Sagarika divides her savings of $10000$ rupees to invest across two schemes A and B. Scheme A offers an interest rate of $10\%$ per annum, compounded half-yearly, while scheme B offers a simple interest rate of $12\%$ per annum. If at the end of first year, the value of her investment in scheme B exceeds the value of her investment in scheme A by $2310$ rupees, then the total interest, in rupees, earned by Sagarika during the first year of investment is

Sagarika has $Rs.10,000$ total, split between two schemes:

	Scheme A	Scheme B
Interest Type	Compound Interest (CI)	Simple Interest (SI)
Rate	$10\%$ p.a., compounded half-yearly	$12\%$ p.a.

"Compounded half-yearly" means instead of applying $10\%$ once a year, the bank applies $5\%$ every 6 months (half of $10\%$), and you earn interest on interest after each 6-month period. So in 1 year, there are 2 compounding periods.

We need to find the total interest earned from both schemes combined after 1 year.

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Let the amount invested in Scheme B $= x$ rupees

Then the amount invested in Scheme A $= (10000 - x)$ rupees

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Total amount from Scheme A after 1 year (CI, half-yearly):

We divide the rate by 2 (interest applied every half-year) and raise to power 2 (two half-year periods in 1 year):

$A_1 = (10000 - x)\left(1 + \dfrac{10}{2 \times 100}\right)^2 = (10000 - x)\left(\dfrac{21}{20}\right)^2 = (10000 - x) \times \dfrac{441}{400}$

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Total amount from Scheme B after 1 year (SI):

$A_2 = x + \dfrac{x \times 12 \times 1}{100} = \dfrac{112x}{100}$

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Value of investment in B exceeds that in A by $Rs.2310$:

$A_2 - A_1 = 2310$

$\dfrac{112x}{100} - (10000 - x) \times \dfrac{441}{400} = 2310$

Taking common denominator of $400$:

$\dfrac{448x - 441(10000 - x)}{400} = 2310$

Distributing: $441(10000 - x) = 4410000 - 441x$

$\dfrac{448x - 4410000 + 441x}{400} = 2310$

$\dfrac{889x - 4410000}{400} = 2310$

$889x - 4410000 = 924000$

$889x = 5334000$

$x = \dfrac{5334000}{889} = 6000$

Since $889 \times 6 = 5334$, so $889 \times 6000 = 5334000$ ✓

So Scheme B $= Rs.6000$ and Scheme A $= Rs.4000$

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Simple Interest from Scheme B:

$SI = \dfrac{6000 \times 12 \times 1}{100} = 720$

Compound Interest from Scheme A (Total Amount $-$ Principal):

$CI = 4000 \times \dfrac{441}{400} - 4000 = 4000\left(\dfrac{441 - 400}{400}\right) = 4000 \times \dfrac{41}{400} = 410$

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Total Interest:

$SI + CI = 720 + 410 = \boxed{1130}$

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Verification: $A_1 = 4000 \times \dfrac{441}{400} = 4410$ and $A_2 = \dfrac{112 \times 6000}{100} = 6720$

$A_2 - A_1 = 6720 - 4410 = 2310$ ✓