Algebra Examples

$\frac{5}{6}$ , $\frac{8}{12}$ , $\frac{4}{7}$

Step 1

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

Fractions with the same denominator:

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

Fractions with different denominators such as, $LCM (\frac{a}{b}, \frac{c}{d})$ :

1: Find the LCM of $b$ and $d = LCM (b, d)$

2: Multiply the numerator and denominator of the first fraction $\frac{a}{b}$ by $\frac{LCM (b, d)}{b}$

3: Multiply the numerator and denominator of the second fraction $\frac{c}{d}$ by $\frac{LCM (b, d)}{d}$

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

Step 2

Find the LCM for the denominators of $\frac{5}{6}, \frac{2}{3}, \frac{4}{7}$ .

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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

$6$ has factors of $2$ and $3$ .

$2 \cdot 3$

Step 2.3

Since $3$ has no factors besides $1$ and $3$ .

$3$ is a prime number

Step 2.4

Since $7$ has no factors besides $1$ and $7$ .

$7$ is a prime number

Step 2.5

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

$2 \cdot 3 \cdot 7$

Step 2.6

Multiply $2 \cdot 3 \cdot 7$ .

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Step 2.6.1

Multiply $2$ by $3$ .

$6 \cdot 7$

Step 2.6.2

Multiply $6$ by $7$ .

$42$

Step 3

Multiply each number by $\frac{n}{n}$ , where $n$ is a number that makes the denominator $42$ .

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Step 3.1

Divide $42$ by $6$ .

$7$

Step 3.2

Multiply the numerator and denominator of $\frac{5}{6}$ by $7$ .

$\frac{5 \cdot 7}{6 \cdot 7}$

Step 3.3

Multiply $5$ by $7$ .

$\frac{35}{6 \cdot 7}$

Step 3.4

Multiply $6$ by $7$ .

$\frac{35}{42}$

Step 3.5

Divide $42$ by $3$ .

$14$

Step 3.6

Multiply the numerator and denominator of $\frac{2}{3}$ by $14$ .

$\frac{2 \cdot 14}{3 \cdot 14}$

Step 3.7

Multiply $2$ by $14$ .

$\frac{28}{3 \cdot 14}$

Step 3.8

Multiply $3$ by $14$ .

$\frac{28}{42}$

Step 3.9

Divide $42$ by $7$ .

$6$

Step 3.10

Multiply the numerator and denominator of $\frac{4}{7}$ by $6$ .

$\frac{4 \cdot 6}{7 \cdot 6}$

Step 3.11

Multiply $4$ by $6$ .

$\frac{24}{7 \cdot 6}$

Step 3.12

Multiply $7$ by $6$ .

$\frac{24}{42}$

Step 3.13

Write the new list with the same denominators.

$\frac{35}{42}, \frac{28}{42}, \frac{24}{42}$

Step 4

Find the LCM for $35, 28, 24$ .

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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

$35$ has factors of $5$ and $7$ .

$5 \cdot 7$

Step 4.3

The prime factors for $28$ are $2 \cdot 2 \cdot 7$ .

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Step 4.3.1

$28$ has factors of $2$ and $14$ .

$2 \cdot 14$

Step 4.3.2

$14$ has factors of $2$ and $7$ .

$2 \cdot 2 \cdot 7$

Step 4.4

The prime factors for $24$ are $2 \cdot 2 \cdot 2 \cdot 3$ .

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Step 4.4.1

$24$ has factors of $2$ and $12$ .

$2 \cdot 12$

Step 4.4.2

$12$ has factors of $2$ and $6$ .

$2 \cdot 2 \cdot 6$

Step 4.4.3

$6$ has factors of $2$ and $3$ .

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

Step 4.5

The LCM of $35, 28, 24$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

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

Step 4.6

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

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Step 4.6.1

Multiply $2$ by $2$ .

$4 \cdot 2 \cdot 3 \cdot 5 \cdot 7$

Step 4.6.2

Multiply $4$ by $2$ .

$8 \cdot 3 \cdot 5 \cdot 7$

Step 4.6.3

Multiply $8$ by $3$ .

$24 \cdot 5 \cdot 7$

Step 4.6.4

Multiply $24$ by $5$ .

$120 \cdot 7$

Step 4.6.5

Multiply $120$ by $7$ .

$840$

Step 5

The answer can be found by taking the LCM of $35, 28, 24$ and dividing by the LCM of $6, 3, 7$ .

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Step 5.1

Divide the LCM of $35, 28, 24$ by the LCM of $6, 3, 7$ .

$\frac{840}{42}$

Step 5.2

Divide $840$ by $42$ .

$20$