To solve this problem, we'll start with the information given and work our way step by step to find the original numbers.
Step 1: Express the numbers in terms of a variable.
Given that the ratio of the two numbers is [tex]\(7:5\)[/tex], we can let:
- The first number be [tex]\(7x\)[/tex]
- The second number be [tex]\(5x\)[/tex]
Step 2: Set up the equation for the new ratio after subtracting 10 from each number.
According to the problem, after subtracting 10 from each of these numbers, the new ratio becomes [tex]\(3:2\)[/tex]. Hence, we have:
[tex]\[
\frac{7x - 10}{5x - 10} = \frac{3}{2}
\][/tex]
Step 3: Formulate the equation and solve for [tex]\(x\)[/tex].
Start by cross-multiplying to clear the fraction:
[tex]\[
2(7x - 10) = 3(5x - 10)
\][/tex]
Expand both sides:
[tex]\[
14x - 20 = 15x - 30
\][/tex]
Rearrange the equation to solve for [tex]\(x\)[/tex]:
[tex]\[
14x - 15x = -30 + 20
\][/tex]
[tex]\[
-x = -10
\][/tex]
[tex]\[
x = 10
\][/tex]
Step 4: Find the original numbers.
Now that we have [tex]\(x = 10\)[/tex], we can find the original numbers:
- First number: [tex]\(7x = 7 \times 10 = 70\)[/tex]
- Second number: [tex]\(5x = 5 \times 10 = 50\)[/tex]
Thus, the two numbers are [tex]\(70\)[/tex] and [tex]\(50\)[/tex].
