\[\frac{1}{4}+\frac{5}{12}-\frac{1}{13}-\frac{7}{8}\]

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Find the Least Common Denominator (LCD) of

\frac{1}{4},\frac{5}{12},\frac{1}{13},\frac{7}{8}

. In other words, find the Least Common Multiple (LCM) of

4,12,13,8

Method 1: By Listing Multiples

List the multiples of each number.

Multiples of 4

: 4, 8, 12, ... , 304, 308, 312, ...

Multiples of 12

: 12, 24, 36, ... , 288, 300, 312, ...

Multiples of 13

: 13, 26, 39, ... , 286, 299, 312, ...

Multiples of 8

: 8, 16, 24, ... , 296, 304, 312, ...

Find the smallest number that is shared by all rows above. This is the LCM.

LCM =

312

Method 2: By Prime Factors

List the prime factors of each number.

Prime Factors of 4

: 2, 2

Prime Factors of 12

: 2, 2, 3

Prime Factors of 13

: 13

Prime Factors of 8

: 2, 2, 2

Find the union of these primes.

2,2,2,3,13

Multiply these numbers:

2\times 2\times 2\times 3\times 13=312

. This is the LCM.

LCM =

312

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LCD =

312

Make the denominators the same as the LCD.

\frac{1\times 78}{4\times 78}+\frac{5\times 26}{12\times 26}-\frac{1\times 24}{13\times 24}-\frac{7\times 39}{8\times 39}

Simplify. Denominators are now the same.

\frac{78}{312}+\frac{130}{312}-\frac{24}{312}-\frac{273}{312}

Join the denominators.

\frac{78+130-24-273}{312}

Simplify.

-\frac{89}{312}

Done

Decimal Form: -0.285256

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