Simplify 3m-2/5n+5m-7+5 1/2*n+3n-2/5p-5n+8p

$3 m - \frac{2}{5} n + 5 m - 7 + 5 \frac{1}{2} \cdot n + 3 n - \frac{2}{5} p - 5 n + 8 p$

Step 1

Convert

$5 \frac{1}{2}$ to an improper fraction.

A mixed number is an addition of its whole and fractional parts.

$3 m - \frac{2}{5} n + 5 m - 7 + (5 + \frac{1}{2}) \cdot n + 3 n - \frac{2}{5} p - 5 n + 8 p$

Add

$5$ and

$\frac{1}{2}$ .

To write

$5$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$3 m - \frac{2}{5} n + 5 m - 7 + (5 \cdot \frac{2}{2} + \frac{1}{2}) \cdot n + 3 n - \frac{2}{5} p - 5 n + 8 p$

Combine

$5$ and

$\frac{2}{2}$ .

$3 m - \frac{2}{5} n + 5 m - 7 + (\frac{5 \cdot 2}{2} + \frac{1}{2}) \cdot n + 3 n - \frac{2}{5} p - 5 n + 8 p$

Combine the numerators over the common denominator.

$3 m - \frac{2}{5} n + 5 m - 7 + \frac{5 \cdot 2 + 1}{2} \cdot n + 3 n - \frac{2}{5} p - 5 n + 8 p$

Simplify the numerator.

Multiply

$5$ by

$2$ .

$3 m - \frac{2}{5} n + 5 m - 7 + \frac{10 + 1}{2} \cdot n + 3 n - \frac{2}{5} p - 5 n + 8 p$

Add

$10$ and

$1$ .

$3 m - \frac{2}{5} n + 5 m - 7 + \frac{11}{2} \cdot n + 3 n - \frac{2}{5} p - 5 n + 8 p$

Step 2

Simplify each term.

Combine

$n$ and

$\frac{2}{5}$ .

$3 m - \frac{n \cdot 2}{5} + 5 m - 7 + \frac{11}{2} \cdot n + 3 n - \frac{2}{5} p - 5 n + 8 p$

Move

$2$ to the left of

$n$ .

$3 m - \frac{2 \cdot n}{5} + 5 m - 7 + \frac{11}{2} \cdot n + 3 n - \frac{2}{5} p - 5 n + 8 p$

Combine

$\frac{11}{2}$ and

$n$ .

$3 m - \frac{2 n}{5} + 5 m - 7 + \frac{11 n}{2} + 3 n - \frac{2}{5} p - 5 n + 8 p$

Combine

$p$ and

$\frac{2}{5}$ .

$3 m - \frac{2 n}{5} + 5 m - 7 + \frac{11 n}{2} + 3 n - \frac{p \cdot 2}{5} - 5 n + 8 p$

Move

$2$ to the left of

$p$ .

$3 m - \frac{2 n}{5} + 5 m - 7 + \frac{11 n}{2} + 3 n - \frac{2 p}{5} - 5 n + 8 p$

Step 3

Add

$3 m$ and

$5 m$ .

$8 m - \frac{2 n}{5} - 7 + \frac{11 n}{2} + 3 n - \frac{2 p}{5} - 5 n + 8 p$

Step 4

To write

$- \frac{2 n}{5}$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$8 m - \frac{2 n}{5} \cdot \frac{2}{2} + \frac{11 n}{2} - 7 + 3 n - \frac{2 p}{5} - 5 n + 8 p$

Step 5

To write

$\frac{11 n}{2}$ as a fraction with a common denominator, multiply by

$\frac{5}{5}$ .

$8 m - \frac{2 n}{5} \cdot \frac{2}{2} + \frac{11 n}{2} \cdot \frac{5}{5} - 7 + 3 n - \frac{2 p}{5} - 5 n + 8 p$

Step 6

Write each expression with a common denominator of

$10$ , by multiplying each by an appropriate factor of

$1$ .

Multiply

$\frac{2 n}{5}$ by

$\frac{2}{2}$ .

$8 m - \frac{2 n \cdot 2}{5 \cdot 2} + \frac{11 n}{2} \cdot \frac{5}{5} - 7 + 3 n - \frac{2 p}{5} - 5 n + 8 p$

Multiply

$5$ by

$2$ .

