Let the two numbers be represented by the ratio $9:7$. This means we can write the numbers as $9x$ and $7x$, where $x$ is a common factor.

Difference Between the Numbers

The problem states that the difference between these two numbers is 6000.

The difference between the larger number ($9x$) and the smaller number ($7x$) is $9x - 7x$.

We can set up the equation:

$\qquad 9x - 7x = 6000$

Simplifying the equation, we get:

$\qquad 2x = 6000$

Finding the Common Factor

To find the value of $x$, we divide both sides of the equation by 2:

$\qquad x = \frac{6000}{2}$

$\qquad x = 3000$

Calculating the Two Numbers

Now that we have the value of $x$, we can find the actual values of the two numbers:

• The first number is $9x = 9 \times 3000 = 27000$.
• The second number is $7x = 7 \times 3000 = 21000$.

We can quickly check if their difference is indeed 6000: $27000 - 21000 = 6000$. This confirms our value of $x$ is correct.

Calculating the Sum

The question asks for the sum of the two numbers. We add the two numbers we found:

Sum = First Number + Second Number

Sum = $27000 + 21000$

Sum = $48000$

Final Sum

The sum of the two numbers is 48000.