Answer :
Sure, let's simplify each given expression step by step.
### (a) [tex]\(\frac{2 x}{3}+\frac{5-x}{4}\)[/tex]
1. Find the common denominator, which is 12 in this case.
2. Rewrite each fraction:
[tex]\[ \frac{2x}{3} = \frac{2x \cdot 4}{3 \cdot 4} = \frac{8x}{12} \][/tex]
[tex]\[ \frac{5 - x}{4} = \frac{(5 - x) \cdot 3}{4 \cdot 3} = \frac{15 - 3x}{12} \][/tex]
3. Add the two fractions:
[tex]\[ \frac{8x}{12} + \frac{15 - 3x}{12} = \frac{8x + 15 - 3x}{12} = \frac{5x + 15}{12} \][/tex]
4. Simplify the fraction if possible:
[tex]\[ \frac{5x + 15}{12} = \frac{5(x + 3)}{12} \][/tex]
Thus, the simplified form is [tex]\( \frac{5 x}{12} + \frac{5}{4} \)[/tex].
### (b) [tex]\(\frac{x-y}{8}-\frac{3 x-2 y}{12}\)[/tex]
1. Find the common denominator, which is 24.
2. Rewrite each fraction:
[tex]\[ \frac{x - y}{8} = \frac{(x - y) \cdot 3}{8 \cdot 3} = \frac{3(x - y)}{24} = \frac{3x - 3y}{24} \][/tex]
[tex]\[ \frac{3x - 2y}{12} = \frac{(3x - 2y) \cdot 2}{12 \cdot 2} = \frac{2(3x - 2y)}{24} = \frac{6x - 4y}{24} \][/tex]
3. Subtract the two fractions:
[tex]\[ \frac{3x - 3y}{24} - \frac{6x - 4y}{24} = \frac{3x - 3y - 6x + 4y}{24} = \frac{-3x + y}{24} \][/tex]
Thus, the simplified form is [tex]\(\frac{-x}{8} + \frac{y}{24}\)[/tex].
### (c) [tex]\(\frac{4(2 a - b)}{3} - \frac{2(3 a + b)}{5}\)[/tex]
1. Simplify each fraction:
[tex]\[ \frac{4(2a - b)}{3} = \frac{8a - 4b}{3} \][/tex]
[tex]\[ \frac{2(3a + b)}{5} = \frac{6a + 2b}{5} \][/tex]
2. Find a common denominator, which is 15.
3. Rewrite the fractions:
[tex]\[ \frac{8a - 4b}{3} = \frac{(8a - 4b) \cdot 5}{3 \cdot 5} = \frac{40a - 20b}{15} \][/tex]
[tex]\[ \frac{6a + 2b}{5} = \frac{(6a + 2b) \cdot 3}{5 \cdot 3} = \frac{18a + 6b}{15} \][/tex]
4. Subtract the fractions:
[tex]\[ \frac{40a - 20b}{15} - \frac{18a + 6b}{15} = \frac{40a - 20b - 18a - 6b}{15} = \frac{22a - 26b}{15} \][/tex]
Thus, the simplified form is [tex]\(\frac{22a}{15} - \frac{26b}{15}\)[/tex].
### (d) [tex]\(\frac{h+f}{3}-\frac{f+k}{2}+\frac{4 h-k}{5}\)[/tex]
1. Find the common denominator, which is 30.
2. Rewrite each fraction:
[tex]\[ \frac{h + f}{3} = \frac{(h + f) \cdot 10}{3 \cdot 10} = \frac{10h + 10f}{30} \][/tex]
[tex]\[ \frac{f + k}{2} = \frac{(f + k) \cdot 15}{2 \cdot 15} = \frac{15f + 15k}{30} \][/tex]
[tex]\[ \frac{4h - k}{5} = \frac{(4h - k) \cdot 6}{5 \cdot 6} = \frac{24h - 6k}{30} \][/tex]
3. Combine the fractions:
[tex]\[ \frac{10h + 10f}{30} - \frac{15f + 15k}{30} + \frac{24h - 6k}{30} = \frac{10h + 10f - 15f - 15k + 24h - 6k}{30} = \frac{34h - 5f - 21k}{30} \][/tex]
Thus, the simplified form is [tex]\(-\frac{f}{6} + \frac{17h}{15} - \frac{7k}{10}\)[/tex].
### (e) [tex]\(3 q - \frac{4 p - 3 q}{5} - \frac{q - 4 p}{6}\)[/tex]
1. Simplify each fraction:
[tex]\[ 3q = \frac{3q \cdot 30}{30} = \frac{90q}{30} \][/tex]
[tex]\[ \frac{4p - 3q}{5} = \frac{(4p - 3q) \cdot 6}{5 \cdot 6} = \frac{24p - 18q}{30} \][/tex]
[tex]\[ \frac{q - 4p}{6} = \frac{(q - 4p) \cdot 5}{6 \cdot 5} = \frac{5q - 20p}{30} \][/tex]
2. Combine the fractions:
[tex]\[ \frac{90q}{30} - \frac{24p - 18q}{30} - \frac{5q - 20p}{30} = \frac{90q - 24p + 18q - 5q + 20p}{30} = \frac{-24p + 103q}{30} \][/tex]
Thus, the simplified form is [tex]\(-\frac{2p}{15} + \frac{103q}{30}\)[/tex].
