An amount of Rs 10000 is deposited in bank A for a certain number of years at a simple interest of 5% per annum. On maturity, the total amount received is deposited in bank B for another 5 years at a simple interest of 6% per annum. If the interests received from bank A and bank B are in the ratio 10 : 13, then the investment period, in years, in bank A is

We are told that, 10000 is deposited in bank A for a certain number of years at a simple interest of 5% per annum. 
Let us say that the number of years is x
Total value of the deposit after x years is, $$10000\left(1+x\left(0.05\right)\right)$$

On maturity, the total amount received is deposited in bank B for another 5 years at a simple interest of 6% per annum
Here we know the years and the interest rate, 
$$10000\left(1+x\left(0.05\right)\right)\left(1+5\left(0.06\right)\right)$$
$$10000\left(1+\left(0.05\right)x\right)\left(1.3\right)$$

Interest received from Bank A is $$\left(x\left(0.05\right)\right)10000$$
Interest received from Bank B is $$0.3\left(10000\left(1+x\left(0.05\right)\right)\right)$$

This ratio is given to be 10:13. 

$$\dfrac{x\left(0.05\right)}{0.3\left(1+x\left(0.05\right)\right)}=\dfrac{10}{13}$$

$$0.65x=3+0.15x$$
$$0.5x=3$$
$$x=6$$

Hence the number of years the money was invested in Bank A is 6 years.