To determine how long it will take for India’s GDP growth rate to surpass that of the United States’, we need to consider the concept of relative growth rates and the compounding effect over time.

Let’s denote:
– \( G_{India} \) as India’s GDP growth rate (7% per year)
– \( G_{US} \) as United States’ GDP growth rate (2% per year)
– \( GDP_{India}(0) \) as India’s current GDP
– \( GDP_{US}(0) \) as United States’ current GDP

The condition we’re interested in is when India’s GDP (\( GDP_{India}(t) \)) overtakes United States’ GDP (\( GDP_{US}(t) \)).

Using the formula for GDP growth compounded annually:

\[ GDP_{India}(t) = GDP_{India}(0) \times (1 + G_{India})^t \]
\[ GDP_{US}(t) = GDP_{US}(0) \times (1 + G_{US})^t \]

We need to find \( t \) such that:

\[ GDP_{India}(0) \times (1 + 0.07)^t > GDP_{US}(0) \times (1 + 0.02)^t \]

Simplifying this inequality:

\[ (1.07)^t > \frac{GDP_{US}(0)}{GDP_{India}(0)} \times (1.02)^t \]

Taking the natural logarithm on both sides gives:

\[ t \times \ln(1.07) > \ln\left( \frac{GDP_{US}(0)}{GDP_{India}(0)} \times (1.02)^t \right) \]

Solving for \( t \):

\[ t > \frac{\ln\left( \frac{GDP_{US}(0)}{GDP_{India}(0)} \times (1.02)^t \right)}{\ln(1.07)} \]

This equation calculates the minimum number of years required for India’s GDP growth rate to surpass that of the United States’. The exact number of years will depend on the initial GDP values of both countries and the growth rates specified.