### (f) [tex]\(\frac{4(x-5)}{7}-\left[\frac{5(x-y)}{6}+\frac{7x-y}{21}\right]\)[/tex]
1. Simplify each fraction:
[tex]\[ \frac{4(x - 5)}{7} = \frac{4x - 20}{7} \][/tex]
[tex]\[ \frac{5(x - y)}{6} = \frac{5(x - y)}{6} \][/tex]
[tex]\[ \frac{7x - y}{21} = \frac{(7x - y) \cdot 1}{21 \cdot 1} = \frac{7x - y}{21} \][/tex]
2. Find common denominators where necessary:
[tex]\[ \frac{4x - 20}{7} = \frac{4x - 20}{7} \][/tex]
[tex]\[ \frac{5(x - y)}{6} = \frac{5x - 5y}{6} \][/tex]
[tex]\[ \frac{7x - y}{21} = \frac{(7x - y) \cdot 1}{21} = \frac{7x - y}{21} \][/tex]
3. Convert to a common denominator, which is 42 for all fractions.
[tex]\[ \frac{4x - 20}{7} = \frac{(4x - 20) \cdot 6}{7 \cdot 6} = \frac{24x - 120}{42} \][/tex]
[tex]\[ \frac{5x - 5y}{6} = \frac{(5x - 5y) \cdot 7}{6 \cdot 7} = \frac{35x - 35y}{42} \][/tex]
[tex]\[ \frac{7x - y}{21} = \frac{(7x - y) \cdot 2}{21 \cdot 2} = \frac{14x - 2y}{42} \][/tex]
4. Combine the initial term and the combined terms inside the brackets:
[tex]\[ \frac{4(x - 5)}{7} - \left[\frac{5(x - y)}{6} + \frac{7x - y}{21}\right] = \frac{24x - 120}{42} - \left[\frac{35x - 35y + 14x - 2y}{42}\right] \][/tex]
5. Final subtraction:
[tex]\[ \frac{24x - 120}{42} - \frac{49x - 37y}{42} = \frac{24x - 120 - 49x + 37y}{42} = \frac{-25x + 37y - 120}{42} \][/tex]
Thus, the simplified form is [tex]\(-\frac{25x}{42} + \frac{37y}{42} - \frac{20}{7}\)[/tex].
### (a) [tex]\(\frac{2 x}{3}+\frac{5-x}{4}\)[/tex]
1. Find the common denominator, which is 12 in this case.
2. Rewrite each fraction:
[tex]\[ \frac{2x}{3} = \frac{2x \cdot 4}{3 \cdot 4} = \frac{8x}{12} \][/tex]
[tex]\[ \frac{5 - x}{4} = \frac{(5 - x) \cdot 3}{4 \cdot 3} = \frac{15 - 3x}{12} \][/tex]
3. Add the two fractions:
[tex]\[ \frac{8x}{12} + \frac{15 - 3x}{12} = \frac{8x + 15 - 3x}{12} = \frac{5x + 15}{12} \][/tex]
4. Simplify the fraction if possible:
[tex]\[ \frac{5x + 15}{12} = \frac{5(x + 3)}{12} \][/tex]
Thus, the simplified form is [tex]\( \frac{5 x}{12} + \frac{5}{4} \)[/tex].
### (b) [tex]\(\frac{x-y}{8}-\frac{3 x-2 y}{12}\)[/tex]
1. Find the common denominator, which is 24.
2. Rewrite each fraction:
[tex]\[ \frac{x - y}{8} = \frac{(x - y) \cdot 3}{8 \cdot 3} = \frac{3(x - y)}{24} = \frac{3x - 3y}{24} \][/tex]
[tex]\[ \frac{3x - 2y}{12} = \frac{(3x - 2y) \cdot 2}{12 \cdot 2} = \frac{2(3x - 2y)}{24} = \frac{6x - 4y}{24} \][/tex]
3. Subtract the two fractions:
[tex]\[ \frac{3x - 3y}{24} - \frac{6x - 4y}{24} = \frac{3x - 3y - 6x + 4y}{24} = \frac{-3x + y}{24} \][/tex]
Thus, the simplified form is [tex]\(\frac{-x}{8} + \frac{y}{24}\)[/tex].
