Solved on Dec 17, 2023

Simplify the complex fraction $\frac{\frac{2}{5 t}-\frac{3}{3 t}}{\frac{1}{2 t}+\frac{1}{2 t}}$ . (1 point) $-\frac{3}{5}$ , $-4$ , $-\frac{5}{3}$ , $-\frac{1}{4}$

STEP 1

Assumptions
1. We are given a complex fraction to simplify.
2. The numerator of the complex fraction is $\frac{2}{5t} - \frac{3}{3t}$ .
3. The denominator of the complex fraction is $\frac{1}{2t} + \frac{1}{2t}$ .
4. We need to perform addition and subtraction of rational expressions.

STEP 2

First, we simplify the numerator by finding a common denominator for the two fractions within it.
$\text{Common denominator for the numerator} = 5t \times 3t$

STEP 3

Rewrite each fraction in the numerator with the common denominator.
$\frac{2}{5t} = \frac{2 \times 3t}{5t \times 3t} = \frac{6t}{15t^2}$
$\frac{3}{3t} = \frac{3 \times 5t}{3t \times 5t} = \frac{15t}{15t^2}$

STEP 4

Subtract the two fractions in the numerator using the common denominator.
$\frac{2}{5t} - \frac{3}{3t} = \frac{6t}{15t^2} - \frac{15t}{15t^2}$

STEP 5

Perform the subtraction in the numerator.
$\frac{6t}{15t^2} - \frac{15t}{15t^2} = \frac{6t - 15t}{15t^2}$

STEP 6

Combine like terms in the numerator.
$\frac{6t - 15t}{15t^2} = \frac{-9t}{15t^2}$

STEP 7

Simplify the numerator by canceling out the common factor of $t$ .
$\frac{-9t}{15t^2} = \frac{-9}{15t}$

STEP 8

Further simplify the numerator by reducing the fraction.
$\frac{-9}{15t} = \frac{-3}{5t}$

STEP 9

Now, we simplify the denominator by adding the two fractions within it.
$\frac{1}{2t} + \frac{1}{2t} = \frac{1 + 1}{2t}$

STEP 10

Perform the addition in the denominator.
$\frac{1 + 1}{2t} = \frac{2}{2t}$

STEP 11

Simplify the denominator by reducing the fraction.
$\frac{2}{2t} = \frac{1}{t}$

STEP 12

Now that we have simplified both the numerator and the denominator, we rewrite the complex fraction.
$\frac{\frac{-3}{5t}}{\frac{1}{t}}$

STEP 13

To divide by a fraction, we multiply by its reciprocal.
$\frac{\frac{-3}{5t}}{\frac{1}{t}} = \frac{-3}{5t} \times \frac{t}{1}$

STEP 14

Multiply the fractions.
$\frac{-3}{5t} \times \frac{t}{1} = \frac{-3 \times t}{5t \times 1}$

STEP 15

Cancel the common factor of $t$ .
$\frac{-3 \times t}{5t \times 1} = \frac{-3}{5}$
The simplified complex fraction is $-\frac{3}{5}$ .

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Simplify the complex fraction $\frac{\frac{2}{5 t}-\frac{3}{3 t}}{\frac{1}{2 t}+\frac{1}{2 t}}$ . (1 point) $-\frac{3}{5}$ , $-4$ , $-\frac{5}{3}$ , $-\frac{1}{4}$

STEP 1

Assumptions
1. We are given a complex fraction to simplify.
2. The numerator of the complex fraction is $\frac{2}{5t} - \frac{3}{3t}$ .
3. The denominator of the complex fraction is $\frac{1}{2t} + \frac{1}{2t}$ .
4. We need to perform addition and subtraction of rational expressions.

STEP 2

First, we simplify the numerator by finding a common denominator for the two fractions within it.
$\text{Common denominator for the numerator} = 5t \times 3t$

STEP 3

Rewrite each fraction in the numerator with the common denominator.
$\frac{2}{5t} = \frac{2 \times 3t}{5t \times 3t} = \frac{6t}{15t^2}$
$\frac{3}{3t} = \frac{3 \times 5t}{3t \times 5t} = \frac{15t}{15t^2}$

STEP 4

Subtract the two fractions in the numerator using the common denominator.
$\frac{2}{5t} - \frac{3}{3t} = \frac{6t}{15t^2} - \frac{15t}{15t^2}$

STEP 5

Perform the subtraction in the numerator.
$\frac{6t}{15t^2} - \frac{15t}{15t^2} = \frac{6t - 15t}{15t^2}$

STEP 6

Combine like terms in the numerator.
$\frac{6t - 15t}{15t^2} = \frac{-9t}{15t^2}$

STEP 7

Simplify the numerator by canceling out the common factor of $t$ .
$\frac{-9t}{15t^2} = \frac{-9}{15t}$

STEP 8

Further simplify the numerator by reducing the fraction.
$\frac{-9}{15t} = \frac{-3}{5t}$

STEP 9

Now, we simplify the denominator by adding the two fractions within it.
$\frac{1}{2t} + \frac{1}{2t} = \frac{1 + 1}{2t}$

STEP 10

Perform the addition in the denominator.
$\frac{1 + 1}{2t} = \frac{2}{2t}$

STEP 11

Simplify the denominator by reducing the fraction.
$\frac{2}{2t} = \frac{1}{t}$

STEP 12

Now that we have simplified both the numerator and the denominator, we rewrite the complex fraction.
$\frac{\frac{-3}{5t}}{\frac{1}{t}}$

STEP 13

To divide by a fraction, we multiply by its reciprocal.
$\frac{\frac{-3}{5t}}{\frac{1}{t}} = \frac{-3}{5t} \times \frac{t}{1}$

STEP 14

Multiply the fractions.
$\frac{-3}{5t} \times \frac{t}{1} = \frac{-3 \times t}{5t \times 1}$

STEP 15

Cancel the common factor of $t$ .
$\frac{-3 \times t}{5t \times 1} = \frac{-3}{5}$
The simplified complex fraction is $-\frac{3}{5}$ .

