Simplify the following: (3a+5b)/4 + (9+b)/8 (2x)/6 + (2x-3y)/3 + (x+y)/2 (2a+4b)/4 - 3(a-4b)/6 (3e)/4 + (e2)/3 (5+4)/8 - (25-2)/4

Here are the solutions to the problems:

1. Simplify $\frac{3a+5b}{4} + \frac{9a+b}{8}$

Step 1: Find a common denominator for 4 and 8, which is 8. $\frac{3a+5b}{4} + \frac{9a+b}{8} = \frac{2(3a+5b)}{2 \cdot 4} + \frac{9a+b}{8}$

Step 2: Rewrite the first fraction with the common denominator. $= \frac{6a+10b}{8} + \frac{9a+b}{8}$

Step 3: Add the numerators and combine like terms. $= \frac{6a+10b+9a+b}{8} = \frac{(6a+9a)+(10b+b)}{8}$ $= \frac{15a+11b}{8}$ The simplified expression is $\boxed{\frac{15a+11b}{8}}$.

2. Simplify $\frac{2x}{6} + \frac{2x-3y}{3} + \frac{x+y}{2}$

Step 1: Find a common denominator for 6, 3, and 2, which is 6. $\frac{2x}{6} + \frac{2x-3y}{3} + \frac{x+y}{2} = \frac{2x}{6} + \frac{2(2x-3y)}{2 \cdot 3} + \frac{3(x+y)}{3 \cdot 2}$

Step 2: Rewrite each fraction with the common denominator. $= \frac{2x}{6} + \frac{4x-6y}{6} + \frac{3x+3y}{6}$

Step 3: Add the numerators and combine like terms. $= \frac{2x+4x-6y+3x+3y}{6} = \frac{(2x+4x+3x)+(-6y+3y)}{6}$ $= \frac{9x-3y}{6}$

Step 4: Factor out the common term in the numerator and simplify the fraction. $= \frac{3(3x-y)}{6} = \frac{3x-y}{2}$ The simplified expression is $\boxed{\frac{3x-y}{2}}$.

3. Simplify $\frac{2a+4b}{4} - \frac{3(a-4b)}{6}$

Step 1: Simplify each fraction individually. $\frac{2a+4b}{4} = \frac{2(a+2b)}{4} = \frac{a+2b}{2}$ $\frac{3(a-4b)}{6} = \frac{a-4b}{2}$

Step 2: Substitute the simplified fractions back into the expression. $\frac{a+2b}{2} - \frac{a-4b}{2}$

Step 3: Subtract the numerators since they have a common denominator. $= \frac{(a+2b)-(a-4b)}{2} = \frac{a+2b-a+4b}{2}$

Step 4: Combine like terms. $= \frac{(a-a)+(2b+4b)}{2} = \frac{6b}{2}$

Step 5: Simplify the fraction. $= 3b$ The simplified expression is $\boxed{3b}$.

4. Simplify $\frac{3e+e^2}{4} + \frac{e}{3}$ (Assuming the operation between the two fractions is addition, as it is a common convention when an operator is missing in such problems.)

Step 1: Find a common denominator for 4 and 3, which is 12. $\frac{3e+e^2}{4} + \frac{e}{3} = \frac{3(3e+e^2)}{3 \cdot 4} + \frac{4e}{4 \cdot 3}$

Step 2: Rewrite each fraction with the common denominator. $= \frac{9e+3e^2}{12} + \frac{4e}{12}$

Step 3: Add the numerators and combine like terms. $= \frac{9e+3e^2+4e}{12} = \frac{3e^2+(9e+4e)}{12}$ $= \frac{3e^2+13e}{12}$ The simplified expression is $\boxed{\frac{3e^2+13e}{12}}$.

5. Simplify $\frac{5 \cdot 5+4}{8} - \frac{25-2}{4}$

Step 1: Calculate the numerators. $5 \cdot 5+4 = 25+4 = 29$ $25-2 = 23$

Step 2: Substitute the calculated numerators back into the expression. $\frac{29}{8} - \frac{23}{4}$

Step 3: Find a common denominator for 8 and 4, which is 8. $= \frac{29}{8} - \frac{2 \cdot 23}{2 \cdot 4}$

Step 4: Rewrite the second fraction with the common denominator. $= \frac{29}{8} - \frac{46}{8}$

Step 5: Subtract the numerators. $= \frac{29-46}{8} = \frac{-17}{8}$ The simplified expression is $\boxed{-\frac{17}{8}}$.

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