Basic Math Examples

\frac{4}{5}

$\frac{4}{5}$ ,

\frac{1}{2}

$\frac{1}{2}$ ,

\frac{3}{4}

$\frac{3}{4}$ ,

\frac{6}{7}

$\frac{6}{7}$ ,

\frac{2}{3}

$\frac{2}{3}$ ,

\frac{5}{6}

$\frac{5}{6}$

Step 1

Write all of the fractions over the common denominator.

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Step 1.1

Find the LCM for the denominators of

\frac{4}{5}, \frac{1}{2}, \frac{3}{4}, \frac{6}{7}, \frac{2}{3}, \frac{5}{6}

$\frac{4}{5}, \frac{1}{2}, \frac{3}{4}, \frac{6}{7}, \frac{2}{3}, \frac{5}{6}$ .

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Step 1.1.1

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 1.1.2

Since

5

$5$ has no factors besides

1

$1$ and

5

$5$ .

5

$5$ is a prime number

Step 1.1.3

Since

2

$2$ has no factors besides

1

$1$ and

2

$2$ .

2

$2$ is a prime number

Step 1.1.4

4

$4$ has factors of

2

$2$ and

2

$2$ .

2 \cdot 2

$2 \cdot 2$

Step 1.1.5

Since

7

$7$ has no factors besides

1

$1$ and

7

$7$ .

7

$7$ is a prime number

Step 1.1.6

Since

3

$3$ has no factors besides

1

$1$ and

3

$3$ .

3

$3$ is a prime number

Step 1.1.7

6

$6$ has factors of

2

$2$ and

3

$3$ .

2 \cdot 3

$2 \cdot 3$

Step 1.1.8

The LCM of

5, 2, 4, 7, 3, 6

$5, 2, 4, 7, 3, 6$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

2 \cdot 2 \cdot 3 \cdot 5 \cdot 7

$2 \cdot 2 \cdot 3 \cdot 5 \cdot 7$

Step 1.1.9

Multiply

2 \cdot 2 \cdot 3 \cdot 5 \cdot 7

$2 \cdot 2 \cdot 3 \cdot 5 \cdot 7$ .

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Step 1.1.9.1

Multiply

2

$2$ by

2

$2$ .

4 \cdot 3 \cdot 5 \cdot 7

$4 \cdot 3 \cdot 5 \cdot 7$

Step 1.1.9.2

Multiply

4

$4$ by

3

$3$ .

12 \cdot 5 \cdot 7

$12 \cdot 5 \cdot 7$

Step 1.1.9.3

Multiply

12

$12$ by

5

$5$ .

60 \cdot 7

$60 \cdot 7$

Step 1.1.9.4

Multiply

60

$60$ by

7

$7$ .

420

$420$

420

$420$

420

$420$

Step 1.2

Multiply each number by

\frac{n}{n}

$\frac{n}{n}$ , where

n

$n$ is a number that makes the denominator

420

$420$ .

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Step 1.2.1

Divide

420

$420$ by

5

$5$ .

84

$84$

Step 1.2.2

Multiply the numerator and denominator of

\frac{4}{5}

$\frac{4}{5}$ by

84

$84$ .

\frac{4 \cdot 84}{5 \cdot 84}

$\frac{4 \cdot 84}{5 \cdot 84}$

Step 1.2.3

Multiply

4

$4$ by

84

$84$ .

\frac{336}{5 \cdot 84}

$\frac{336}{5 \cdot 84}$

Step 1.2.4

Multiply

5

$5$ by

84

$84$ .

\frac{336}{420}

$\frac{336}{420}$

Step 1.2.5

Divide

420

$420$ by

2

$2$ .

210

$210$

Step 1.2.6

Multiply the numerator and denominator of

\frac{1}{2}

$\frac{1}{2}$ by

210

$210$ .

\frac{1 \cdot 210}{2 \cdot 210}

$\frac{1 \cdot 210}{2 \cdot 210}$

Step 1.2.7

Multiply

210

$210$ by

1

$1$ .

\frac{210}{2 \cdot 210}

$\frac{210}{2 \cdot 210}$

Step 1.2.8

Multiply

2

$2$ by

210

$210$ .

