Final answer:
The question is about solving a system of equations derived from the given ratio of two numbers and the condition for their modified ratio. By setting up the ratios as x/y = 2/3 and (x-2)/(y-8) = 3/2 and solving the equations, the numbers are found to be 10 and 15.
Step-by-step explanation:
The two numbers based on their ratio and a condition applied to their modified ratio when certain values are subtracted from them. Let's assume the two numbers are x and y. Given, the ratio of the two numbers is 2/3, we can write it as x/y = 2/3. The condition states that when we subtract 2 from x and 8 from y, the new ratio is the reciprocal of the original ratio (3/2). This gives us the equation (x-2)/(y-8) = 3/2.
The two equations to solve are:
- x/y = 2/3
- (x-2)/(y-8) = 3/2
To find x and y we can set up a system of equations:
- 3x = 2y
- 2(x - 2) = 3(y - 8)
By simplifying the second equation, we get:
- 2x - 4 = 3y - 24
Now, we'll solve the system of equations using substitution or elimination. From the first equation, we can express y in terms of x: y = 3x/2. Substituting this into the second equation gives us:
2x - 4 = 3(3x/2 - 8)
Simplifying further, we get to:
x = 10, y = 15
Therefore, the two numbers are 10 and 15, matching option D: 10, 15.