Question

solve the equation below. whats the value of g?
now, rewrite the expression on the left without parentheses. we have 1 group of 15g + 5 in the numerator, which can just be written as 15g + 5. then, rewrite 5 in the denominator.
\\(\frac{2(15g + 5)}{10}\\) = \\(-5g + 3\\)
\\(\frac{15g + 5}{\square}\\) = \\(-5g + 3\\)
options: 2, 5, 10

Explanation:

Step1: Analyze the denominator simplification

The original numerator is \(2(15g + 5)\) and denominator is \(10\). We can simplify the fraction \(\frac{2(15g + 5)}{10}\) by dividing numerator and denominator by their greatest common divisor, which is \(2\)? Wait, no, wait. Wait, the problem says "rewrite 5 in the denominator". Wait, let's check the given text: "We have 1 group of \(15g + 5\) in the numerator, which can just be written as \(15g + 5\). Then, rewrite 5 in the denominator." Wait, maybe there was a miscalculation earlier. Wait, the original fraction was \(\frac{2(15g + 5)}{10}\). Let's factor the denominator: \(10 = 2\times5\). The numerator is \(2(15g + 5)\). So we can cancel the 2 in the numerator and denominator: \(\frac{2(15g + 5)}{2\times5}=\frac{15g + 5}{5}\). Ah, so the denominator after simplifying the fraction \(\frac{2(15g + 5)}{10}\) is \(5\)? Wait, no, wait. Wait, the problem is showing that after rewriting the numerator as \(15g + 5\) (maybe they factored out the 2? Wait, no, the text says "1 group of \(15g + 5\) in the numerator", so maybe the \(2\) and the denominator's factor? Wait, let's look at the boxes. The first box is the denominator, and the options are 2,5,10. Wait, the original fraction is \(\frac{2(15g + 5)}{10}\). Let's simplify that fraction. Divide numerator and denominator by 2: \(\frac{15g + 5}{5}\). So the denominator here is 5? Wait, no, the numerator becomes \(15g + 5\) (because \(2\) times \(15g + 5\) divided by 2 is \(15g + 5\)) and the denominator is \(10\) divided by 2, which is \(5\)? Wait, no, \(\frac{2(15g + 5)}{10}=\frac{15g + 5}{5}\) (since \(2\div2 = 1\) and \(10\div2 = 5\)). So the denominator is 5? Wait, but the problem is asking for the denominator in the box. Wait, the expression after rewriting is \(\frac{15g + 5}{\square}=-5g + 3\). From the simplification \(\frac{2(15g + 5)}{10}=\frac{15g + 5}{5}\) (because \(2\) in numerator and \(10 = 2\times5\) in denominator, so cancel 2), so the denominator is 5. Wait, but let's check the options: 2,5,10. So the correct denominator is 5? Wait, no, wait. Wait, the original fraction is \(\frac{2(15g + 5)}{10}\). Let's compute that: \(\frac{2(15g + 5)}{10}=\frac{30g + 10}{10}=\frac{30g}{10}+\frac{10}{10}=3g + 1\). But the problem says to rewrite the numerator as \(15g + 5\) (so maybe they did \(\frac{2(15g + 5)}{10}=\frac{15g + 5}{5}\) by dividing numerator and denominator by 2). So the denominator is 5. So the box should be 5. Wait, but let's confirm. The text says "rewrite 5 in the denominator". So the denominator is 5. So the answer for the box is 5. Then, after that, we can solve for \(g\). Let's proceed.

Step2: Solve for \(g\) using the simplified equation

Now that we have \(\frac{15g + 5}{5}=-5g + 3\). First, simplify the left side: \(\frac{15g}{5}+\frac{5}{5}=3g + 1\). So the equation becomes \(3g + 1=-5g + 3\).

Step3: Add \(5g\) to both sides

\(3g + 5g + 1=-5g + 5g + 3\)
\(8g + 1 = 3\)

Step4: Subtract 1 from both sides

\(8g + 1 - 1=3 - 1\)
\(8g = 2\)

Step5: Divide both sides by 8

\(g=\frac{2}{8}=\frac{1}{4}=0.25\)

Wait, but the first part was about the denominator. The question in the image first asks for the denominator (the box) and then solve for \(g\). Let's first solve the denominator part. The original fraction is \(\frac{2(15g + 5)}{10}\). We can simplify this fraction by dividing numerator and denominator by 2: \(\frac{2(15g + 5)\div2}{10\div2}=\frac{15g + 5}{5}\). So the denominator is 5. So the box should be 5. Then, solving for \(g\):

From \(\frac{15g + 5}{5}=-5g + 3\), multiply both sides by 5 to eliminate the denominator:
\(15g + 5 = 5(-5g + 3)\)
\(15g + 5=-25g + 15\)

Add \(25g\) to both sides:
\(15g + 25g + 5=-25g + 25g + 15\)
\(40g + 5 = 15\)

Subtract 5 from both sides:
\(40g + 5 - 5=15 - 5\)
\(40g = 10\)

Divide both sides by 40:
\(g=\frac{10}{40}=\frac{1}{4}=0.25\)

But the first part is the denominator. The options are 2,5,10. So the denominator is 5. So the box is 5. Then, solving for \(g\) gives \(g = \frac{1}{4}\) or 0.25.

Wait, the problem first asks to fill the denominator box, then solve for \(g\). Let's confirm the denominator:

Original fraction: \(\frac{2(15g + 5)}{10}\). Simplify numerator and denominator by dividing by 2: \(\frac{15g + 5}{5}\). So denominator is 5. So the box is 5. Then, solving the equation:

\(\frac{15g + 5}{5}=-5g + 3\)
Multiply both sides by 5: \(15g + 5 = -25g + 15\)
Add \(25g\) to both sides: \(40g + 5 = 15\)
Subtract 5: \(40g = 10\)
Divide by 40: \(g = \frac{10}{40} = \frac{1}{4}\)

So the value of \(g\) is \(\frac{1}{4}\) or 0.25.

But the first part (the denominator) is 5. Let's check the options: 2,5,10. So the correct denominator is 5. Then, solving for \(g\) gives \(g = \frac{1}{4}\).

Answer:

The denominator box is 5, and the value of \(g\) is \(\frac{1}{4}\) (or 0.25). But if we are to answer the value of \(g\) after solving, the answer is \(\frac{1}{4}\) (or 0.25).