What is (1/3)+(1/6)+(1/9)+(1/12)+(1/15) ?

The solution to (1/3)+(1/6)+(1/9)+(1/12)+(1/15) is 137/180

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Popular Pre Algebra >

(1/3)+(1/6)+(1/9)+(1/12)+(1/15)

Solution

(1/3) + (1/6) + (1/9) + (1/12) + (1/15)

Solution

\frac{137}{180}

Decimal

0.76111 \dots

Solution steps

(1/3) + (1/6) + (1/9) + (1/12) + (1/15)

Follow the PEMDAS order of operations

Calculate within parentheses

(1/3) : \frac{1}{3}

1/3

1/3 = \frac{1}{3}

= \frac{1}{3}

= \frac{1}{3} + (1/6) + (1/9) + (1/12) + (1/15)

Calculate within parentheses

(1/6) : \frac{1}{6}

1/6

1/6 = \frac{1}{6}

= \frac{1}{6}

= \frac{1}{3} + \frac{1}{6} + (1/9) + (1/12) + (1/15)

Calculate within parentheses

(1/9) : \frac{1}{9}

1/9

1/9 = \frac{1}{9}

= \frac{1}{9}

= \frac{1}{3} + \frac{1}{6} + \frac{1}{9} + (1/12) + (1/15)

Calculate within parentheses

(1/12) : \frac{1}{12}

1/12

1/12 = \frac{1}{12}

= \frac{1}{12}

= \frac{1}{3} + \frac{1}{6} + \frac{1}{9} + \frac{1}{12} + (1/15)

Calculate within parentheses

(1/15) : \frac{1}{15}

1/15

1/15 = \frac{1}{15}

= \frac{1}{15}

= \frac{1}{3} + \frac{1}{6} + \frac{1}{9} + \frac{1}{12} + \frac{1}{15}

Add and subtract (left to right)

\frac{1}{3} + \frac{1}{6} + \frac{1}{9} + \frac{1}{12} + \frac{1}{15} : \frac{137}{180}

\frac{1}{3} + \frac{1}{6} + \frac{1}{9} + \frac{1}{12} + \frac{1}{15}

\frac{1}{3} + \frac{1}{6} = \frac{1}{2}

\frac{1}{3} + \frac{1}{6}

Least Common Multiplier of

3, 6 : 6

3, 6

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

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

3

6

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

6

For

\frac{1}{3} :

multiply the denominator and numerator by

2

\frac{1}{3} = \frac{1 \cdot 2}{3 \cdot 2} = \frac{2}{6}

= \frac{2}{6} + \frac{1}{6}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 + 1}{6}

Add the numbers:

2 + 1 = 3

= \frac{3}{6}

Cancel the common factor:

3

= \frac{1}{2}

= \frac{1}{2} + \frac{1}{9} + \frac{1}{12} + \frac{1}{15}

\frac{1}{2} + \frac{1}{9} = \frac{11}{18}

\frac{1}{2} + \frac{1}{9}

Least Common Multiplier of

2, 9 : 18

2, 9

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

9 : 3 \cdot 3

9

9

divides by

39 = 3 \cdot 3

= 3 \cdot 3

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

2

9

= 2 \cdot 3 \cdot 3

Multiply the numbers:

2 \cdot 3 \cdot 3 = 18

= 18

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

18

For

\frac{1}{2} :

multiply the denominator and numerator by

9

\frac{1}{2} = \frac{1 \cdot 9}{2 \cdot 9} = \frac{9}{18}

For

\frac{1}{9} :

multiply the denominator and numerator by

2

\frac{1}{9} = \frac{1 \cdot 2}{9 \cdot 2} = \frac{2}{18}

= \frac{9}{18} + \frac{2}{18}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{9 + 2}{18}

Add the numbers:

9 + 2 = 11

= \frac{11}{18}

= \frac{11}{18} + \frac{1}{12} + \frac{1}{15}

\frac{11}{18} + \frac{1}{12} = \frac{25}{36}

\frac{11}{18} + \frac{1}{12}

Least Common Multiplier of

18, 12 : 36

18, 12

Least Common Multiplier (LCM)

Prime factorization of

18 : 2 \cdot 3 \cdot 3

18

18

divides by

218 = 9 \cdot 2

= 2 \cdot 9

9

divides by

39 = 3 \cdot 3

= 2 \cdot 3 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3 \cdot 3

Prime factorization of

12 : 2 \cdot 2 \cdot 3

12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3

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

18

12

= 2 \cdot 2 \cdot 3 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 \cdot 3 = 36

= 36

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

36

For

\frac{11}{18} :

multiply the denominator and numerator by

2

\frac{11}{18} = \frac{11 \cdot 2}{18 \cdot 2} = \frac{22}{36}

For

\frac{1}{12} :

multiply the denominator and numerator by

3

\frac{1}{12} = \frac{1 \cdot 3}{12 \cdot 3} = \frac{3}{36}

= \frac{22}{36} + \frac{3}{36}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{22 + 3}{36}

Add the numbers:

22 + 3 = 25

= \frac{25}{36}

= \frac{25}{36} + \frac{1}{15}

\frac{25}{36} + \frac{1}{15} = \frac{137}{180}

\frac{25}{36} + \frac{1}{15}

Least Common Multiplier of

36, 15 : 180

36, 15

Least Common Multiplier (LCM)

Prime factorization of

36 : 2 \cdot 2 \cdot 3 \cdot 3

36

36

divides by

236 = 18 \cdot 2

= 2 \cdot 18

18

divides by

218 = 9 \cdot 2

= 2 \cdot 2 \cdot 9

9

divides by

39 = 3 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3 \cdot 3

Prime factorization of

15 : 3 \cdot 5

15

15

divides by

315 = 5 \cdot 3

= 3 \cdot 5

3, 5

are all prime numbers, therefore no further factorization is possible

= 3 \cdot 5

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

36

15

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

Multiply the numbers:

2 \cdot 2 \cdot 3 \cdot 3 \cdot 5 = 180

= 180

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

180

For

\frac{25}{36} :

multiply the denominator and numerator by

5

\frac{25}{36} = \frac{25 \cdot 5}{36 \cdot 5} = \frac{125}{180}

For

\frac{1}{15} :

multiply the denominator and numerator by

12

\frac{1}{15} = \frac{1 \cdot 12}{15 \cdot 12} = \frac{12}{180}

= \frac{125}{180} + \frac{12}{180}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{125 + 12}{180}

Add the numbers:

125 + 12 = 137

= \frac{137}{180}

= \frac{137}{180}

= \frac{137}{180}

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Frequently Asked Questions (FAQ)

What is (1/3)+(1/6)+(1/9)+(1/12)+(1/15) ?
The solution to (1/3)+(1/6)+(1/9)+(1/12)+(1/15) is 137/180

(1/3)+(1/6)+(1/9)+(1/12)+(1/15)

Solution

Solution

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Frequently Asked Questions (FAQ)

What is (1/3)+(1/6)+(1/9)+(1/12)+(1/15) ?

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