$8 m - \frac{2 n \cdot 2}{10} + \frac{11 n}{2} \cdot \frac{5}{5} - 7 + 3 n - \frac{2 p}{5} - 5 n + 8 p$

Multiply

$\frac{11 n}{2}$ by

$\frac{5}{5}$ .

$8 m - \frac{2 n \cdot 2}{10} + \frac{11 n \cdot 5}{2 \cdot 5} - 7 + 3 n - \frac{2 p}{5} - 5 n + 8 p$

Multiply

$2$ by

$5$ .

$8 m - \frac{2 n \cdot 2}{10} + \frac{11 n \cdot 5}{10} - 7 + 3 n - \frac{2 p}{5} - 5 n + 8 p$

Step 7

Combine the numerators over the common denominator.

$8 m + \frac{- 2 n \cdot 2 + 11 n \cdot 5}{10} - 7 + 3 n - \frac{2 p}{5} - 5 n + 8 p$

Step 8

Simplify each term.

Simplify the numerator.

Factor

$n$ out of

$- 2 n \cdot 2 + 11 n \cdot 5$ .

Factor

$n$ out of

$- 2 n \cdot 2$ .

$8 m + \frac{n (- 2 \cdot 2) + 11 n \cdot 5}{10} - 7 + 3 n - \frac{2 p}{5} - 5 n + 8 p$

Factor

$n$ out of

$11 n \cdot 5$ .

$8 m + \frac{n (- 2 \cdot 2) + n (11 \cdot 5)}{10} - 7 + 3 n - \frac{2 p}{5} - 5 n + 8 p$

Factor

$n$ out of

$n (- 2 \cdot 2) + n (11 \cdot 5)$ .

$8 m + \frac{n (- 2 \cdot 2 + 11 \cdot 5)}{10} - 7 + 3 n - \frac{2 p}{5} - 5 n + 8 p$

Multiply

$- 2$ by

$2$ .

$8 m + \frac{n (- 4 + 11 \cdot 5)}{10} - 7 + 3 n - \frac{2 p}{5} - 5 n + 8 p$

Multiply

$11$ by

$5$ .

$8 m + \frac{n (- 4 + 55)}{10} - 7 + 3 n - \frac{2 p}{5} - 5 n + 8 p$

Add

$- 4$ and

$55$ .

$8 m + \frac{n \cdot 51}{10} - 7 + 3 n - \frac{2 p}{5} - 5 n + 8 p$

Move

$51$ to the left of

$n$ .

$8 m + \frac{51 n}{10} - 7 + 3 n - \frac{2 p}{5} - 5 n + 8 p$

Step 9

To write

$3 n$ as a fraction with a common denominator, multiply by

$\frac{10}{10}$ .

$8 m + \frac{51 n}{10} + 3 n \cdot \frac{10}{10} - 7 - \frac{2 p}{5} - 5 n + 8 p$

Step 10

Simplify terms.

Combine

$3 n$ and

$\frac{10}{10}$ .

$8 m + \frac{51 n}{10} + \frac{3 n \cdot 10}{10} - 7 - \frac{2 p}{5} - 5 n + 8 p$

Combine the numerators over the common denominator.

$8 m + \frac{51 n + 3 n \cdot 10}{10} - 7 - \frac{2 p}{5} - 5 n + 8 p$

Step 11

Simplify the numerator.

Factor

$3 n$ out of

$51 n + 3 n \cdot 10$ .

Factor

$3 n$ out of

$51 n$ .

$8 m + \frac{3 n (17) + 3 n \cdot 10}{10} - 7 - \frac{2 p}{5} - 5 n + 8 p$

Factor

$3 n$ out of

$3 n \cdot 10$ .

$8 m + \frac{3 n (17) + 3 n (10)}{10} - 7 - \frac{2 p}{5} - 5 n + 8 p$

Factor

$3 n$ out of

$3 n (17) + 3 n (10)$ .

$8 m + \frac{3 n (17 + 10)}{10} - 7 - \frac{2 p}{5} - 5 n + 8 p$

Add

$17$ and

$10$ .

