X and Y sharing profits in the ratio of 7: 3 admit Z for $\frac{3}{7}$ shares in the new firm which he takes $\frac{2}{7}$ from X and $\frac{1}{7}$ from Y. The new ratio of X, Y and Z will be :

Question

Accepted Answer

The correct option is 3:
To determine the new profit-sharing ratio upon the admission of a new partner (Z), we must calculate the remaining share for the old partners (X and Y) after deducting the portion they sacrificed for Z. X and Y were sharing profits in the ratio of 7:3, meaning X's original share was $7/10$ and Y's was $3/10$. Z is admitted for a $3/7$ share, which he takes as $2/7$ from X and $1/7$ from Y. The new share for X is calculated as his Old Share minus his Sacrifice: $7/10 - 2/7$. By finding the common denominator, 70, this becomes $49/70 - 20/70 = 29/70$. Similarly, Y's new share is Old Share minus Sacrifice: $3/10 - 1/7$, which is $21/70 - 10/70 = 11/70$. Z's share of $3/7$ can be expressed with a denominator of 70 as $30/70$. Therefore, the new profit-sharing ratio of X, Y, and Z is $29/70 : 11/70 : 30/70$, or simply $29:11:30$.

The incorrect option is 1:
The ratio 7:03:03 is incorrect because it fails to account for the reduction in X's and Y's old shares due to the sacrifice they made for Z. This option appears to represent X and Y's original shares (7:3) and adds Z's total share (3/7) to create a new ratio, but this is a fundamentally flawed calculation. In the admission of a partner, the old partners' shares must decrease, and simply appending the new partner's numerator to the old ratio is not the correct accounting procedure. This simplification ignores the required algebraic steps of subtraction and finding a common denominator for the entire new ratio.

The incorrect option is 2:
The ratio 14:06:08 is a significantly incorrect simplification and does not reflect the proper calculation of the new ratio. This option seems to have incorrectly scaled or modified the original ratio (7:3) and Z's share ($3/7$), likely due to an error in finding the common denominator. Specifically, $14:6$ is a multiple of the old ratio, but the new partner's share (8/28, if simplified from $8$) does not correctly sum up with the remaining shares. This ratio also does not account for the specific amounts of sacrifice ($2/7$ and $1/7$) made by X and Y.

The incorrect option is 4:
The ratio 14:06:15 is incorrect because the calculations for X's and Z's shares are wrong, while Y's share is likely correct (11/70 is close to $6/28$ or $14/70$). A major flaw in reaching this conclusion would be an error in X's sacrifice or an incorrect conversion of Z's share to the common denominator. The first two terms ($14:6$) are a simplification of the original ratio, and the last term (15) is an arbitrary value not derived from the correct formula of $29:11:30$. This result suggests a mistake in subtracting the sacrificed portion from X's share.