Question

How do you solve \[\dfrac{{n - 6}}{{n - 7}} = \dfrac{9}{2}\] ?

Answer 1

Hint: In the given problem we need to solve this for ‘x’. Using cross multiplication we will obtain a simple linear equation. We can solve this using the transposition method. The common transposition method is to do the same thing (mathematically) to both sides of the equation, with the aim of bringing like terms together and isolating the variable (or the unknown quantity). That is we group the ‘x’ terms one side and constants on the other side of the equation.

Complete step by step solution:
Given, \[\dfrac{{n - 6}}{{n - 7}} = \dfrac{9}{2}\] .
Cross multiplying we have,
 \[2(n - 6) = 9(n - 7)\]
Expanding the brackets we have,
 \[2n - 12 = 9n - 63\]
We transpose \[ - 12\] which is present in the left hand side of the equation to the right hand side of the equation by adding \[12\] on the right hand side of the equation.
 \[2n = 9n - 63 + 12\]
We transpose \[9n\] to the left hand side of the equation by subtracting \[9n\] on the left hand side of the equation.
 \[2n - 9n = - 63 + 12\]
 \[ - 7n = - 51\] .
Dividing the whole equation by \[ - 7\]
 \[ \Rightarrow n = \dfrac{{ - 51}}{{ - 7}}\] .
 \[ \Rightarrow n = \dfrac{{51}}{7}\] This is the exact form.
 \[ \Rightarrow n = 7.285\] . This is the decimal form.
So, the correct answer is “ \[ \ n = \dfrac{{51}}{7}\]”.

Note: We can check whether the obtained solution is correct or wrong. All we need to do is substituting the value of ‘n’ in the given problem.
 \[\dfrac{{\left( {\dfrac{{51}}{7}} \right) - 6}}{{\left( {\dfrac{{51}}{7}} \right) - 7}} = \dfrac{9}{2}\]
 \[\dfrac{{\left( {\dfrac{{51 - 42}}{7}} \right)}}{{\left( {\dfrac{{51 - 49}}{7}} \right)}} = \dfrac{9}{2}\]
 \[\dfrac{{\left( {\dfrac{9}{7}} \right)}}{{\left( {\dfrac{2}{7}} \right)}} = \dfrac{9}{2}\]
 \[ \Rightarrow \dfrac{9}{2} = \dfrac{9}{2}\]
Hence the obtained answer is correct.
In the above we did the transpose of addition and subtraction. Similarly if we have multiplication we use division to transpose. If we have division we use multiplication to transpose. Follow the same procedure for these kinds of problems.