If z = x + y and x is 2.5 percent of z, where x, y, and z are positive numbers, which of the following is the ratio of y to x?
The ratio of y to x is 39 to 1.
In the equation z = x + y, if x is 2.5 percent of z, we can express this relationship mathematically. By substituting the value of x in terms of z, we find that y is equivalent to 39 times the value of x, leading to the conclusion that the correct ratio is 39 to 1.
This ratio suggests that for every unit of x, there are 41 units of y. However, if x is 2.5 percent of z, the calculation shows that y must be significantly less than this ratio, indicating that this option exceeds the actual value.
A ratio of 40 to 1 would mean y is 40 times larger than x. This is slightly closer to the correct answer, but still does not account for the precise calculation derived from the given percentages and the relationship between x, y, and z.
This is the correct ratio, as derived from the equation where x equals 2.5 percent of z. Given x + y = z, we can express y as z - x, and substituting x as 0.025z leads us to find that y equals 0.975z. Therefore, the ratio of y to x simplifies to 39 to 1.
This ratio implies y is 35 times the value of x. Similar to the other incorrect options, this does not hold true when calculated based on the percentage relationship provided. The actual relationship yields a higher ratio, showing this option is incorrect.
This option asserts that y is 32 times larger than x, which is inconsistent with the calculations based on the given percentage. The derived ratio is much closer to 39 to 1, making this choice incorrect.
The relationship established by the equation z = x + y, with x being 2.5 percent of z, leads to the conclusion that y is 39 times the value of x. This precise calculation highlights the importance of understanding mathematical relationships and ratios in problem-solving, particularly when dealing with percentages and their implications in algebraic expressions.
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