The sum of the numerator and denominator of a fraction is 17. If 3 is added to the numerator, the value of the fraction will be 1. What is the fraction?
n = numerator
d = denominator
The sum of the numerator and denominator of a fraction is 17 mean:
n + d = 17
If 3 is added to the numerator, the value of the fraction will be 1 mean:
( n + 3 ) / d = 1
Now you have system of two equations:
n + d = 17
( n + 3 ) / d = 1
n + d = 17
Subtract d to both sides
n + d - d = 17 - d
n = 17 - d
( n + 3 ) / d = 1
Multiply both sides by d
n + 3 = d
Subtract 3 to both sides
n + 3 - 3 = d - 3
n = d - 3
n = n
17 - d = d - 3
Add d to both sides
17 - d + d = d - 3 + d
17 = 2 d - 3
Add 3 to both sides
17 + 3 = 2 d - 3 + 3
20 = 2 d
Divide both sides by 2
10 = d
d = 10
n = d - 3 = 10 - 3 = 7
Your fraction:
7 / 10
Proof:
n + d = 7 + 10 = 17
( n + 3 ) / d = ( 7 + 3 ) / 10 = 10 / 10 = 1
To find the fraction, let's set up an equation using the given information.
Let's assume the numerator of the fraction is represented by the variable 'x'. Since 3 is added to the numerator, the new numerator will be (x + 3).
The denominator of the fraction can be represented by the expression (17 - x), as the sum of the numerator and denominator is 17.
According to the given information, if 3 is added to the numerator, the fraction becomes 1. Mathematically, this can be expressed as:
(x + 3) / (17 - x) = 1
To solve this equation, we can cross-multiply:
(x + 3) = 17 - x
Expanding the equation:
x + 3 = 17 - x
Combining like terms:
2x + 3 = 17
Subtracting 3 from both sides:
2x = 14
Dividing both sides by 2:
x = 7
Now that we know the value of x (numerator), we can substitute it back into one of the original equations to find the denominator:
Denominator = 17 - x
Denominator = 17 - 7
Denominator = 10
So, the fraction is 7/10.