Simplify the complex fraction:
x+(4x/y)/(7/3x)
I know that the answer is 3x^2(y+4)/7y. I don't know how to get this answer. Can someone please help me walk through this and figure out how to simplify it? I would be so grateful.
Okay it's a complex fraction, so it's (x+(4x/y)) / (7/3x). I'm not sure how else I would type this.
To simplify the complex fraction, we need to follow a few steps:
Step 1: Simplify the numerator of the expression.
The numerator consists of two terms: x and 4x/y. The second term is a fraction, so we need to find a common denominator to combine the terms.
To find the common denominator, we multiply y in the second term by y/y to get 4x * (y/y) / y, which gives us (4xy/y) / y.
Combining the terms in the numerator, we have x + (4xy/y) / y.
Step 2: Simplify the denominator of the expression.
The denominator is 7 / (3x). To simplify this, we can multiply by the reciprocal of (3x), which is (1/(3x)).
Multiplying the reciprocal to the denominator gives us 7 * (1/(3x)).
Step 3: Combine the simplified numerator and denominator.
Now that the numerator and denominator are both simplified, we can combine them.
The expression becomes (x + (4xy/y) / y) / (7 * (1/(3x))).
To combine the numerators, we can multiply the whole numerator by y to get rid of the fraction:
(x * y + (4xy/y)) / y
The 4xy/y term cancels out, leaving us with (x * y) / y.
The expression is then (xy + (xy/y)) / y.
For the denominator, multiplying the reciprocal gives us:
(7 * (3x)) / 1, which simplifies to 21x.
Our expression is now (xy + (xy/y)) / (21x).
Step 4: Simplify further if possible.
Now we can simplify the expression by canceling out the common term xy in the numerator:
(xy + (xy/y)) / (21x) = (xy(1 + (1/y))) / (21x).
Finally, we can simplify further by canceling out the common factor of x in the numerator and denominator:
(xy(1 + (1/y))) / (21x) = (y(1 + (1/y))) / 21.
Multiplying (1 + (1/y)) gives us (y + 1) / y, so our final simplified expression is:
(y + 1) / (21y).
Therefore, the simplified form of the complex fraction x + (4x/y) / (7 / (3x)) is (y + 1) / (21y).
x+(4x/y)/(7/(3x)) = x + (4x/y)(3x/7) = x + (12x^2)/(7y)
From here there is no way to get the alleged answer.
I suspect a typo in either the question or the answer.
Aha! The parens make a big difference!
(x+(4x/y)) / (7/3x)
= (xy+4x)/y * 3x/7
= (7(xy+4x) + 3x^2)/(7y)
= (7xy+28x+3x^2)/(7y)
Still no way to get to 3x^2(y+4)/7y
Hmmm.
(3x^2(y+4)/7y)*(7/(3x)) = xy(y+4)
so, xy(y+4)/(7/3x) = 3x^2(y+4)/7y
Still not sure how to get from xy(y+4) to (x + 4x/y)