Finite Math Examples

63000 - 27000

$63000 - 27000$ ,

3 \div 36000

$3 \div 36000$ ,

\frac{1}{12000} \cdot 2

$\frac{1}{12000} \cdot 2$

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

36000, \frac{1}{12000}, \frac{1}{6000}

$36000, \frac{1}{12000}, \frac{1}{6000}$ .

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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

The prime factors for

12000

$12000$ are

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

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

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Step 2.3.1

$12000$ has factors of

$2$ and

$6000$ .

$2 \cdot 6000$

Step 2.3.2

$6000$ has factors of

$2$ and

$3000$ .

$2 \cdot 2 \cdot 3000$

Step 2.3.3

$3000$ has factors of

$2$ and

$1500$ .

$2 \cdot 2 \cdot 2 \cdot 1500$

Step 2.3.4

$1500$ has factors of

$2$ and

$750$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 750$

Step 2.3.5

$750$ has factors of

$2$ and

$375$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 375$

Step 2.3.6

$375$ has factors of

$3$ and

$125$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 125$

Step 2.3.7

$125$ has factors of

$5$ and

$25$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5 \cdot 25$

Step 2.3.8

$25$ has factors of

$5$ and

$5$ .

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

Step 2.4

The prime factors for

$6000$ are

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

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Step 2.4.1

$6000$ has factors of

$2$ and

$3000$ .

$2 \cdot 3000$

Step 2.4.2

$3000$ has factors of

$2$ and

$1500$ .

$2 \cdot 2 \cdot 1500$

Step 2.4.3

$1500$ has factors of

$2$ and

$750$ .

$2 \cdot 2 \cdot 2 \cdot 750$

Step 2.4.4

$750$ has factors of

$2$ and

$375$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 375$

Step 2.4.5

$375$ has factors of

$3$ and

$125$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 125$

Step 2.4.6

$125$ has factors of

$5$ and

$25$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5 \cdot 25$

Step 2.4.7

$25$ has factors of

$5$ and

$5$ .

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

Step 2.5

The LCM of

$1, 12000, 6000$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

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

Step 2.6

Multiply

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

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Step 2.6.1

Multiply

$2$ by

$2$ .

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

Step 2.6.2

Multiply

$4$ by

$2$ .

$8 \cdot 2 \cdot 2 \cdot 3 \cdot 5 \cdot 5 \cdot 5$

Step 2.6.3

Multiply

$8$ by

$2$ .

$16 \cdot 2 \cdot 3 \cdot 5 \cdot 5 \cdot 5$

Step 2.6.4

Multiply

$16$ by

$2$ .

$32 \cdot 3 \cdot 5 \cdot 5 \cdot 5$

Step 2.6.5

Multiply

$32$ by

$3$ .

$96 \cdot 5 \cdot 5 \cdot 5$

Step 2.6.6

Multiply

$96$ by

$5$ .

$480 \cdot 5 \cdot 5$

Step 2.6.7

Multiply

$480$ by

$5$ .

$2400 \cdot 5$

Step 2.6.8

Multiply

$2400$ by

$5$ .

$12000$

Step 3

Multiply each number by

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

$n$ is a number that makes the denominator

$12000$ .

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Step 3.1

Multiply the numerator and denominator of

$36000$ by

$12000$ .

$\frac{36000 \cdot 12000}{1 \cdot 12000}$

Step 3.2

Multiply

$36000$ by

$12000$ .

$\frac{432000000}{1 \cdot 12000}$

Step 3.3

Multiply

$12000$ by

$1$ .

$\frac{432000000}{12000}$

Step 3.4

Divide

$12000$ by

$12000$ .

$1$

Step 3.5

Multiply the numerator and denominator of

$\frac{1}{12000}$ by

$1$ .

$\frac{1 \cdot 1}{12000 \cdot 1}$

Step 3.6

Multiply

$1$ by

$1$ .

$\frac{1}{12000 \cdot 1}$

Step 3.7

Multiply

$12000$ by

$1$ .

$\frac{1}{12000}$

Step 3.8

Divide

$12000$ by

$6000$ .

$2$

Step 3.9

Multiply the numerator and denominator of

$\frac{1}{6000}$ by

$2$ .

$\frac{1 \cdot 2}{6000 \cdot 2}$

Step 3.10

Multiply

$2$ by

$1$ .

$\frac{2}{6000 \cdot 2}$

Step 3.11

Multiply

$6000$ by

$2$ .