$8 m + \frac{3 n \cdot 27}{10} - 7 - \frac{2 p}{5} - 5 n + 8 p$

Multiply

$27$ by

$3$ .

$8 m + \frac{81 n}{10} - 7 - \frac{2 p}{5} - 5 n + 8 p$

Step 12

To write

$- 5 n$ as a fraction with a common denominator, multiply by

$\frac{10}{10}$ .

$8 m - 7 - \frac{2 p}{5} + \frac{81 n}{10} - 5 n \cdot \frac{10}{10} + 8 p$

Step 13

Simplify terms.

Combine

$- 5 n$ and

$\frac{10}{10}$ .

$8 m - 7 - \frac{2 p}{5} + \frac{81 n}{10} + \frac{- 5 n \cdot 10}{10} + 8 p$

Combine the numerators over the common denominator.

$8 m - 7 - \frac{2 p}{5} + \frac{81 n - 5 n \cdot 10}{10} + 8 p$

Step 14

Simplify each term.

Simplify the numerator.

Factor

$n$ out of

$81 n - 5 n \cdot 10$ .

Factor

$n$ out of

$81 n$ .

$8 m - 7 - \frac{2 p}{5} + \frac{n \cdot 81 - 5 n \cdot 10}{10} + 8 p$

Factor

$n$ out of

$- 5 n \cdot 10$ .

$8 m - 7 - \frac{2 p}{5} + \frac{n \cdot 81 + n (- 5 \cdot 10)}{10} + 8 p$

Factor

$n$ out of

$n \cdot 81 + n (- 5 \cdot 10)$ .

$8 m - 7 - \frac{2 p}{5} + \frac{n (81 - 5 \cdot 10)}{10} + 8 p$

Multiply

$- 5$ by

$10$ .

$8 m - 7 - \frac{2 p}{5} + \frac{n (81 - 50)}{10} + 8 p$

Subtract

$50$ from

$81$ .

$8 m - 7 - \frac{2 p}{5} + \frac{n \cdot 31}{10} + 8 p$

Move

$31$ to the left of

$n$ .

$8 m - 7 - \frac{2 p}{5} + \frac{31 n}{10} + 8 p$

Step 15

To write

$8 p$ as a fraction with a common denominator, multiply by

$\frac{5}{5}$ .

$8 m - 7 + \frac{31 n}{10} - \frac{2 p}{5} + 8 p \cdot \frac{5}{5}$

Step 16

Simplify terms.

Combine

$8 p$ and

$\frac{5}{5}$ .

$8 m - 7 + \frac{31 n}{10} - \frac{2 p}{5} + \frac{8 p \cdot 5}{5}$

Combine the numerators over the common denominator.

$8 m - 7 + \frac{31 n}{10} + \frac{- 2 p + 8 p \cdot 5}{5}$

Step 17

Simplify the numerator.

Factor

$2 p$ out of

$- 2 p + 8 p \cdot 5$ .

Factor

$2 p$ out of

$- 2 p$ .

$8 m - 7 + \frac{31 n}{10} + \frac{2 p (- 1) + 8 p \cdot 5}{5}$

Factor

$2 p$ out of

$8 p \cdot 5$ .

$8 m - 7 + \frac{31 n}{10} + \frac{2 p (- 1) + 2 p (4 \cdot 5)}{5}$

Factor

$2 p$ out of

$2 p (- 1) + 2 p (4 \cdot 5)$ .

$8 m - 7 + \frac{31 n}{10} + \frac{2 p (- 1 + 4 \cdot 5)}{5}$

Multiply

$4$ by

$5$ .

$8 m - 7 + \frac{31 n}{10} + \frac{2 p (- 1 + 20)}{5}$

Add

$- 1$ and

$20$ .

$8 m - 7 + \frac{31 n}{10} + \frac{2 p \cdot 19}{5}$

Multiply

$19$ by

$2$ .

$8 m - 7 + \frac{31 n}{10} + \frac{38 p}{5}$

Step 18

Move

$- 7$ .

$8 m + \frac{31 n}{10} + \frac{38 p}{5} - 7$

Simplify 3m-2/5n+5m-7+5 1/2*n+3n-2/5p-5n+8p

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