the denominator of a rational number is greater than the numerator by 6. if the numerator is decreased by 2 and the denominator is increased by 4 . the rational is 1/5 .Find the rational number.
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Let's solve this step-by-step.
Let's assume the numerator of the rational number is "x".
According to the given information, the denominator is greater than the numerator by 6. So, the denominator would be "x + 6".
Now, if we decrease the numerator by 2, the new numerator would be "x - 2".
Similarly, if we increase the denominator by 4, the new denominator would be "x + 6 + 4".
According to the problem, the rational number after these adjustments is 1/5.
So, we can set up the following equation:
(x - 2)/(x + 10) = 1/5
To solve this equation, we can cross-multiply:
5(x - 2) = x + 10
Simplifying the equation:
5x - 10 = x + 10
4x = 20
Dividing both sides by 4:
x = 5
Therefore, the numerator of the rational number is 5. And since the denominator is greater than the numerator by 6, the denominator would be 5 + 6 = 11.
So, the rational number is 5/11.
To solve this problem, we can set up an equation using the given information and solve for the unknown rational number.
Let's denote the numerator of the rational number as "x" and the denominator as "y".
According to the problem, the denominator is greater than the numerator by 6, so we can write the equation as:
y = x + 6 (equation 1)
It is also stated that if we decrease the numerator by 2 and increase the denominator by 4, we get the rational number 1/5.
So, we can set up another equation based on this information:
(x - 2) / (y + 4) = 1/5 (equation 2)
Now we have a system of two equations (equations 1 and 2) that we can solve simultaneously to find the values of x and y.
To solve the system, we can substitute the value of y from equation 1 into equation 2 and solve for x:
(x - 2) / ((x + 6) + 4) = 1/5
Simplifying the expression:
(x - 2) / (x + 10) = 1/5
Cross-multiplying:
5(x - 2) = (x + 10)
Expanding:
5x - 10 = x + 10
Simplifying:
4x = 20
Dividing both sides by 4:
x = 5
Now that we have the value of x, we can substitute it back into equation 1 to find the value of y:
y = x + 6
y = 5 + 6
y = 11
Therefore, the rational number is 5/11.