Kumar lent an amount to Arif at a simple interest rate of $10\%$ p.a. for $3$ years, and Arif lent this amount to Naresh at a simple interest rate of $20\%$ p.a. for $3$ years. If the interest Arif received was ₹$1,560$ after $3$ years, then what was the amount that Kumar had lent to Arif?

Let the principal amount lent by Kumar to Arif be $P$.The interest received by Arif from Naresh is calculated at a rate of $20\%$ p.a. for $3$ years:
$$\text{Interest received by Arif}=\frac{P\times 20\times 3}{100}=\frac{60P}{100}$$
Given that the interest Arif received was ₹$1,560$:
$$\frac{60P}{100}=1560$$
$$60P=1560\times 100$$
$$P=\frac{156000}{60}$$
$$P=2600$$
Note: If the question means the net profit/interest earned by Arif (interest received minus interest paid), then:
$$\text{Net Interest}=\text{Interest Received}-\text{Interest Paid}$$
$$\text{Net Interest}=\frac{P\times 20\times 3}{100}-\frac{P\times 10\times 3}{100}=\frac{30P}{100}$$
Given net interest = $1560$:
$$\frac{30P}{100}=1560$$
$$30P=156000$$
$$P=\frac{156000}{30}=5200$$
Thus, interpreting "the interest Arif received" as his net gain from the transaction, the amount is ₹$5,200$.