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When x is added to each of 13, 19, 16 and 23, then the numbers so obtained, in this order, are in proportion. Then, if 5x: y:: y: (8x-4), and y > 0, what is the value of y? 613
42
37
41
30
30
For proportions (a+x)/(b+x) = (c+x)/(d+x), the shortcut is x = (ad-bc)/(b+c-a-d). Calculate x first, then apply the mean proportion formula y squared = 5x(8x-4).
For proportions (a+x)/(b+x) = (c+x)/(d+x), the shortcut is x = (ad-bc)/(b+c-a-d). Calculate x first, then apply the mean proportion formula y squared = 5x(8x-4).
Solve for x
Given (13+x)/(19+x) = (16+x)/(23+x). Cross multiplying: (13+x)(23+x) = (19+x)(16+x). Expanding gives 299 + 36x + x┬▓ = 304 + 35x + x┬▓. Subtracting x┬▓ from both sides, we get 36x - 35x = 304 - 299, so x = 5.
Solve for y
Substitute x=5 into 5x: y:: y: (8x-4). This becomes 25: y:: y: (8*5-4), which is 25: y:: y: 36. Thus, y┬▓ = 25 * 36. Taking the square root, y = 5 * 6 = 30.
A: 42 is incorrect as it does not satisfy the proportion y squared = 900. B: 37 is incorrect as it is not the square root of 900. C: 41 is incorrect because it deviates from the calculated value of y = 30.
D is correct because substituting x=5 into the mean proportion formula yields y squared = 900, which results in y = 30.
In proportion problems with added variables, always write the equation as a fraction to quickly cancel out the x squared terms.
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