A certain sum of money amount to Rs 2200 at 5% interest Rs 2320 at 8% interest in the same period of time. The period of time is = ?

Answer: Option A

$$\begin{array}{rl}& \text{P}+\text{SI = }\frac{\text{P}\times \text{R}\times \text{T}}{100}+\text{P}\\ & \Rightarrow 2200\text{ = }\frac{\text{P}\times 5\times \text{T}}{100}+\text{P}\end{array}$$
⇒ 2200 × 100 = 5PT + 100P ......... (i)
$$\Rightarrow 2320=\frac{\text{P}\times 8\times \text{T}}{100}+\text{P}$$
⇒ 2320 × 100 = 8PT + 100P
⇒ 2320 × 100 = 3PT + 5PT + 100P ............(ii)
Value of equation (i) put equation (ii)
⇒ 2320 × 100 = 3PT + 2200 × 100
⇒ 3PT = 120 × 100
⇒ PT = 4000
Value of PT in equation (i)
⇒ 2200 × 100 = 5 × 4000 + 100P
⇒ 220000 - 20000 = 100P
$$\begin{array}{rl}& \Rightarrow \text{P = }\frac{200000}{100}\\ & \Rightarrow \text{P = Rs 2000}\\ & \text{Using this formula}\\ & \left(\because \text{SI = }\frac{\text{P}\times \text{R}\times \text{T}}{100}\right)\\ & \therefore \text{200}=\frac{2000\times 5\times \text{T}}{100}\\ & \Rightarrow \text{T}=\frac{200}{100}\text{ = 2 years}\\ & \\ & \mathbf{Alternate}:\\ & \left(8-5\right)\%=2320-2200\\ & \Rightarrow 3\%=120\\ & \Rightarrow 1\%=40\\ & \Rightarrow 5\%=200\\ & \text{Principal = 2200 - 200}\\ & \text{ = Rs}\text{. 2000}\\ & \text{SI = }\frac{\text{P}\times \text{R}\times \text{T}}{100}\\ & \Rightarrow \text{200}=\frac{2000\times 5\times \text{T}}{100}\\ & \Rightarrow \text{T}=\frac{200}{100}\text{ = 2 years}\end{array}$$