I'm having trouble understanding how to do complex fractions.. My book doesn't explain it very well..
12.
(x+3/12)/(4x-5/15)
14.
(2/x^2 + 1/x)/(4/x^2 - 1/x)
16.
(1/y + 3/y^2)/(y + 27/y^2)
If you could please explain how to begin the process I would appreciate it.
12.
3 / 12 = 3 / ( 3 * 4 ) = 1 / 4
5 / 15 = 5 / ( 5 * 3 ) = 1 / 3
( x + 3 / 12 ) / ( 4 x - 5 / 15 ) =
( x + 1 / 4 ) / ( 4 x - 1 / 3 ) =
( 4 x / 4 + 1 / 4 ) / ( 4 x * 3 / 3 - 1 / 3 ) =
[ ( 4 x + 1 ) / 4 ] / [ ( 12 x - 1 ) / 3 ] =
( 4 x + 1 ) * 3 / [ 4 * ( 12 x - 1 ) ] =
( 3 / 4 ) ( 4 x + 1 ) / ( 12 x - 1 ) =
3 * ( 4 x + 1 ) / [ 4 * ( 12 x - 1 ) ] =
( 12 x + 3 ) / ( 48 x + 4 )
14.
( 2 / x ^ 2 + 1 / x ) / ( 4 / x ^ 2 - 1 / x ) =
[ 2 / x ^ 2 + 1 * x / ( x * x ) ] / [ 4 / x ^ 2 - 1 * x / ( x * x ) ] =
( 2 / x ^ 2 + x / x ^ 2 ) / ( 4 / x ^ 2 - x / x ^ 2 ) =
[ ( 2 + x ) / x ^ 2 ] / [ ( 4 - x ) / x ^ 2 ] =
[ x ^ 2 * ( 2 + x ) / x ^ 2 ] / [ x ^ 2 * ( 4 - x ) / x ^ 2 ] =
( 2 + x ) / ( 4 - x )
16.
( 1 / y + 3 / y ^ 2 ) / ( y + 27 / y ^ 2 ) =
[ 1 * y / ( y * y ) + 3 / y ^ 2 ) ] / [ y * y ^ 2 / y ^ 2 + 27 / y ^ 2 ) ] =
( y / y ^ 2 + 3 / y ^ 2 ) / ( y ^ 3 / y ^ 2 + 27 / y ^ 2 ) =
[ ( y + 3 ) / y ^ 2 ] / [ ( y ^ 3 + 27 ) / y ^ 2 ) =
[ y ^ 2 * ( y + 3 ) / y ^ 2 ] / [ y ^ 2 ( y ^ 3 + 27 ) / y ^ 2 ) =
( y + 3 ) / ( y ^ 3 + 27 ) =
( y + 3 ) / [ ( y + 3 ) * ( y ^ 2 - 3 y + 9 ) =
1 / ( y ^ 2 - 3 y + 9 )
12.
(x+3/12)/(4x-5/15)
first reduce those fractions to lowest terms
= (x + 1/4) / (4x - 1/3)
multipy top and bottom by 12 , the LCD
= (12x + 3)/(48x+4) or 3(4x+1)/(4(12x+1) )
14.
(2/x^2 + 1/x)/(4/x^2 - 1/x)
do it the same way, multiply top and bottom by x^2 , the LCD
= (2 + x)/(4 - x)
all done!
try the last one using the same method
Thank you both!
To simplify complex fractions, you need to simplify the numerator and denominator separately, and then divide the simplified numerator by the simplified denominator. Here's how to approach each of the complex fractions you provided:
12.
To simplify (x+3/12)/(4x-5/15), you can start by finding the least common denominator (LCD), which is 60 in this case since 12 and 15 have 60 as a multiple. Then, multiply both the numerator and denominator by the LCD to eliminate the fractions.
Multiply the numerator:
(x+3/12) * 60 = (60/12)x + (3/12) * 60 = 5x + 15
Multiply the denominator:
(4x-5/15) * 60 = (60/15)(4x) - (5/15) * 60 = 4(4x) - 20 = 16x - 20
Now, the fraction becomes:
(5x + 15)/(16x - 20)
14.
For (2/x^2 + 1/x)/(4/x^2 - 1/x), begin by finding the least common denominator, which is x^2 in this case. Multiply each term in the numerator and denominator by this LCD.
Multiply the numerator:
(2/x^2 + 1/x) * x^2 = 2 + x
Multiply the denominator:
(4/x^2 - 1/x) * x^2 = 4 - x^2
Now, the fraction becomes:
(2 + x)/(4 - x^2)
16.
For (1/y + 3/y^2)/(y + 27/y^2), you can simplify it in a similar way. Find the least common denominator, which is y^2 in this case.
Multiply the numerator:
(1/y + 3/y^2) * y^2 = y^2/y + 3
Multiply the denominator:
(y + 27/y^2) * y^2 = y^3 + 27
Now, the fraction becomes:
(y^2/y + 3)/(y^3 + 27)
These are the simplified forms of the given complex fractions. Remember to simplify further if possible by factoring, canceling out common factors, or applying any additional rules that may be applicable.