Jada solved the equation $-\frac{4}{9}=\frac{x}{108}$ for $x$ using the steps below. What was Jada's error?
$\begin{aligned}
-\frac{4}{9} & =\frac{x}{108} \\
-\frac{4}{9}\left(-\frac{9}{4}\right) & =\frac{x}{108}\left(-\frac{9}{4}\right) \\
x & =-\frac{1}{48}
\end{aligned}$
A. Jada should have multiplied both sides of the equation by 108.
B. Jada should have multiplied both sides of the equation by $-\frac{4}{9}$.
C. The product of $-\frac{9}{4}$ and $\frac{1}{108}$ is not equal to $-\frac{1}{48}$.
Answer (2)
To solve for x in the equation − 9 4 = 108 x , multiply both sides by 108.
This gives x = − 9 4 × 108 = − 48 .
Jada incorrectly multiplied both sides by − 4 9 , which does not directly isolate x .
Jada's error was that she should have multiplied both sides of the equation by 108, so the answer is: Jada should have multiplied both sides of the equation by 108. Jada should have multiplied both sides of the equation by 108.
Explanation
Analyzing the Problem We are given the equation − 9 4 = 108 x and Jada's attempt to solve for x . We need to identify the error in Jada's steps. Let's first solve the equation correctly. To isolate x , we need to multiply both sides of the equation by 108.
Solving for x Multiplying both sides of the equation by 108, we get: − 9 4 × 108 = 108 x × 108 − 9 4 × 108 = x − 9 432 = x x = − 48
Analyzing Jada's Steps Now let's analyze Jada's steps. Jada multiplied both sides of the equation by − 4 9 . This is not the correct operation to isolate x . While multiplying by the reciprocal can be useful in some situations, in this case, it complicates the equation rather than simplifying it directly to solve for x .
Verifying the Product Let's check Jada's calculation: − 9 4 ( − 4 9 ) = 108 x ( − 4 9 ) . This simplifies to 1 = 108 x ( − 4 9 ) . Then, x = 108 ( − 9 4 ) = − 48 . However, Jada obtained x = − 48 1 . Let's verify if the product of − 4 9 and 108 1 is equal to − 48 1 .
− 4 9 × 108 1 = − 432 9 = − 48 1 So, the product is indeed − 48 1 . However, Jada made an error in the initial step by multiplying by − 4 9 instead of 108.
Identifying the Error Therefore, Jada's error was that she should have multiplied both sides of the equation by 108 to isolate x .
Examples
When solving for a variable in an equation, it's like unwrapping a present. You need to undo each step to reveal the gift (the variable). In this case, to find 'x', we needed to multiply both sides by 108, just like carefully unwrapping each layer of wrapping paper to get to the present inside. This principle is used in many real-world scenarios, such as calculating ingredient amounts in recipes or determining distances on maps.
Jada made a mistake by multiplying both sides of the equation − 9 4 = 108 x by − 4 9 instead of 108. The correct approach is to multiply both sides by 108 to isolate x correctly. The answer is: Jada should have multiplied both sides of the equation by 108.
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