Question

If \[\dfrac{{a + 1}}{{b - 1}} = \dfrac{5}{1}\] and \[\dfrac{{a - 1}}{{b + 1}} = \dfrac{1}{1}\] , how do you find the value of \[\dfrac{b}{a}\] ?

Answer 1

Hint: Here in this question, we have to find the value of ration \[\dfrac{b}{a}\] by using the two given ratios, for finding this we first we have to find the relation of a and b in the both ratios using the basic arithmetic operation and next by the substitution method. On simplification we get the required solution or ratio.

Complete step by step solution:
 a ratio is a comparison of two or more numbers that indicates their sizes in relation to each other. A ratio compares two quantities by division, with the dividend or number being divided termed the antecedent and the divisor or number that is dividing termed the consequent.
Consider the given two ratios:
  \[\dfrac{{a + 1}}{{b - 1}} = \dfrac{5}{1}\] ---------(1)
  \[\dfrac{{a - 1}}{{b + 1}} = \dfrac{1}{1}\] ---------(2)
Now, we have to find the relation of a and b in the both ratios
Consider, the first ratio
  \[ \Rightarrow \dfrac{{a + 1}}{{b - 1}} = \dfrac{5}{1}\]
On cross multiplying, we get
  \[ \Rightarrow 1\left( {a + 1} \right) = 5\left( {b - 1} \right)\]
On removing the parenthesis
  \[ \Rightarrow a + 1 = 5b - 5\]
On rearranging this, we get
  \[ \Rightarrow a - 5b = - 5 - 1\]
  \[ \Rightarrow a - 5b = - 6\] ------(3)
Now, consider the second ratio
  \[ \Rightarrow \dfrac{{a - 1}}{{b + 1}} = \dfrac{1}{1}\]
On cross multiplying, we get
  \[ \Rightarrow a - 1 = b + 1\]
On rearranging this, we get
  \[ \Rightarrow a = b + 1 + 1\]
  \[ \Rightarrow a = b + 2\] --------(4)
Substitute, equation (4) in (3), then
  \[ \Rightarrow b + 2 - 5b = - 6\]
Take constant value on RHS
  \[ \Rightarrow b - 5b = - 6 - 2\]
  \[ \Rightarrow - 4b = - 8\]
Divide both side by -4, we get
  \[ \Rightarrow b = \dfrac{{ - 8}}{{ - 4}}\]
On simplification, we get
  \[ \Rightarrow b = 2\]
We found the value of b substitute b in equation (3) or equation (4), to find the value of a
Consider equation (4)
  \[ \Rightarrow a = b + 2\]
  \[ \Rightarrow a = 2 + 2\]
  \[ \Rightarrow a = 4\]
Therefore, the value of \[\,\dfrac{b}{a} = \dfrac{2}{4}\]
Divide both numerator and denominator by 2
  \[ \Rightarrow \dfrac{b}{a} = \dfrac{1}{2}\]
Hence, the value of \[\dfrac{b}{a}\] is \[\dfrac{1}{2}\] .
So, the correct answer is “ \[\dfrac{1}{2}\] ”.

Note: The given equation is of the form of the algebraic equation. Here we have to find the equation for the variable \[\dfrac{b}{a}\] . If we want to write the equation for the variable, we transfer the other variable or terms or constants to the other side. While shifting or transferring the terms the sign will change. Hence we obtain the required result for the given question.