Question

If $7x - 15y = 4x + y$ ,use properties of proportion to find the value of $\dfrac{{9x + 5y}}{{9x - 5y}}$ .

Answer 1

Hint: From the properties of proportion,using componendo and dividendo rule,we know that if a:b=c:d then $(a + b):(a - b) = (c + d):(c - d)$ .

Complete step-by-step answer:
Step1: Let us find the ratio of x and y from the given equation.
 $\eqalign{
  & Given,\,\,7x - 15y = 4x + y \cr
  & \Rightarrow 3x = 16y \cr
  & \Rightarrow ,x:y = 16:3 \cr} $
Step2: To get 9x in the ratio,multiplying first element by 9 in both sides,
$ \Rightarrow $ $9x:y = (9 \times 16):3 = 144:3$
Step3: To get 5y in the ratio,multiplying second element by 5 in both sides,
$ \Rightarrow $ $9x:5y = 144:(5 \times 3) = 144:15 = 48:5$
Step4:Using the rule stated in hint,
$ \Rightarrow $ $(9x + 5y):(9x - 5y) = (48 + 5):(48 - 5) = 53:43$
Therefore $\dfrac{{9x + 5y}}{{9x - 5y}} = \dfrac{{53}}{{43}} = 1\dfrac{{10}}{{43}}$ .

Note: Here we use both componendo as well as dividendo rule as per question.If we justify the rule, $3:2 = 6:4,\,\,then\,\,\dfrac{{3 + 2}}{{3 - 2}} = \dfrac{5}{1} = \dfrac{{10}}{2} = \dfrac{{6 + 4}}{{6 - 4}}$ . Hence the rule holds.We can prove this rule by taking the common ratio value equal to some fixed number.