$A$ and $B$ are partners in a firm sharing profits in the ratio of $5:3$. They admit $C$ as a new partner for $\frac{1}{7}$th share in the profits. The new profit sharing ratio will be $4:2:1$. The sacrificing ratio of $A$ and $B$ is

We need to find how much $A$ and $B$ each gave up (sacrificed) from their original shares to make room for $C$.

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$A$ and $B$ share profits in the ratio $5:3$

Total parts $= 5 + 3 = 8$

$A$'s original share $= \frac{5}{8}$

$B$'s original share $= \frac{3}{8}$

💡 We convert ratios to fractions to see what portion of the total profit each partner gets.

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The new ratio is $4:2:1$ $(A:B:C)$

Total parts $= 4 + 2 + 1 = 7$

$A$'s new share $= \frac{4}{7}$

$B$'s new share $= \frac{2}{7}$

$C$'s new share $= \frac{1}{7}$

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Sacrifice $=$ Original Share $-$ New Share

$A$'s sacrifice:

$= \frac{5}{8} - \frac{4}{7}$

To subtract, we need a common denominator (LCM of $8$ and $7 = 56$):

$= \frac{5 \times 7}{56} - \frac{4 \times 8}{56} = \frac{35}{56} - \frac{32}{56} = \frac{3}{56}$

$B$'s sacrifice:

$= \frac{3}{8} - \frac{2}{7}$

$= \frac{3 \times 7}{56} - \frac{2 \times 8}{56} = \frac{21}{56} - \frac{16}{56} = \frac{5}{56}$

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Sacrificing Ratio $= A$'s sacrifice $: B$'s sacrifice

$= \frac{3}{56} : \frac{5}{56}$

Since denominators are the same:

$= 3:5$

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💡 $A$ sacrificed less ($3$ parts) and $B$ sacrificed more ($5$ parts) to bring in $C$.

The sacrificing ratio of $A$ and $B$ is $3:5$