} = n$ , because multiplying by the same number that divides the variable cancels the division. On the right side, $8 \cdot (-32) = -256$ (a positive times a negative produces a negative product). The solution is:

$n = -256$

Finally, verify by substituting $-256$ for $n$ in the original equation:

$\frac{-256}{8} \stackrel{?}{=} -32$

-32 = -32 checkmark

Because both sides are equal, $n = -256$ is confirmed as the correct solution. This example illustrates how the phrase "divided by" in a sentence signals a division equation, which is then solved by multiplying both sides by the divisor — the inverse operation of the division that acts on the variable.

Updated 2026-04-21

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OpenStax Elementary Algebra

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