Solve 5(1/2 - 1/5) + (1 1/3 + 1/9) * zeta - \{(5/3 * 1/8) + 9/6\}

Question

$$5(\frac{1}{2}-\frac{1}{5})+(1\frac{1}{3}+\frac{1}{9})\zeta-\{(\frac{5}{3}\frac{1}{8})+\frac{9}{6}\}$$

Evaluate

$\frac{13\zeta }{9}-\frac{5}{24}$

Solution Steps

Least common multiple of $2$ and $5$ is $10$. Convert $\frac{1}{2}$ and $\frac{1}{5}$ to fractions with denominator $10$.

$$5\left(\frac{5}{10}-\frac{2}{10}\right)+\left(\frac{1\times 3+1}{3}+\frac{1}{9}\right)\zeta -\left(\frac{5}{3}\times \frac{1}{8}+\frac{9}{6}\right)$$

Since $\frac{5}{10}$ and $\frac{2}{10}$ have the same denominator, subtract them by subtracting their numerators.

$$5\times \frac{5-2}{10}+\left(\frac{1\times 3+1}{3}+\frac{1}{9}\right)\zeta -\left(\frac{5}{3}\times \frac{1}{8}+\frac{9}{6}\right)$$

Subtract $2$ from $5$ to get $3$.

$$5\times \frac{3}{10}+\left(\frac{1\times 3+1}{3}+\frac{1}{9}\right)\zeta -\left(\frac{5}{3}\times \frac{1}{8}+\frac{9}{6}\right)$$

Express $5\times \frac{3}{10}$ as a single fraction.

$$\frac{5\times 3}{10}+\left(\frac{1\times 3+1}{3}+\frac{1}{9}\right)\zeta -\left(\frac{5}{3}\times \frac{1}{8}+\frac{9}{6}\right)$$

Multiply $5$ and $3$ to get $15$.

$$\frac{15}{10}+\left(\frac{1\times 3+1}{3}+\frac{1}{9}\right)\zeta -\left(\frac{5}{3}\times \frac{1}{8}+\frac{9}{6}\right)$$

Reduce the fraction $\frac{15}{10}$ to lowest terms by extracting and canceling out $5$.

$$\frac{3}{2}+\left(\frac{1\times 3+1}{3}+\frac{1}{9}\right)\zeta -\left(\frac{5}{3}\times \frac{1}{8}+\frac{9}{6}\right)$$

Multiply $1$ and $3$ to get $3$.

$$\frac{3}{2}+\left(\frac{3+1}{3}+\frac{1}{9}\right)\zeta -\left(\frac{5}{3}\times \frac{1}{8}+\frac{9}{6}\right)$$

Add $3$ and $1$ to get $4$.

$$\frac{3}{2}+\left(\frac{4}{3}+\frac{1}{9}\right)\zeta -\left(\frac{5}{3}\times \frac{1}{8}+\frac{9}{6}\right)$$

Least common multiple of $3$ and $9$ is $9$. Convert $\frac{4}{3}$ and $\frac{1}{9}$ to fractions with denominator $9$.

$$\frac{3}{2}+\left(\frac{12}{9}+\frac{1}{9}\right)\zeta -\left(\frac{5}{3}\times \frac{1}{8}+\frac{9}{6}\right)$$

Since $\frac{12}{9}$ and $\frac{1}{9}$ have the same denominator, add them by adding their numerators.

$$\frac{3}{2}+\frac{12+1}{9}\zeta -\left(\frac{5}{3}\times \frac{1}{8}+\frac{9}{6}\right)$$

Add $12$ and $1$ to get $13$.

$$\frac{3}{2}+\frac{13}{9}\zeta -\left(\frac{5}{3}\times \frac{1}{8}+\frac{9}{6}\right)$$

Multiply $\frac{5}{3}$ times $\frac{1}{8}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{3}{2}+\frac{13}{9}\zeta -\left(\frac{5\times 1}{3\times 8}+\frac{9}{6}\right)$$

Do the multiplications in the fraction $\frac{5\times 1}{3\times 8}$.

$$\frac{3}{2}+\frac{13}{9}\zeta -\left(\frac{5}{24}+\frac{9}{6}\right)$$

Reduce the fraction $\frac{9}{6}$ to lowest terms by extracting and canceling out $3$.

$$\frac{3}{2}+\frac{13}{9}\zeta -\left(\frac{5}{24}+\frac{3}{2}\right)$$

Least common multiple of $24$ and $2$ is $24$. Convert $\frac{5}{24}$ and $\frac{3}{2}$ to fractions with denominator $24$.

$$\frac{3}{2}+\frac{13}{9}\zeta -\left(\frac{5}{24}+\frac{36}{24}\right)$$

Since $\frac{5}{24}$ and $\frac{36}{24}$ have the same denominator, add them by adding their numerators.

$$\frac{3}{2}+\frac{13}{9}\zeta -\frac{5+36}{24}$$

Add $5$ and $36$ to get $41$.

$$\frac{3}{2}+\frac{13}{9}\zeta -\frac{41}{24}$$

Least common multiple of $2$ and $24$ is $24$. Convert $\frac{3}{2}$ and $\frac{41}{24}$ to fractions with denominator $24$.

$$\frac{36}{24}+\frac{13}{9}\zeta -\frac{41}{24}$$

Since $\frac{36}{24}$ and $\frac{41}{24}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{36-41}{24}+\frac{13}{9}\zeta $$

Subtract $41$ from $36$ to get $-5$.

$$-\frac{5}{24}+\frac{13}{9}\zeta $$

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