To find the two numbers based on the conditions given, let’s define the two numbers as x and y. According to the problem, the ratio of these two numbers is:
Ratio 1: \( \frac{x}{y} = 23 \
This implies that:
- x = 23y
Next, if we subtract 2 from the first number and 8 from the second, the new ratio can be expressed as:
New Ratio: \( \frac{x – 2}{y – 8} = \frac{1}{23} \
This gives us a second equation:
- Cross-multiplying yields: \( 23(x – 2) = y – 8 \)
Now, substituting our earlier expression for x (which is \( x = 23y \)) into this new equation, we can rewrite it as:
- \( 23(23y – 2) = y – 8 \)
This simplifies to:
- \( 529y – 46 = y – 8 \)
Combining like terms, we move y to one side:
- \( 529y – y = 46 – 8 \)
- \( 528y = 38 \)
- \( y = \frac{38}{528} = \frac{19}{264} \)
To find x, we substitute back:
- \( x = 23y = 23 \times \frac{19}{264} = \frac{437}{264} \)
Thus, the two numbers are:
- x = \frac{437}{264}
- y = \frac{19}{264}
In a practical sense, this set of values satisfies the original conditions of the problem, confirming that the initial ratio of the two numbers is 23, and the adjusted ratio results in its reciprocal.