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How many pairs of positive integers (x,y) have the property that the ratio x : 4 equals the ratio 9 : y ?

 Aug 9, 2023
 #1
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Read the pairs of positive integers (x, y) from left to right:

 

x : 4 = 9 : y

 

(36, 1) , (18, 2) , (12, 3) , (9, 4) , (6, 6) , (4, 9) , (3, 12) , (2, 18) , (1, 36)> = 9 such pairs

 

The pairs mean: [x : 4] = [9 : y] = [36 : 4] =[9 : 1] 

 Aug 9, 2023
 #2
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To solve this, we can set up a proportion equation:

 

x / 4 = 9 / y

 

Cross-multiplying gives:

 

xy = 36

 

Now, we need to find the pairs of positive integers (x, y) that satisfy this equation. The pairs that work are:

 

(1, 36), (2, 18), (3, 12), (4, 9), (6, 6), (9, 4), (12, 3), (18, 2), (36, 1)

 

So, there are 9 pairs of positive integers (x, y) that satisfy the given property.

 

Therefore, the required pair of positive integers (x,y) satisfied the given property is 9.

 Aug 10, 2023

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