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Simplify the complex fraction 25t−33t12t+12t\frac{\frac{2}{5 t}-\frac{3}{3 t}}{\frac{1}{2 t}+\frac{1}{2 t}}2t1​+2t1​5t2​−3t3​​. (1 point) −35-\frac{3}{5}−53​, −4-4−4, −53-\frac{5}{3}−35​, −14-\frac{1}{4}−41​

STEP 1

STEP 2

First, we simplify the numerator by finding a common denominator for the two fractions within it.Common denominator for the numerator=5t×3t\text{Common denominator for the numerator} = 5t \times 3tCommon denominator for the numerator=5t×3t

STEP 3

STEP 4

Subtract the two fractions in the numerator using the common denominator.25t−33t=6t15t2−15t15t2\frac{2}{5t} - \frac{3}{3t} = \frac{6t}{15t^2} - \frac{15t}{15t^2}5t2​−3t3​=15t26t​−15t215t​

STEP 5

Perform the subtraction in the numerator.6t15t2−15t15t2=6t−15t15t2\frac{6t}{15t^2} - \frac{15t}{15t^2} = \frac{6t - 15t}{15t^2}15t26t​−15t215t​=15t26t−15t​

STEP 6

Combine like terms in the numerator.6t−15t15t2=−9t15t2\frac{6t - 15t}{15t^2} = \frac{-9t}{15t^2}15t26t−15t​=15t2−9t​

STEP 7

Simplify the numerator by canceling out the common factor of ttt.−9t15t2=−915t\frac{-9t}{15t^2} = \frac{-9}{15t}15t2−9t​=15t−9​

STEP 8

Further simplify the numerator by reducing the fraction.−915t=−35t\frac{-9}{15t} = \frac{-3}{5t}15t−9​=5t−3​

STEP 9

Now, we simplify the denominator by adding the two fractions within it.12t+12t=1+12t\frac{1}{2t} + \frac{1}{2t} = \frac{1 + 1}{2t}2t1​+2t1​=2t1+1​

STEP 10

Perform the addition in the denominator.1+12t=22t\frac{1 + 1}{2t} = \frac{2}{2t}2t1+1​=2t2​

STEP 11

Simplify the denominator by reducing the fraction.22t=1t\frac{2}{2t} = \frac{1}{t}2t2​=t1​

STEP 12

Now that we have simplified both the numerator and the denominator, we rewrite the complex fraction.−35t1t\frac{\frac{-3}{5t}}{\frac{1}{t}}t1​5t−3​​

STEP 13

To divide by a fraction, we multiply by its reciprocal.−35t1t=−35t×t1\frac{\frac{-3}{5t}}{\frac{1}{t}} = \frac{-3}{5t} \times \frac{t}{1}t1​5t−3​​=5t−3​×1t​

STEP 14

Multiply the fractions.−35t×t1=−3×t5t×1\frac{-3}{5t} \times \frac{t}{1} = \frac{-3 \times t}{5t \times 1}5t−3​×1t​=5t×1−3×t​

STEP 15

Cancel the common factor of ttt.−3×t5t×1=−35\frac{-3 \times t}{5t \times 1} = \frac{-3}{5}5t×1−3×t​=5−3​The simplified complex fraction is −35-\frac{3}{5}−53​.

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Simplify the complex fraction $\frac{\frac{2}{5 t}-\frac{3}{3 t}}{\frac{1}{2 t}+\frac{1}{2 t}}$ . (1 point) $-\frac{3}{5}$ , $-4$ , $-\frac{5}{3}$ , $-\frac{1}{4}$

First, we simplify the numerator by finding a common denominator for the two fractions within it.
$\text{Common denominator for the numerator} = 5t \times 3t$

Subtract the two fractions in the numerator using the common denominator.
$\frac{2}{5t} - \frac{3}{3t} = \frac{6t}{15t^2} - \frac{15t}{15t^2}$

Perform the subtraction in the numerator.
$\frac{6t}{15t^2} - \frac{15t}{15t^2} = \frac{6t - 15t}{15t^2}$

Combine like terms in the numerator.
$\frac{6t - 15t}{15t^2} = \frac{-9t}{15t^2}$

Simplify the numerator by canceling out the common factor of $t$ .
$\frac{-9t}{15t^2} = \frac{-9}{15t}$

Further simplify the numerator by reducing the fraction.
$\frac{-9}{15t} = \frac{-3}{5t}$

Now, we simplify the denominator by adding the two fractions within it.
$\frac{1}{2t} + \frac{1}{2t} = \frac{1 + 1}{2t}$

Perform the addition in the denominator.
$\frac{1 + 1}{2t} = \frac{2}{2t}$

Simplify the denominator by reducing the fraction.
$\frac{2}{2t} = \frac{1}{t}$

Now that we have simplified both the numerator and the denominator, we rewrite the complex fraction.
$\frac{\frac{-3}{5t}}{\frac{1}{t}}$

To divide by a fraction, we multiply by its reciprocal.
$\frac{\frac{-3}{5t}}{\frac{1}{t}} = \frac{-3}{5t} \times \frac{t}{1}$

Multiply the fractions.
$\frac{-3}{5t} \times \frac{t}{1} = \frac{-3 \times t}{5t \times 1}$

Cancel the common factor of $t$ .
$\frac{-3 \times t}{5t \times 1} = \frac{-3}{5}$
The simplified complex fraction is $-\frac{3}{5}$ .