\frac{210}{420}

$\frac{210}{420}$

Step 1.2.9

Divide

420

$420$ by

4

$4$ .

105

$105$

Step 1.2.10

Multiply the numerator and denominator of

\frac{3}{4}

$\frac{3}{4}$ by

105

$105$ .

\frac{3 \cdot 105}{4 \cdot 105}

$\frac{3 \cdot 105}{4 \cdot 105}$

Step 1.2.11

Multiply

3

$3$ by

105

$105$ .

\frac{315}{4 \cdot 105}

$\frac{315}{4 \cdot 105}$

Step 1.2.12

Multiply

4

$4$ by

105

$105$ .

\frac{315}{420}

$\frac{315}{420}$

Step 1.2.13

Divide

420

$420$ by

7

$7$ .

60

$60$

Step 1.2.14

Multiply the numerator and denominator of

\frac{6}{7}

$\frac{6}{7}$ by

60

$60$ .

\frac{6 \cdot 60}{7 \cdot 60}

$\frac{6 \cdot 60}{7 \cdot 60}$

Step 1.2.15

Multiply

6

$6$ by

60

$60$ .

\frac{360}{7 \cdot 60}

$\frac{360}{7 \cdot 60}$

Step 1.2.16

Multiply

7

$7$ by

60

$60$ .

\frac{360}{420}

$\frac{360}{420}$

Step 1.2.17

Divide

420

$420$ by

3

$3$ .

140

$140$

Step 1.2.18

Multiply the numerator and denominator of

\frac{2}{3}

$\frac{2}{3}$ by

140

$140$ .

\frac{2 \cdot 140}{3 \cdot 140}

$\frac{2 \cdot 140}{3 \cdot 140}$

Step 1.2.19

Multiply

2

$2$ by

140

$140$ .

\frac{280}{3 \cdot 140}

$\frac{280}{3 \cdot 140}$

Step 1.2.20

Multiply

3

$3$ by

140

$140$ .

\frac{280}{420}

$\frac{280}{420}$

Step 1.2.21

Divide

420

$420$ by

6

$6$ .

70

$70$

Step 1.2.22

Multiply the numerator and denominator of

\frac{5}{6}

$\frac{5}{6}$ by

70

$70$ .

\frac{5 \cdot 70}{6 \cdot 70}

$\frac{5 \cdot 70}{6 \cdot 70}$

Step 1.2.23

Multiply

5

$5$ by

70

$70$ .

\frac{350}{6 \cdot 70}

$\frac{350}{6 \cdot 70}$

Step 1.2.24

Multiply

6

$6$ by

70

$70$ .

\frac{350}{420}

$\frac{350}{420}$

Step 1.2.25

Write the new list with the same denominators.

\frac{336}{420}, \frac{210}{420}, \frac{315}{420}, \frac{360}{420}, \frac{280}{420}, \frac{350}{420}

$\frac{336}{420}, \frac{210}{420}, \frac{315}{420}, \frac{360}{420}, \frac{280}{420}, \frac{350}{420}$

\frac{336}{420}, \frac{210}{420}, \frac{315}{420}, \frac{360}{420}, \frac{280}{420}, \frac{350}{420}

$\frac{336}{420}, \frac{210}{420}, \frac{315}{420}, \frac{360}{420}, \frac{280}{420}, \frac{350}{420}$

\frac{336}{420}, \frac{210}{420}, \frac{315}{420}, \frac{360}{420}, \frac{280}{420}, \frac{350}{420}

$\frac{336}{420}, \frac{210}{420}, \frac{315}{420}, \frac{360}{420}, \frac{280}{420}, \frac{350}{420}$

Step 2

Since the denominators are equal, arrange the numerators.

\frac{210}{420}, \frac{280}{420}, \frac{315}{420}, \frac{336}{420}, \frac{350}{420}, \frac{360}{420}

$\frac{210}{420}, \frac{280}{420}, \frac{315}{420}, \frac{336}{420}, \frac{350}{420}, \frac{360}{420}$

Step 3

Replace the fractions with the original fractions.

$\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}$