Answer:
p : q = 5 : 1
Explanation:
You want the ratio p : q, such that ...
- (p-5) : (q-5) = 9 : 1
- (p+20) : (q+20) = 7 : 3
Equations
Cross-multiplying the given relations gives us the equations ...
p -5 = 9(q -5)
3(p +20) = 7(q +20)
Solution
Using the first equation to write an expression for p, we have ...
p = 9q -40
Substituting this into the second equation gives ...
3((9q -40) +20) = 7(q +20)
27q -60 = 7q +140
20q = 200
q = 10
p = 9(10) -40 = 50
Then the desired ratio is ...
p : q = 50 : 10 = 5 : 1
In simplest terms, the ratio is p : q = 5 : 1.
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Additional comment
After subtracting 5 from p and q, the ratio p' : q' = 9 : 1 has a difference of 8 ratio units. After adding (5+20) = 25 to each of these numbers, their ratio is 7:3, which numbers have a difference of 4. Doubling these ratio values to 14 : 6 gives them a difference of 8. We note that 14 is 5 ratio units more than 9, so the 25 that we added must be equivalent to 5 ratio units. The 5 that we subtracted is then 1 ratio unit, and the original ratio was p:q = (9+1):(1+1) = 10:2 = 5:1.
Here, we have taken advantage of the fact that adding or subtracting the same value from each element of the ratio does not change their difference.
The original ratio is 50:10. After subtracting 5, it becomes 45:5, and after adding 20 it becomes 70:30.
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