Suppose that x and y are positive integers, where x y. Let n = x/y, and suppose that n is not an integer.The value n can be expressed as an integer part and a fractional part, where the fractional part is between 0 and 1. Using this representation, the fractional part would be equal to:

Question

Suppose that x and y are positive integers, where x > y. Let n = x/y, and suppose that n is not an integer.The value n can be expressed as an integer part and a fractional part, where the fractional part is between 0 and 1. Using this representation, the fractional part would be equal to:

BlackTom AI · Accepted Answer

We start by restating the problem in our own words to ensure clarity: x and y are positive integers with x > y, n = x/y is not an integer, so n has a fractional part, and we want to express that fractional part. First, recall the standard division algorithm: for integers x and y with y > 0, we can write x = qy + r, where q = floor(x/y) is the integer part of the quotient and r = x mod y is the remainder, with 0 ≤ r < y. The fractional part of x/y is then r/y, since x/y = q + r/y. Now evaluate each option: Option a: x mod y - This is the remainder r itself, not the fractional part. The fractional part is a value between 0 and 1, whereas x mod y ranges from 0 to y‑1. Therefore this is not the fractional part. Option b: (x mod y)/y - This matches r/y, which is exactly the fractional part of x/y. Since 0 ≤ r < y, dividing by y yields a number in [0,1), as required. This is the correct form for the fractional part. Option c: (y mod x)/y - Here we would take the remainder of y divided by x and then divide by y. This does not relate to the quotient x/y’s fractional part and can produce values outside the needed r/y form. This is incorrect. Option d: (x mod y)/x - This uses the remainder divided by x, not by y. The fractional part is always r/y, not r/x, so this misplaces the denominator and is incorrect. Option e: y mod x - This is the remainder when y is divided by x, which is an entirely different quantity from the fractional part r/y. Moreover, since y < x, y mod x = y, which would not be between 0 and 1. This is incorrect. Option f: (y mod x)/x - Similar to option e, but divided by x; still not matching the fractional part r/y and does not align with the division algorithm for x/y. This is incorrect. In summary, only the expression (x mod y)/y yields a value in [0,1) that correctly represents the fractional part of x/y, given that x > y and x/y is not an integer. The other options misplace the denominator, compute an unrelated remainder, or fail to produce a fraction between 0 and 1.

FIT1058_S1_2025 Week 7 Pre-reading Quiz

Suppose that x and y are positive integers, where x > y. Let n = x/y, and suppose that n is not an integer.The value n can be expressed as an integer part and a fractional part, where the fractional part is between 0 and 1. Using this representation, the fractional part would be equal to:

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