1Cademy - Translating and Solving The quotient of $$y$$ and $$-4$$ is $$68$$

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Translating and Solving 'The quotient of $y$ and $-4$ is $68$ '

To translate and solve the sentence "The quotient of $y$ and $-4$ is $68$ ," 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 quotient of $y$ and $-4$ " uses the keyword quotient to signal division, giving $\frac{y}{-4}$ .
Translate the right side: The right side is simply $68$ .

Combining these elements yields the equation $\frac{y}{-4} = 68$ .

To solve the equation, isolate the variable by multiplying both sides by $-4$ (the Multiplication Property of Equality):

$(-4) \cdot \frac{y}{-4} = (-4) \cdot 68$

On the left side, $(-4) \cdot \frac{y}{-4} = \frac{-4y}{-4} = y$ , because multiplying by the same number that divides the variable cancels the division. On the right side, $(-4)(68) = -272$ (a negative times a positive produces a negative product). The solution is:

$y = -272$

Finally, verify by substituting $-272$ for $y$ in the original equation:

$\frac{-272}{-4} \stackrel{?}{=} 68$

$68 = 68 \checkmark$

Because both sides are equal, $y = -272$ is confirmed as the correct solution. This example illustrates how the keyword "quotient" in a sentence signals a division equation — just like "divided by" — and the equation is solved by multiplying both sides by the negative divisor. When two negative numbers are divided, their quotient is positive, which is why $\frac{-272}{-4}$ yields the positive value $68$ .

Updated 2026-04-21

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OpenStax Elementary Algebra

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