Understanding the Problem

We are asked to find the greater of two numbers. We are given two pieces of information:

• The difference between the two numbers is 1140.
• 6% of one number is equal to 10% of the other number.

Setting up the Equations

Let's denote the two numbers as $x$ and $y$. Since we need to find the greater number, let's assume $x$ is the greater number and $y$ is the smaller number.

From the first piece of information, the difference is 1140:

$x - y = 1140 \quad (1)$

From the second piece of information, 6% of one number equals 10% of the other. Since $x$ is the greater number, it's likely that 6% of the greater number is equal to 10% of the smaller number (because a smaller percentage of a larger number can equal a larger percentage of a smaller number). Let's write this as:

$6\% \text{ of } x = 10\% \text{ of } y$

Converting percentages to decimals:

$0.06x = 0.10y$

We can simplify this equation:

$6x = 10y$

Dividing both sides by 2:

$3x = 5y \quad (2)$

Solving the Equations

Now we have a system of two linear equations with two variables:

1. $x - y = 1140$
2. $3x = 5y$

We can solve this system using substitution or elimination. Let's use substitution. First, let's express $y$ in terms of $x$ from equation (2):

$y = \frac{3x}{5}$

Now, substitute this expression for $y$ into equation (1):

$x - \left(\frac{3x}{5}\right) = 1140$

To solve for $x$, find a common denominator:

$\frac{5x}{5} - \frac{3x}{5} = 1140$

$\frac{5x - 3x}{5} = 1140$

$\frac{2x}{5} = 1140$

Multiply both sides by 5:

$2x = 1140 \times 5$

$2x = 5700$

Divide by 2:

$x = \frac{5700}{2}$

$x = 2850$

Finding the Smaller Number (Optional Check)

We can also find the value of $y$ to check our work. Using equation (1):

$y = x - 1140$

$y = 2850 - 1140$

$y = 1710$

Verifying the Conditions

Let's check if these numbers satisfy the conditions:

• Difference: Is $2850 - 1710 = 1140$? Yes, it is.
• Percentage Relationship: Is 6% of 2850 equal to 10% of 1710?
  • $0.06 \times 2850 = 171$
  • $0.10 \times 1710 = 171$
  Yes, both are equal to 171.

Our calculations are correct.

Conclusion

We defined $x$ as the greater number and found its value to be 2850. The smaller number $y$ is 1710.

Therefore, the greater number is 2850.