$\frac{2}{12000}$

Step 3.12

Write the new list with the same denominators.

$\frac{432000000}{12000}, \frac{1}{12000}, \frac{2}{12000}$

Step 4

Find the LCM for

$432000000, 1, 2$ .

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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

$432000000$ are

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

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Step 4.2.1

$432000000$ has factors of

$2$ and

$216000000$ .

$2 \cdot 216000000$

Step 4.2.2

$216000000$ has factors of

$2$ and

$108000000$ .

$2 \cdot 2 \cdot 108000000$

Step 4.2.3

$108000000$ has factors of

$2$ and

$54000000$ .

$2 \cdot 2 \cdot 2 \cdot 54000000$

Step 4.2.4

$54000000$ has factors of

$2$ and

$27000000$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 27000000$

Step 4.2.5

$27000000$ has factors of

$2$ and

$13500000$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 13500000$

Step 4.2.6

$13500000$ has factors of

$2$ and

$6750000$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 6750000$

Step 4.2.7

$6750000$ has factors of

$2$ and

$3375000$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3375000$

Step 4.2.8

$3375000$ has factors of

$2$ and

$1687500$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 1687500$

Step 4.2.9

$1687500$ has factors of

$2$ and

$843750$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 843750$

Step 4.2.10

$843750$ has factors of

$2$ and

$421875$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 421875$

Step 4.2.11

$421875$ has factors of

$3$ and

$140625$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 140625$

Step 4.2.12

$140625$ has factors of

$3$ and

$46875$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 46875$

Step 4.2.13

$46875$ has factors of

$3$ and

$15625$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 15625$

Step 4.2.14

$15625$ has factors of

$5$ and

$3125$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 5 \cdot 3125$

Step 4.2.15

$3125$ has factors of

$5$ and

$625$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdot 625$

Step 4.2.16

$625$ has factors of

$5$ and

$125$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdot 5 \cdot 125$

Step 4.2.17

$125$ has factors of

$5$ and

$25$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 25$

Step 4.2.18

$25$ has factors of

$5$ and

$5$ .

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

Step 4.3

The number

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

Not prime

Step 4.4

Since

$2$ has no factors besides

$1$ and

$2$ .

$2$ is a prime number

Step 4.5

The LCM of

$432000000, 1, 2$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

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

Step 4.6

Multiply

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

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Step 4.6.1

Multiply

$2$ by

$2$ .

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

Step 4.6.2

Multiply

$4$ by

$2$ .

$8 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5$

Step 4.6.3

Multiply

$8$ by

$2$ .

$16 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5$

Step 4.6.4

Multiply

$16$ by

$2$ .

$32 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5$

Step 4.6.5

Multiply

$32$ by

$2$ .

$64 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5$

Step 4.6.6

Multiply

$64$ by

$2$ .

$128 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5$

Step 4.6.7

Multiply

$128$ by

$2$ .

$256 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5$

Step 4.6.8

Multiply

$256$ by

$2$ .

$512 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5$

Step 4.6.9

Multiply

$512$ by

$2$ .

$1024 \cdot 3 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5$

Step 4.6.10

Multiply

$1024$ by

$3$ .

$3072 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5$

Step 4.6.11

Multiply

$3072$ by

$3$ .

$9216 \cdot 3 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5$

Step 4.6.12

Multiply

$9216$ by

$3$ .

$27648 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5$

Step 4.6.13

Multiply

$27648$ by

$5$ .

$138240 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5$

Step 4.6.14

Multiply

$138240$ by

$5$ .

$691200 \cdot 5 \cdot 5 \cdot 5 \cdot 5$

Step 4.6.15

Multiply

$691200$ by

$5$ .

$3456000 \cdot 5 \cdot 5 \cdot 5$

Step 4.6.16

Multiply

$3456000$ by

$5$ .

$17280000 \cdot 5 \cdot 5$

Step 4.6.17

Multiply

$17280000$ by

$5$ .

$86400000 \cdot 5$

Step 4.6.18

Multiply

$86400000$ by

$5$ .

$432000000$

Step 5

The answer can be found by taking the LCM of

$432000000, 1, 2$ and dividing by the LCM of

$1, 12000, 6000$ .

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Step 5.1

Divide the LCM of

$432000000, 1, 2$ by the LCM of

$1, 12000, 6000$ .

$\frac{432000000}{12000}$

Step 5.2

Divide

$432000000$ by

$12000$ .

$36000$