The sum of the cost prices of three items was $₹24,150$. The person sold the first item at a profit of $72\%$, the second item at a profit of $40\%$, and the third item at a profit of $126\%$. The total profit earned from the sale of these three items was $₹19,432$. If the cost price of the second item was $₹175$ more than what it actually was, then the ratio of the cost prices of the second and the third items, respectively, would have been $4 : 5$. What was the difference (in Rs.) between the profits earned from the sale of the first and the second items?

Let the cost prices of the first, second and third items be $a$, $b$ and $c$ (in Rs.).

$a + b + c = 24150$

Total profit $= 72\%$ of $a$ $+$ $40\%$ of $b$ $+$ $126\%$ of $c$:

$0.72a + 0.40b + 1.26c = 19432$

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If the second item cost Rs. $175$ more, the ratio of second to third would be $4 : 5$:

$\dfrac{b + 175}{c} = \dfrac{4}{5}$

$5b + 875 = 4c$

$c = \dfrac{5b + 875}{4}$

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Put $a = 24150 - b - c$ in the profit equation:

$0.72(24150 - b - c) + 0.40b + 1.26c = 19432$

$17388 - 0.72b - 0.72c + 0.40b + 1.26c = 19432$

$0.54c - 0.32b = 2044$

$27c - 16b = 102200$

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Put $c = \dfrac{5b + 875}{4}$:

$27 \times \dfrac{5b + 875}{4} - 16b = 102200$

$27(5b + 875) - 64b = 408800$

$135b + 23625 - 64b = 408800$

$71b = 385175$

$b = 5425$

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$c = \dfrac{5 \times 5425 + 875}{4}$

$= \dfrac{28000}{4}$

$= 7000$

$a = 24150 - 5425 - 7000$

$= 11725$

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Profit on the first item $= 0.72 \times 11725 = 8442$

Profit on the second item $= 0.40 \times 5425 = 2170$

Difference $= 8442 - 2170 = 6272$

Therefore, the required difference is Rs. $6272$.