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Multiply \( \frac{2}{5} \times \frac{3}{4} p \) and \( \frac{2}{5} \times \frac{5}{6} w \) to get \( \frac{3}{10} p + \frac{1}{3} w \), then multiply \( \frac{1}{5} \times \frac{1}{2} p \) and \( \frac{1}{5} \times 6 \frac{2}{3} w \) to get \( \frac{1}{10} p - \frac{4}{3} w \). Combine the terms: \( \frac{3}{10} p + \frac{1}{10} p = \frac{2}{5} p \) and \( \frac{1}{3} w - \frac{4}{3} w = -w \). The final result is \( \frac{2}{5} p - w \).

11 answers

Reformat the expression as follows:

  1. Multiply the two expressions: \[ \frac{2}{5} \times \frac{3}{4} p = \frac{3}{10} p \] \[ \frac{2}{5} \times \frac{5}{6} w = \frac{1}{3} w \] Thus, combining these results gives: \[ \frac{3}{10} p + \frac{1}{3} w \]

  2. Next, multiply the second set of expressions: \[ \frac{1}{5} \times \frac{1}{2} p = \frac{1}{10} p \] \[ \frac{1}{5} \times 6 \frac{2}{3} w = \frac{1}{5} \times \frac{20}{3} w = \frac{4}{3} w \] Combining these results gives: \[ \frac{1}{10} p - \frac{4}{3} w \]

  3. Now combine the terms from both results: For \( p \) terms: \[ \frac{3}{10} p + \frac{1}{10} p = \frac{4}{10} p = \frac{2}{5} p \] For \( w \) terms: \[ \frac{1}{3} w - \frac{4}{3} w = -\frac{3}{3} w = -w \]

  4. Therefore, the final result is: \[ \frac{2}{5} p - w \]

shorter

Here’s a more concise version:

  1. Multiply to get the first result: \[ \frac{2}{5} \times \frac{3}{4} p = \frac{3}{10} p, \quad \frac{2}{5} \times \frac{5}{6} w = \frac{1}{3} w \] Thus, \[ \frac{3}{10} p + \frac{1}{3} w. \]

  2. Multiply to get the second result: \[ \frac{1}{5} \times \frac{1}{2} p = \frac{1}{10} p, \quad \frac{1}{5} \times 6 \frac{2}{3} w = \frac{4}{3} w \] So, \[ \frac{1}{10} p - \frac{4}{3} w. \]

  3. Combine the results: \[ \frac{3}{10} p + \frac{1}{10} p = \frac{2}{5} p, \quad \frac{1}{3} w - \frac{4}{3} w = -w. \]

  4. Final result: \[ \frac{2}{5} p - w. \]

even shorter

Here’s a more streamlined version:

  1. First Multiplication: \[ \frac{2}{5} \times \frac{3}{4} p = \frac{3}{10} p, \quad \frac{2}{5} \times \frac{5}{6} w = \frac{1}{3} w \quad \Rightarrow \quad \frac{3}{10} p + \frac{1}{3} w \]

  2. Second Multiplication: \[ \frac{1}{5} \times \frac{1}{2} p = \frac{1}{10} p, \quad \frac{1}{5} \times 6 \frac{2}{3} w = \frac{4}{3} w \quad \Rightarrow \quad \frac{1}{10} p - \frac{4}{3} w \]

  3. Combine: \[ \left(\frac{3}{10} + \frac{1}{10}\right)p + \left(\frac{1}{3} - \frac{4}{3}\right)w = \frac{2}{5} p - w \]

  4. Final Result: \[ \frac{2}{5} p - w \]

format it so i can copy and paste it

Sure! Here's the formatted version for easy copying:

  1. First Multiplication: \[ \frac{2}{5} \times \frac{3}{4} p = \frac{3}{10} p, \quad \frac{2}{5} \times \frac{5}{6} w = \frac{1}{3} w \quad \Rightarrow \quad \frac{3}{10} p + \frac{1}{3} w \]

  2. Second Multiplication: \[ \frac{1}{5} \times \frac{1}{2} p = \frac{1}{10} p, \quad \frac{1}{5} \times 6 \frac{2}{3} w = \frac{4}{3} w \quad \Rightarrow \quad \frac{1}{10} p - \frac{4}{3} w \]

  3. Combine: \[ \left(\frac{3}{10} + \frac{1}{10}\right)p + \left(\frac{1}{3} - \frac{4}{3}\right)w = \frac{2}{5} p - w \]

  4. Final Result: \[ \frac{2}{5} p - w \]

reformat it so i can copy it

Certainly! Here is the content formatted for easy copying:

  1. First Multiplication: \[ \frac{2}{5} \times \frac{3}{4} p = \frac{3}{10} p, \quad \frac{2}{5} \times \frac{5}{6} w = \frac{1}{3} w \quad \Rightarrow \quad \frac{3}{10} p + \frac{1}{3} w \]

  2. Second Multiplication: \[ \frac{1}{5} \times \frac{1}{2} p = \frac{1}{10} p, \quad \frac{1}{5} \times 6 \frac{2}{3} w = \frac{4}{3} w \quad \Rightarrow \quad \frac{1}{10} p - \frac{4}{3} w \]

  3. Combine: \[ \left(\frac{3}{10} + \frac{1}{10}\right) p + \left(\frac{1}{3} - \frac{4}{3}\right) w = \frac{2}{5} p - w \]

  4. Final Result: \[ \frac{2}{5} p - w \]

Feel free to copy and paste this text directly!

write it so i can copy the whole thing

Of course! Here it is in plain text format so you can easily copy the whole thing:

1. First Multiplication:
   \\[
   \frac{2}{5} \times \frac{3}{4} p = \frac{3}{10} p, \quad \frac{2}{5} \times \frac{5}{6} w = \frac{1}{3} w \quad \Rightarrow \quad \frac{3}{10} p + \frac{1}{3} w
   \\]

2. Second Multiplication:
   \\[
   \frac{1}{5} \times \frac{1}{2} p = \frac{1}{10} p, \quad \frac{1}{5} \times 6 \frac{2}{3} w = \frac{4}{3} w \quad \Rightarrow \quad \frac{1}{10} p - \frac{4}{3} w
   \\]

3. Combine:
   \\[
   \left(\frac{3}{10} + \frac{1}{10}\right) p + \left(\frac{1}{3} - \frac{4}{3}\right) w = \frac{2}{5} p - w
   \\]

4. Final Result:
   \\[
   \frac{2}{5} p - w
   \\]

You can copy everything inside the code block directly.