The numerator of a rational number is 17 more than its denominator. The numerator of its absolute is 7 more than its denominator. Find the rational number.
No
To solve this problem, let's set up equations based on the given information.
Let's assume that the numerator of the rational number is represented by "n" and the denominator is represented by "d".
According to the problem, the numerator of the rational number is 17 more than its denominator. Therefore, we can write the first equation as:
n = d + 17
Next, we are told that the numerator of its absolute value is 7 more than its denominator. The absolute value of a number disregards its sign, so if the rational number is negative, we can eliminate the negative sign without changing its value. This means that the numerator and denominator (after removing the sign) will be the same, with the numerator 7 more. Thus, our second equation becomes:
|n| = |d| + 7
Now, let's consider the absolute value equation. This equation indicates that both sides are either positive or negative, resulting in two possible scenarios.
First, suppose both sides of the equation are positive:
n = d + 7
Since n = d + 17 (from the first equation), we can substitute this into the second equation to find:
d + 17 = d + 7
By canceling out the d terms:
17 = 7
This equation is false, meaning that both sides cannot be positive simultaneously.
Secondly, let's suppose both sides are negative:
-n = -d + 7
Using our claim that the numerator is 17 more than the denominator (n = d + 17), we can substitute this into the second equation:
-(d + 17) = -d + 7
Expanding the negative sign:
-d - 17 = -d + 7
By canceling out the d terms:
-17 = 7
Again, this equation is false, proving that both sides cannot be negative at the same time.
Since neither case satisfies the equations, there is no solution that fulfills both conditions simultaneously. Consequently, there is no rational number that satisfies the given conditions.