Question

solve the proportion.\\(\frac{3b - 2}{6}=\frac{b - 3}{12}\\)\\(b=square\\) (type an integer or a simplified fraction.)

Explanation:

Step1: Cross - multiply the proportion

To solve the proportion \(\frac{3b - 2}{6}=\frac{b - 3}{12}\), we use the cross - multiplication property of proportions. If \(\frac{a}{b}=\frac{c}{d}\), then \(a\times d=b\times c\). So, we have \((3b - 2)\times12=6\times(b - 3)\).

Step2: Expand both sides

Expand the left - hand side: \(12\times(3b-2)=12\times3b-12\times2 = 36b-24\).
Expand the right - hand side: \(6\times(b - 3)=6b-18\).
Now our equation is \(36b-24 = 6b-18\).

Step3: Isolate the variable terms

Subtract \(6b\) from both sides of the equation: \(36b-6b-24=6b - 6b-18\), which simplifies to \(30b-24=-18\).
Then add 24 to both sides of the equation: \(30b-24 + 24=-18 + 24\), so \(30b=6\).

Step4: Solve for \(b\)

Divide both sides of the equation by 30: \(b=\frac{6}{30}\).
Simplify the fraction \(\frac{6}{30}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 6. So, \(b=\frac{6\div6}{30\div6}=\frac{1}{5}\).

Answer:

\(\frac{1}{5}\)