### (c) [tex]\(\frac{4(2 a - b)}{3} - \frac{2(3 a + b)}{5}\)[/tex]
1. Simplify each fraction:
[tex]\[ \frac{4(2a - b)}{3} = \frac{8a - 4b}{3} \][/tex]
[tex]\[ \frac{2(3a + b)}{5} = \frac{6a + 2b}{5} \][/tex]
2. Find a common denominator, which is 15.
3. Rewrite the fractions:
[tex]\[ \frac{8a - 4b}{3} = \frac{(8a - 4b) \cdot 5}{3 \cdot 5} = \frac{40a - 20b}{15} \][/tex]
[tex]\[ \frac{6a + 2b}{5} = \frac{(6a + 2b) \cdot 3}{5 \cdot 3} = \frac{18a + 6b}{15} \][/tex]
4. Subtract the fractions:
[tex]\[ \frac{40a - 20b}{15} - \frac{18a + 6b}{15} = \frac{40a - 20b - 18a - 6b}{15} = \frac{22a - 26b}{15} \][/tex]
Thus, the simplified form is [tex]\(\frac{22a}{15} - \frac{26b}{15}\)[/tex].
### (d) [tex]\(\frac{h+f}{3}-\frac{f+k}{2}+\frac{4 h-k}{5}\)[/tex]
1. Find the common denominator, which is 30.
2. Rewrite each fraction:
[tex]\[ \frac{h + f}{3} = \frac{(h + f) \cdot 10}{3 \cdot 10} = \frac{10h + 10f}{30} \][/tex]
[tex]\[ \frac{f + k}{2} = \frac{(f + k) \cdot 15}{2 \cdot 15} = \frac{15f + 15k}{30} \][/tex]
[tex]\[ \frac{4h - k}{5} = \frac{(4h - k) \cdot 6}{5 \cdot 6} = \frac{24h - 6k}{30} \][/tex]
3. Combine the fractions:
[tex]\[ \frac{10h + 10f}{30} - \frac{15f + 15k}{30} + \frac{24h - 6k}{30} = \frac{10h + 10f - 15f - 15k + 24h - 6k}{30} = \frac{34h - 5f - 21k}{30} \][/tex]
Thus, the simplified form is [tex]\(-\frac{f}{6} + \frac{17h}{15} - \frac{7k}{10}\)[/tex].
### (e) [tex]\(3 q - \frac{4 p - 3 q}{5} - \frac{q - 4 p}{6}\)[/tex]
1. Simplify each fraction:
[tex]\[ 3q = \frac{3q \cdot 30}{30} = \frac{90q}{30} \][/tex]
[tex]\[ \frac{4p - 3q}{5} = \frac{(4p - 3q) \cdot 6}{5 \cdot 6} = \frac{24p - 18q}{30} \][/tex]
[tex]\[ \frac{q - 4p}{6} = \frac{(q - 4p) \cdot 5}{6 \cdot 5} = \frac{5q - 20p}{30} \][/tex]
2. Combine the fractions:
[tex]\[ \frac{90q}{30} - \frac{24p - 18q}{30} - \frac{5q - 20p}{30} = \frac{90q - 24p + 18q - 5q + 20p}{30} = \frac{-24p + 103q}{30} \][/tex]
Thus, the simplified form is [tex]\(-\frac{2p}{15} + \frac{103q}{30}\)[/tex].
### (f) [tex]\(\frac{4(x-5)}{7}-\left[\frac{5(x-y)}{6}+\frac{7x-y}{21}\right]\)[/tex]
1. Simplify each fraction:
[tex]\[ \frac{4(x - 5)}{7} = \frac{4x - 20}{7} \][/tex]
[tex]\[ \frac{5(x - y)}{6} = \frac{5(x - y)}{6} \][/tex]
[tex]\[ \frac{7x - y}{21} = \frac{(7x - y) \cdot 1}{21 \cdot 1} = \frac{7x - y}{21} \][/tex]
2. Find common denominators where necessary:
[tex]\[ \frac{4x - 20}{7} = \frac{4x - 20}{7} \][/tex]
[tex]\[ \frac{5(x - y)}{6} = \frac{5x - 5y}{6} \][/tex]
[tex]\[ \frac{7x - y}{21} = \frac{(7x - y) \cdot 1}{21} = \frac{7x - y}{21} \][/tex]
3. Convert to a common denominator, which is 42 for all fractions.
[tex]\[ \frac{4x - 20}{7} = \frac{(4x - 20) \cdot 6}{7 \cdot 6} = \frac{24x - 120}{42} \][/tex]
[tex]\[ \frac{5x - 5y}{6} = \frac{(5x - 5y) \cdot 7}{6 \cdot 7} = \frac{35x - 35y}{42} \][/tex]
[tex]\[ \frac{7x - y}{21} = \frac{(7x - y) \cdot 2}{21 \cdot 2} = \frac{14x - 2y}{42} \][/tex]
4. Combine the initial term and the combined terms inside the brackets:
[tex]\[ \frac{4(x - 5)}{7} - \left[\frac{5(x - y)}{6} + \frac{7x - y}{21}\right] = \frac{24x - 120}{42} - \left[\frac{35x - 35y + 14x - 2y}{42}\right] \][/tex]
5. Final subtraction:
[tex]\[ \frac{24x - 120}{42} - \frac{49x - 37y}{42} = \frac{24x - 120 - 49x + 37y}{42} = \frac{-25x + 37y - 120}{42} \][/tex]
Thus, the simplified form is [tex]\(-\frac{25x}{42} + \frac{37y}{42} - \frac{20}{7}\)[/tex].
