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

Related Questions