The price relatives and weights of a set of commodities are given below:

Commodity	A	B	C	D
Price relative	125	120	127	119
Weight	$x$	$2x$	$y$	$y+3$

 If sum of the weights is $40$ and the index for the set is $122$, then the numerical values of $x$ and $y$ is

The above data can be written in a table as given below:

 As it is given that sum of weights is 40, we get

$\begin{array}{l}3x+2y+3=40\\ \Rightarrow 3x+2y=37 .... \left(i\right)\end{array}$

As index number is given to be 122, we get

$\frac{\mathrm{\Sigma }Iw}{\mathrm{\Sigma }w}=122$

$\begin{array}{l}\Rightarrow \frac{365x+246y+357}{40}=122\\ \Rightarrow 365x+246y=122\times 40-357\\ \Rightarrow 365x+246y=4523 .... \left(ii\right)\end{array}$

To solve (i) and (ii), multiply (i) by 123,

$369x+246y=4551 .... \left(iii\right)$

Subtracting (ii) from (iii), we get

$4x=28\Rightarrow x=7$

Putting this value of $x$in (i), we get $y=8$

Hence, $x = 7$ and $y = 8$

$\therefore$ Option (b) is the correct answer.