Question

Simplify the complex fraction

Answer 1

\[\frac{ \frac{ 2 }{ 5t} - \frac{ 3 }{ 3t }}{ \frac{ 1 }{ 2t } + \frac{ 1 }{ 2t }}\]

Answer 2

First, simplify the denominator. The two fractions there have a common denominator, so they can be directly added and common terms canceled. Next remember that dividing by a fraction is equivalent to multiplying by the reciprocal of the fraction, which is simply the fraction with numerator and denominator swapped (reciprocal of 1/2 = 2/1 = 2). I think you can figure it out, give it a shot!

Answer 3

Alternate simplification route: multiply top and bottom by something, then simplify denominator.

Answer 4

@tcarroll010

Answer 5

Denominator of your complex fraction: \[\frac{ 1 }{ 2t }+\frac{ 1 }{ 2t }=\frac{ 2 }{ 2t }=\frac{ 1 }{ t }\]That one is easier, because the denominators are already the same.

Numerator: first simplify right fraction: 3/(3t) = 1/t. But it can also be written as 5/(5t), so now they have same denominator as well:\[\frac{ 2 }{ 5t }-\frac{ 3 }{ 3t }=\frac{ 2 }{ 5t }-\frac{ 1 }{ t }=\frac{ 2 }{ 5t }-\frac{ 5 }{ 5t }=\frac{ -3 }{ 5t }=-\frac{ 3 }{ 5t }\]Now divide the two results:\[\frac{ -\frac{ 3 }{ 5t } }{ \frac{ 1 }{ t } }\]If you look at this calculation, you see you have to divide by 1/t. But: this means you can also multiply by the inverse of 1/t, which is t:\[\frac{ -\frac{ 3 }{ 5t } }{ \frac{ 1 }{ t } }=-\frac{ 3 }{ 5t } \cdot t=...\]Do you see what will be the last step?

Answer 6

@ZeHanz so is it a - 5/3

Answer 7

No, just write t as t/1 and multiply both fractions:\[-\frac{ 3 }{ 5t }\cdot \frac{ t }{ 1 }=-\frac{ 3 \cdot t }{ 5 \cdot t \cdot 1 }=-\frac{ 3 }{ 5 }\]

Answer 8

oh ok thank you ! @ZeHanz

Answer 9

YW!

Answer 10

Alternate approach:

\[\frac{\frac{2}{5t}-\frac{3}{3t}}{\frac{1}{2t}+\frac{1}{2t}} * \frac{t}{t} = \frac{\frac{2}{5}-\frac{3}{3}}{\frac{1}{2}+\frac{1}{2}} = \frac{\frac{2}{5}-\frac{3}{3}}{1} = \frac{2}{5} - 1 = -\frac{3}{5}\]

(This was what my "multiply top and bottom by something" suggestion was hinting at)

Answer 11

@whpalmer4: good job!