Finite Math Examples

36

$36$ ,

49 \div 9

$49 \div 9$

Step 1

To find the LCM for a list of fractions, check if denominators are similar or not.

Fractions with the same denominator:

LCM (\frac{a}{b}, \frac{c}{b}) = \frac{LCM (a, c)}{b}

$LCM (\frac{a}{b}, \frac{c}{b}) = \frac{LCM (a, c)}{b}$

Fractions with different denominators such as,

LCM (\frac{a}{b}, \frac{c}{d})

$LCM (\frac{a}{b}, \frac{c}{d})$ :

1: Find the LCM of

b

$b$ and

d = LCM (b, d)

$d = LCM (b, d)$

2: Multiply the numerator and denominator of the first fraction

\frac{a}{b}

$\frac{a}{b}$ by

\frac{LCM (b, d)}{b}

$\frac{LCM (b, d)}{b}$

3: Multiply the numerator and denominator of the second fraction

\frac{c}{d}

$\frac{c}{d}$ by

\frac{LCM (b, d)}{d}

$\frac{LCM (b, d)}{d}$

4: After making the denominators for all the fractions same, in this case, only two fractions, find the LCM of the new numerators

5: The LCM will be the

\frac{LCM (numerators)}{LCM (b, d)}

$\frac{LCM (numerators)}{LCM (b, d)}$

Step 2

Find the LCM for the denominators of

36, \frac{49}{9}

$36, \frac{49}{9}$ .

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Step 2.1

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 2.2

The number

1

$1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.3

9

$9$ has factors of

3

$3$ and

3

$3$ .

3 \cdot 3

$3 \cdot 3$

Step 2.4

Multiply

3

$3$ by

3

$3$ .

9

$9$

9

$9$

Step 3

Multiply each number by

\frac{n}{n}

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

n

$n$ is a number that makes the denominator

9

$9$ .

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Step 3.1

Multiply the numerator and denominator of

36

$36$ by

9

$9$ .

\frac{36 \cdot 9}{1 \cdot 9}

$\frac{36 \cdot 9}{1 \cdot 9}$

Step 3.2

Multiply

36

$36$ by

9

$9$ .

\frac{324}{1 \cdot 9}

$\frac{324}{1 \cdot 9}$

Step 3.3

Multiply

9

$9$ by

1

$1$ .

\frac{324}{9}

$\frac{324}{9}$

Step 3.4

Divide

9

$9$ by

9

$9$ .

1

$1$

Step 3.5

Multiply the numerator and denominator of

\frac{49}{9}

$\frac{49}{9}$ by

1

$1$ .

\frac{49 \cdot 1}{9 \cdot 1}

$\frac{49 \cdot 1}{9 \cdot 1}$

Step 3.6

Multiply

49

$49$ by

1

$1$ .

\frac{49}{9 \cdot 1}

$\frac{49}{9 \cdot 1}$

Step 3.7

Multiply

9

$9$ by

1

$1$ .

\frac{49}{9}

$\frac{49}{9}$

Step 3.8

Write the new list with the same denominators.

\frac{324}{9}, \frac{49}{9}

$\frac{324}{9}, \frac{49}{9}$

\frac{324}{9}, \frac{49}{9}

$\frac{324}{9}, \frac{49}{9}$

Step 4

Find the LCM for

324, 49

$324, 49$ .

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Step 4.1

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 4.2

The prime factors for

324

$324$ are

2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3

$2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3$ .

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Step 4.2.1

324

$324$ has factors of

2

$2$ and

162

$162$ .

2 \cdot 162

$2 \cdot 162$

Step 4.2.2

162

$162$ has factors of

2

$2$ and

81

$81$ .

2 \cdot 2 \cdot 81

$2 \cdot 2 \cdot 81$

Step 4.2.3

81

$81$ has factors of

3

$3$ and

27

$27$ .

2 \cdot 2 \cdot 3 \cdot 27

$2 \cdot 2 \cdot 3 \cdot 27$

Step 4.2.4

27

$27$ has factors of

3

$3$ and

9

$9$ .

2 \cdot 2 \cdot 3 \cdot 3 \cdot 9

$2 \cdot 2 \cdot 3 \cdot 3 \cdot 9$

Step 4.2.5

9

$9$ has factors of

3

$3$ and

3

$3$ .

2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3

$2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3$

2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3

$2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3$

Step 4.3

49

$49$ has factors of

7

$7$ and

7

$7$ .

7 \cdot 7

$7 \cdot 7$

Step 4.4

The LCM of

324, 49

$324, 49$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

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

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

Step 4.5

Multiply

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

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

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Step 4.5.1

Multiply

2

$2$ by

2

$2$ .

4 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 7 \cdot 7

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

Step 4.5.2

Multiply

4

$4$ by

3

$3$ .

12 \cdot 3 \cdot 3 \cdot 3 \cdot 7 \cdot 7

$12 \cdot 3 \cdot 3 \cdot 3 \cdot 7 \cdot 7$

Step 4.5.3

Multiply

12

$12$ by

3

$3$ .

36 \cdot 3 \cdot 3 \cdot 7 \cdot 7

$36 \cdot 3 \cdot 3 \cdot 7 \cdot 7$

Step 4.5.4

Multiply

36

$36$ by

3

$3$ .

108 \cdot 3 \cdot 7 \cdot 7

$108 \cdot 3 \cdot 7 \cdot 7$

Step 4.5.5

Multiply

108

$108$ by

3

$3$ .

324 \cdot 7 \cdot 7

$324 \cdot 7 \cdot 7$

Step 4.5.6

Multiply

324

$324$ by

7

$7$ .

2268 \cdot 7

$2268 \cdot 7$

Step 4.5.7

Multiply

2268

$2268$ by

7

$7$ .

15876

$15876$

15876

$15876$

15876

$15876$

Step 5

The answer can be found by taking the LCM of

324, 49

$324, 49$ and dividing by the LCM of

1, 9

$1, 9$ .

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Step 5.1

Divide the LCM of

324, 49

$324, 49$ by the LCM of

1, 9

$1, 9$ .

\frac{15876}{9}

$\frac{15876}{9}$

Step 5.2

Divide

15876

$15876$ by

9

$9$ .

1764

$1764$

1764

$1764$