This problem involves calculating the ratio of two investment amounts based on profit percentages and a given difference when these percentages are interchanged. We will use algebraic equations to solve for the individual investment amounts and then determine their ratio.

Investment Problem Setup

Let the total amount invested be Rs. 100,000. This amount is divided into two parts and invested in two different schemes.

• Let the first part of the investment be \(x\) rupees.
• Let the second part of the investment be \(y\) rupees.

From the problem statement, we know that the sum of these two investments is the total amount:

\[x + y = 100,000 \quad \text{(Equation 1)}\]

Profit Calculation Scenarios

Original Profit Scheme

In the original scheme:

• The first part (\(x\)) yields a 10% profit.
• The second part (\(y\)) yields a 12% profit.

The total profit in this scenario, let's call it \(P_1\), can be expressed as:

\[P_1 = 0.10x + 0.12y\]

Interchanged Profit Scheme

If the profit percentages are interchanged:

• The first part (\(x\)) yields a 12% profit.
• The second part (\(y\)) yields a 10% profit.

The total profit in this interchanged scenario, let's call it \(P_2\), can be expressed as:

\[P_2 = 0.12x + 0.10y\]

Formulating the Difference Equation

The problem states that if the profit percentages are interchanged, the person would have got Rs. 120 less. This means the original profit \(P_1\) is Rs. 120 more than the interchanged profit \(P_2\).

Therefore, we can write the equation:

\[P_1 - P_2 = 120\]

Substitute the expressions for \(P_1\) and \(P_2\) into this equation:

\[(0.10x + 0.12y) - (0.12x + 0.10y) = 120\]

Now, simplify the equation by combining like terms:

\[0.10x + 0.12y - 0.12x - 0.10y = 120\]

\[(0.10x - 0.12x) + (0.12y - 0.10y) = 120\]

\[-0.02x + 0.02y = 120\]

To eliminate the decimals, multiply the entire equation by 100:

\[-2x + 2y = 120 \times 100\]

\[-2x + 2y = 12000\]

Divide the entire equation by 2 to simplify further:

\[-x + y = 6000 \quad \text{(Equation 2)}\]

Solving for Investment Amounts

We now have a system of two linear equations with two variables:

1. \(x + y = 100,000\)
2. \(-x + y = 6000\)

We can solve this system using the elimination method. Add Equation 1 and Equation 2:

\[(x + y) + (-x + y) = 100,000 + 6000\]

\[x + y - x + y = 106,000\]

\[2y = 106,000\]

Now, solve for \(y\):

\[y = \frac{106,000}{2}\]

\[y = 53,000\]

Substitute the value of \(y\) back into Equation 1 (\(x + y = 100,000\)) to find \(x\):

\[x + 53,000 = 100,000\]

\[x = 100,000 - 53,000\]

\[x = 47,000\]

So, the two parts of the investment are Rs. 47,000 and Rs. 53,000.

Calculating Investment Ratio

The problem asks for the ratio between his investments in the two schemes. This is the ratio of \(x\) to \(y\).

\[\text{Ratio} = x : y\]

\[\text{Ratio} = 47,000 : 53,000\]

To simplify the ratio, divide both numbers by their greatest common divisor, which is 1000:

\[\text{Ratio} = \frac{47,000}{1000} : \frac{53,000}{1000}\]

\[\text{Ratio} = 47 : 53\]

Therefore, the ratio between his investments in the two schemes is 47 : 53.