If denominator of a fraction is one less than twice its numerator.If 3 is
added to the numerator and 4 is added to the deominator,the fraction beomes
3/5.Find the fraction.
f = fraction
n = numerator
d = denominator
f = n / d
Denominator of a fraction is one less than twice its numerator means
d = 2 n - 1
If 3 is added to the numerator and 4 is added to the deominator,the fraction becomes 3 / 5 means:
( n + 3 ) / ( d + 4 ) = 3 / 5
Replace d with 2 n - 1 in this equation.
( n + 3 ) / ( 2 n - 1 + 4 ) = 3 / 5
( n + 3 ) / ( 2 n + 3 ) = 3 / 5
Cross multiply
5 ( n + 3 ) = 3 ( 2 n + 3 )
5 n + 15 = 6 n + 9
Subtract 5 n to both sides
15 = n + 9
Subtract 9 n to both sides
6 = n
n = 6
d = 2 n - 1
d = 2 ∙ 6 - 1
d = 12 - 1
d = 11
f = n / d = 6 / 11
Check result:
( n + 3 ) / ( d + 4 ) = 3 / 5
( 6 + 3 ) / ( 11 + 4 ) = 3 / 5
9 / 15 = 3 / 5
3 ∙ 3 / 3 ∙ 5 = 3 ∙ 5
Let's solve this step by step.
Let the numerator of the fraction be "x" and the denominator be "y".
According to the given conditions:
1. The denominator is one less than twice the numerator:
y = 2x - 1
2. When 3 is added to the numerator and 4 is added to the denominator, the fraction becomes 3/5:
(x + 3) / (y + 4) = 3/5
Now, let's substitute the first equation into the second equation to solve for "x":
(x + 3) / ((2x - 1) + 4) = 3/5
(x + 3) / (2x + 3) = 3/5
Cross multiplying gives us:
5(x + 3) = 3(2x + 3)
5x + 15 = 6x +9
5x - 6x = 9 - 15
-x = -6
x = 6
Now we can substitute the value of "x" back into the first equation to find "y":
y = 2x - 1
y = (2 * 6) - 1
y = 12 - 1
y = 11
Therefore, the fraction is 6/11.