To find the two numbers based on the given conditions, let’s denote the two numbers as x and y. According to the problem, the numbers are in the ratio of 56, which can be expressed mathematically as:
x/y = 56
From this, we can express x in terms of y:
x = 56y
Now, the problem states that if 8 is subtracted from each number, the ratio becomes 45:
(x – 8)/(y – 8) = 45
Substituting x = 56y into the second equation gives:
(56y – 8)/(y – 8) = 45
Cross-multiplying to eliminate the fraction, we have:
56y – 8 = 45(y – 8)
Expanding the right side:
56y – 8 = 45y – 360
Now, to isolate y, we will move all terms involving y to one side and the constant terms to the other:
56y – 45y = -360 + 8
11y = -352
Dividing both sides by 11 gives:
y = -32
Now substituting back to find x:
x = 56 * -32 = -1792
So, the two numbers are:
x = -1792 and y = -32.
To verify, check the ratios:
The original ratio is:
-1792 / -32 = 56
After subtracting 8 from each:
(-1792 – 8)/(-32 – 8) = -1800 / -40 = 45
Both conditions are satisfied, thus the two numbers are:
-1792 and -32.