Analyzing Company Financial Data for 2016 Income

The question asks us to determine the income of the company in the year 2016 based on the provided table data for subsequent years and specific relationships between 2016 and 2017.

First, let's extract the relevant information for the year 2017 from the table:

• Income in 2017 (\(I_{2017}\)) = Rs. 103.824 million
• Profit Percentage in 2017 (\(P\%_{2017}\)) = -12%

A negative profit percentage indicates a loss. The profit percentage is typically calculated relative to the expenditure:

$$ P\% = \frac{\text{Income} - \text{Expenditure}}{\text{Expenditure}} \times 100 $$

This can be rearranged to express Income in terms of Expenditure and Profit Percentage:

$$ \text{Income} = \text{Expenditure} \times \left(1 + \frac{P\%}{100}\right) $$

For the year 2017, we have:

$$ I_{2017} = E_{2017} \times \left(1 + \frac{-12}{100}\right) $$

$$ 103.824 = E_{2017} \times (1 - 0.12) $$

$$ 103.824 = E_{2017} \times 0.88 $$

We can calculate the Expenditure in 2017 (\(E_{2017}\)) from this equation:

$$ E_{2017} = \frac{103.824}{0.88} $$

$$ E_{2017} = 118.004545... \text{ million Rs.} $$

Next, let's consider the relationship between the expenditure in 2017 and 2016. The question states that the expenditure was increased by 20% in 2017 compared to 2016.

$$ E_{2017} = E_{2016} + 20\% \text{ of } E_{2016} $$

$$ E_{2017} = E_{2016} \times (1 + 0.20) $$

$$ E_{2017} = 1.20 \times E_{2016} $$

We can find the Expenditure in 2016 (\(E_{2016}\)) using the calculated \(E_{2017}\):

$$ E_{2016} = \frac{E_{2017}}{1.20} $$

$$ E_{2016} = \frac{118.004545...}{1.20} $$

$$ E_{2016} = 98.337121... \text{ million Rs.} $$

Now, let's consider the relationship between the profit percentage in 2016 and 2017. The question states that the profit percentage in 2016 was 25% less than the profit percentage in 2017.

$$ P\%_{2016} = P\%_{2017} - 25\% \text{ of } P\%_{2017} $$

$$ P\%_{2016} = P\%_{2017} \times (1 - 0.25) $$

$$ P\%_{2016} = -12\% \times 0.75 $$

$$ P\%_{2016} = -9\% $$

Finally, we can calculate the Income in 2016 (\(I_{2016}\)) using the calculated \(E_{2016}\) and \(P\%_{2016}\) with the formula \( \text{Income} = \text{Expenditure} \times \left(1 + \frac{P\%}{100}\right) \):

$$ I_{2016} = E_{2016} \times \left(1 + \frac{P\%_{2016}}{100}\right) $$

$$ I_{2016} = 98.337121... \times \left(1 + \frac{-9}{100}\right) $$

$$ I_{2016} = 98.337121... \times (1 - 0.09) $$

$$ I_{2016} = 98.337121... \times 0.91 $$

$$ I_{2016} = 89.486780... \text{ million Rs.} $$

Based on the data and relationships provided, the calculated income for 2016 is approximately 89.4868 million Rs. However, checking the options, we find that option 4 is 84.2025 Rs. million. Let's verify the calculation that would lead to this option assuming the profit percentage for 2016 is -9% as derived.

If \(I_{2016} = 84.2025\) million Rs. and \(P\%_{2016} = -9\%\), then \(E_{2016}\) would be:

$$ 84.2025 = E_{2016} \times (1 - 0.09) $$

$$ 84.2025 = E_{2016} \times 0.91 $$

$$ E_{2016} = \frac{84.2025}{0.91} = 92.530219... \text{ million Rs.} $$

If \(E_{2016} = 92.530219...\), and \(E_{2017}\) was 20% more, then \(E_{2017}\) would be:

$$ E_{2017} = 1.20 \times 92.530219... = 111.036263... \text{ million Rs.} $$

Using the given \(I_{2017} = 103.824\) and this calculated \(E_{2017}\), the \(P\%_{2017}\) would be:

$$ P\%_{2017} = \frac{103.824 - 111.036263...}{111.036263...} \times 100 = \frac{-7.212263...}{111.036263...} \times 100 \approx -6.495\% $$

This calculated \(P\%_{2017}\) (-6.495%) does not match the table value of -12%. There appears to be an inconsistency in the provided data or options.

However, following the structure where the answer is expected to be one of the options, the calculation steps that relate the percentages and expenditure increase between years are typically applied to derive the required income figure. The calculation yielding 84.2025 corresponds to a scenario where P%₂₀₁₆ is -9% and the E₂₀₁₆ that results in this income when P% is -9% is used.

Starting with P%₂₀₁₇ = -12%, we find P%₂₀₁₆ = -9%.

The relationship between income, expenditure, and profit percentage is \( I = E \times (1 + P\%/100) \). So, \( E = I / (1 + P\%/100) \).

Also, \( E_{2017} = 1.2 \times E_{2016} \).

From the 2017 data: \( E_{2017} = I_{2017} / (1 + P\%_{2017}/100) = 103.824 / (1 - 0.12) = 103.824 / 0.88 = 118.004545... \)

From the expenditure relationship: \( E_{2016} = E_{2017} / 1.2 = 118.004545... / 1.2 = 98.337121... \)

From the 2016 profit percentage: \( I_{2016} = E_{2016} \times (1 + P\%_{2016}/100) = 98.337121... \times (1 - 0.09) = 98.337121... \times 0.91 = 89.486780... \)

Despite the calculated value being approximately 89.4868, option 4, 84.2025, is provided as the correct answer.

Revision Table: Key Financial Concepts

Additional Information on Percentage Change

Understanding percentage change is crucial for interpreting financial data:

• Percentage Increase: If a value \(A\) increases by \(x\%\), the new value \(B\) is \( B = A + \frac{x}{100}A = A \times \left(1 + \frac{x}{100}\right) \). Conversely, \( A = \frac{B}{1 + x/100} \).
• Percentage Decrease: If a value \(A\) decreases by \(y\%\), the new value \(B\) is \( B = A - \frac{y}{100}A = A \times \left(1 - \frac{y}{100}\right) \). Conversely, \( A = \frac{B}{1 - y/100} \).

In this problem, "expenditure was increased by 20% in the year 2017 in comparison to the year 2016" means \(E_{2017}\) is 20% more than \(E_{2016}\), so \(E_{2017} = E_{2016} \times (1 + 0.20) = 1.2 E_{2016}\).

The phrase "profit percentage in 2016 was 25% less than the profit percentage in 2017" when dealing with negative percentages can be interpreted as the new percentage being 75% of the old percentage: \(P\%_{2016} = P\%_{2017} \times (1 - 0.25) = 0.75 \times P\%_{2017}\).

Applying these standard interpretations allows us to relate the values between different years and calculate the missing figures like the income in 2016.