Simplify the following: 3a+4b/10 + 5a-b/4 6y−5x/10 − 2x+y/5 + x/3 5t/9 − 9/4 4u+5/4 − 3u−5/2 x−3/3 − x+2/5

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Here are the solutions to the problems:

1. Simplify $\frac{3a+4b}{10} + \frac{5a-b}{4}$

Find the least common multiple (LCM) of the denominators 10 and 4.
The LCM is 20.
Rewrite each fraction with the common denominator.
$\frac{3a+4b}{10} = \frac{2(3a+4b)}{2 \times 10} = \frac{6a+8b}{20}$ $\frac{5a-b}{4} = \frac{5(5a-b)}{5 \times 4} = \frac{25a-5b}{20}$
Add the fractions.
$\frac{6a+8b}{20} + \frac{25a-5b}{20} = \frac{6a+8b+25a-5b}{20}$
Combine like terms in the numerator.
$\frac{(6a+25a) + (8b-5b)}{20} = \frac{31a+3b}{20}$
The simplified expression is $\boxed{\frac{31a+3b}{20}}$ .

2. Simplify $\frac{6y-5x}{10} - \frac{2x+y}{5} + \frac{x}{3}$
Find the LCM of the denominators 10, 5, and 3.
The LCM is 30.
Rewrite each fraction with the common denominator.
$\frac{6y-5x}{10} = \frac{3(6y-5x)}{3 \times 10} = \frac{18y-15x}{30}$ $\frac{2x+y}{5} = \frac{6(2x+y)}{6 \times 5} = \frac{12x+6y}{30}$ $\frac{x}{3} = \frac{10x}{10 \times 3} = \frac{10x}{30}$
Combine the fractions.
$\frac{18y-15x}{30} - \frac{12x+6y}{30} + \frac{10x}{30} = \frac{(18y-15x) - (12x+6y) + 10x}{30}$
Distribute the negative sign and combine like terms in the numerator.
$\frac{18y-15x-12x-6y+10x}{30} = \frac{(18y-6y) + (-15x-12x+10x)}{30}$ $\frac{12y - 17x}{30}$
The simplified expression is $\boxed{\frac{12y-17x}{30}}$ .

3. Simplify $\frac{5t}{9} + \frac{t-3}{4} - \frac{t-3}{2}$
Find the LCM of the denominators 9, 4, and 2.
The LCM is 36.
Rewrite each fraction with the common denominator.
$\frac{5t}{9} = \frac{4(5t)}{4 \times 9} = \frac{20t}{36}$ $\frac{t-3}{4} = \frac{9(t-3)}{9 \times 4} = \frac{9t-27}{36}$ $\frac{t-3}{2} = \frac{18(t-3)}{18 \times 2} = \frac{18t-54}{36}$
Combine the fractions.
$\frac{20t}{36} + \frac{9t-27}{36} - \frac{18t-54}{36} = \frac{20t + (9t-27) - (18t-54)}{36}$
Distribute the negative sign and combine like terms in the numerator.
$\frac{20t+9t-27-18t+54}{36} = \frac{(20t+9t-18t) + (-27+54)}{36}$ $\frac{11t+27}{36}$
The simplified expression is $\boxed{\frac{11t+27}{36}}$ .

4. Simplify $\frac{u+5}{4} - \frac{3u-5}{2}$
Find the LCM of the denominators 4 and 2.
The LCM is 4.
Rewrite the second fraction with the common denominator.
$\frac{3u-5}{2} = \frac{2(3u-5)}{2 \times 2} = \frac{6u-10}{4}$
Subtract the fractions.
$\frac{u+5}{4} - \frac{6u-10}{4} = \frac{(u+5) - (6u-10)}{4}$
Distribute the negative sign and combine like terms in the numerator.
$\frac{u+5-6u+10}{4} = \frac{(u-6u) + (5+10)}{4}$ $\frac{-5u+15}{4}$
The simplified expression is $\boxed{\frac{-5u+15}{4}}$ .

5. Simplify $\frac{x-3}{3} - \frac{x+2}{5}$
Find the LCM of the denominators 3 and 5.
The LCM is 15.
Rewrite each fraction with the common denominator.
$\frac{x-3}{3} = \frac{5(x-3)}{5 \times 3} = \frac{5x-15}{15}$ $\frac{x+2}{5} = \frac{3(x+2)}{3 \times 5} = \frac{3x+6}{15}$
Subtract the fractions.
$\frac{5x-15}{15} - \frac{3x+6}{15} = \frac{(5x-15) - (3x+6)}{15}$
Distribute the negative sign and combine like terms in the numerator.
$\frac{5x-15-3x-6}{15} = \frac{(5x-3x) + (-15-6)}{15}$ $\frac{2x-21}{15}$
The simplified expression is $\boxed{\frac{2x-21}{15}}$ .

Simplify the following: 3a+4b/10 + 5a-b/4 6y-5x/10 - 2x+y/5 + x/3 5t/9 - 9/4 4u+5/4 - 3u-5/2 x-3/3 - x+2/5

Step-by-step solution

Find the least common multiple (LCM) of the denominators 10 and 4.

Rewrite each fraction with the common denominator.

Add the fractions.

Combine like terms in the numerator.

Find the LCM of the denominators 10, 5, and 3.

Rewrite each fraction with the common denominator.

Combine the fractions.

Distribute the negative sign and combine like terms in the numerator.

Find the LCM of the denominators 9, 4, and 2.

Rewrite each fraction with the common denominator.

Combine the fractions.

Distribute the negative sign and combine like terms in the numerator.

Find the LCM of the denominators 4 and 2.

Rewrite the second fraction with the common denominator.

Subtract the fractions.

Distribute the negative sign and combine like terms in the numerator.

Find the LCM of the denominators 3 and 5.

Rewrite each fraction with the common denominator.

Subtract the fractions.

Distribute the negative sign and combine like terms in the numerator.

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