Answer:
x : y = 4 : 1
Step-by-step explanation:
You want x:y, given adding 20 to both makes the ratio 5:2, and subtracting 5 from both makes the ratio 5:1.
The given relations tell us ...
(x +20) : (y +20) = 5 : 2
(x -5) : (y -5) = 5 : 1
Cross multiplying gives the equations ...
2(x +20) = 5(y +20) . . . . . . . [eq1]
(x -5) = 5(y -5) . . . . . . . . . . . . [eq2]
Solving [eq2] for x gives ...
x = 5y -20
Substituting that into [eq1], we have ...
2(5y -20 +20) = 5(y +20)
10y = 5y +100
5y = 100
y = 20
x = 5y -10 = 100 -20 = 80
Then x:y = 80:20 = 4:1.
The ratio x:y in lowest terms is 4:1.
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Additional comment
When working with ratios, sometimes it works well to consider the difference of the numerator and denominator. Here, the difference when 20 is added is 5-2 = 3 "ratio units", which is 3/2 times the value y+20. The difference when 5 is subtracted is 5-1 = 4 "ratio units", which is 4 times the value y-5. Adding or subtracting the same number does not change the actual difference (x-y). This tells us ...
(x -y) = 3/2(y+20) = 4(y-5)
3y +60 = 8y -40
100 = 5y
20 = y
x = y + 4(y-5) = 20 +4(15) = 80