Question

If $\dfrac{{a + 1}} {b} + 3 = \dfrac{{4a}} {b}$ , then $b$ can be written as
(A) $b = \dfrac{{3a + 1}}
{3}$
(B) $b = \dfrac{{ - 3a + 1}}
{3}$
(C) $b = \dfrac{{3a - 1}}
{3}$
(D) $b = \dfrac{{ - 2a + 1}}
{3}$

Answer 1

Hint:To write the given expression in terms of b, we can first take LCM of each side of the expression with respect to their denominators which is b. Then apply cross multiplication of the denominator of the left hand side to the right hand side of the expression. In this way we can write the given expression in terms of b.

Complete step by step solution:
The given expression is, $\dfrac{{a + 1}}
{b} + 3 = \dfrac{{4a}}
{b}$
To write the above expression in terms of $b$ , first take the term $b$ as the LCM on the left hand side of the above expression.
$
\dfrac{{a + 1}}
{b} + 3 = \dfrac{{4a}}
{b} \\
\dfrac{{a + 1 + 3b}}
{b} = \dfrac{{4a}}
{b} \\
 $
Multiply each side of the above expression by the term $b$ to cancel the denominators of the fractions of the above expression.
$
\left( {\dfrac{{a + 1 + 3b}}
{b}} \right)b = \left( {\dfrac{{4a}}
{b}} \right)b \\
a + 1 + 3b = 4a \\
 $
Now, subtract the term a from each side to cancel the term a from the left hand side of the above expression.
\[
a + 1 + 3b - a = 4a - a \\
1 + 3b = 3a \\
 \]
Subtract the number 1 from each side to cancel the number 1 from the left hand side of the above expression.
\[
1 + 3b = 3a \\
1 + 3b - 1 = 3a - 1 \\
3b = 3a - 1 \\
 \]
Divide each side by the number 3 to cancel the number 3 from the left hand side of the above expression.
\[
\dfrac{{3b}}
{3} = \dfrac{{3a - 1}}
{3} \\
b = \dfrac{{3a - 1}}
{3} \\
 \]
Thus, the given expression can be written in terms of b as $b = \dfrac{{3a - 1}}
{3}$

Note: The given expression can be written in terms of b by using another method. First we can subtract 3 from each side of the above expression in order to cancel the number 3 from the left hand side of the above expression. Then, we can cross multiply the term b in the left hand side to the right hand side of the expression. In this method we can also write the given expression in terms of b.