Data Sufficiency: Calculating Mr. X's 2000 Earnings

This question asks us to determine how much Mr. X earned in the year 2000, based on the information given in two separate statements. We need to decide if statement 1 alone is sufficient, if statement 2 alone is sufficient, if both are needed, or if neither is sufficient.

Analyzing Statement 1 for Sufficiency

Statement 1 says: Mr. X earned ₹ 7,000 in 2003, which is 10% more than what he earned in 2000.

Let $E_{2000}$ be Mr. X's earnings in the year 2000.

According to Statement 1, his earnings in 2003 were ₹ 7,000.

The statement also says that ₹ 7,000 is 10% more than his earnings in 2000. This can be written as:

Earnings in 2003 = Earnings in 2000 + 10% of Earnings in 2000

Using symbols, this is:

$$7000 = E_{2000} + 0.10 \times E_{2000}$$

$$7000 = E_{2000}(1 + 0.10)$$

$$7000 = 1.10 \times E_{2000}$$

We can solve this equation for $E_{2000}$:

$$E_{2000} = \frac{7000}{1.10}$$

This equation gives a specific numerical value for Mr. X's earnings in 2000. Therefore, Statement 1 alone is sufficient to answer the question.

Analyzing Statement 2 for Sufficiency

Statement 2 says: Mrs. Y, Mr. X's wife, earned half of the amount Mr. X earned, and they together earned ₹ 9,000 in 2000.

Let $E_{X,2000}$ be Mr. X's earnings in 2000 and $E_{Y,2000}$ be Mrs. Y's earnings in 2000.

According to the first part of Statement 2:

Mrs. Y's earnings in 2000 = Half of Mr. X's earnings in 2000

$$E_{Y,2000} = 0.5 \times E_{X,2000}$$

According to the second part of Statement 2:

Mr. X's earnings in 2000 + Mrs. Y's earnings in 2000 = Together earned in 2000

$$E_{X,2000} + E_{Y,2000} = 9000$$

Now we can substitute the first equation into the second equation:

$$E_{X,2000} + (0.5 \times E_{X,2000}) = 9000$$

$$E_{X,2000}(1 + 0.5) = 9000$$

$$1.5 \times E_{X,2000} = 9000$$

We can solve this equation for $E_{X,2000}$:

$$E_{X,2000} = \frac{9000}{1.5}$$

This equation also gives a specific numerical value for Mr. X's earnings in 2000. Therefore, Statement 2 alone is sufficient to answer the question.

Conclusion on Sufficiency

We found that Statement 1 alone is sufficient to determine Mr. X's earnings in 2000, and Statement 2 alone is also sufficient to determine Mr. X's earnings in 2000. Therefore, either statement is sufficient to answer the question.

Revision Table: Understanding Data Sufficiency

Additional Information: Percentage Change and Ratios

Understanding percentage changes and ratios is key to solving problems like these.

• Percentage Increase: If a value $A$ is P% more than a value $B$, then $A = B + (P/100) \times B = B(1 + P/100)$. In Statement 1, 7000 is 10% more than $E_{2000}$, so $7000 = E_{2000}(1 + 10/100) = E_{2000}(1.10)$.
• Ratios and Combined Amounts: If two quantities are in a certain ratio (like Mrs. Y earning half of Mr. X's earnings, which is a ratio of 1:2) and their sum is known, you can find the individual quantities. In Statement 2, let Mr. X earn $2x$ and Mrs. Y earn $x$. Their total is $2x + x = 3x$. If $3x = 9000$, then $x = 3000$. Mr. X's earnings would be $2x = 6000$. This confirms we can find the value.

Both statements presented different ways to establish a relationship or equation involving the unknown earnings in 2000, allowing us to solve for it.