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$Find the Least Common Denominator (LCD) of \frac{11}{12},\frac{13}{16}1211,1613. In other words, find the Least Common Multiple (LCM) of 12,1612,16.</p><p></p><p>Method 1: By Listing Multiples</p><p></p><p>1</p><p></p><p> </p><p></p><p>List the multiples of each number.</p><p></p><p>Multiples of 12</p><p></p><p> : 12, 24, 36, 48, ...</p><p></p><p>Multiples of 16</p><p></p><p> : 16, 32, 48, ...</p><p></p><p>2</p><p></p><p> </p><p></p><p>Find the smallest number that is shared by all rows above. This is the LCM.</p><p></p><p>LCM = 4848</p><p></p><p></p><p>Method 2: By Prime Factors</p><p></p><p>1</p><p></p><p> </p><p></p><p>List the prime factors of each number.</p><p></p><p>Prime Factors of 12</p><p></p><p> : 2, 2, 3</p><p></p><p>Prime Factors of 16</p><p></p><p> : 2, 2, 2, 2</p><p></p><p>2</p><p></p><p> </p><p></p><p>Find the union of these primes.</p><p></p><p>2,2,2,2,32,2,2,2,3</p><p></p><p></p><p>3</p><p></p><p> </p><p></p><p>Multiply these numbers: 2\times 2\times 2\times 2\times 3=482×2×2×2×3=48. This is the LCM.</p><p></p><p>LCM = 4848</p><p></p><p></p><p>LCD = 4848</p><p></p><p></p><p>2</p><p></p><p> </p><p></p><p>Make the denominators the same as the LCD.</p><p></p><p>1+\frac{11\times 4}{12\times 4}+\frac{13\times 3}{16\times 3}1+12×411×4+16×313×3</p><p></p><p></p><p>3</p><p></p><p> </p><p></p><p>Simplify. Denominators are now the same.</p><p></p><p>1+\frac{44}{48}+\frac{39}{48}1+4844+4839</p><p></p><p></p><p>4</p><p></p><p> </p><p></p><p>Join the denominators.</p><p></p><p>1+\frac{44+39}{48}1+4844+39</p><p></p><p></p><p>5</p><p></p><p> </p><p></p><p>Simplify.</p><p></p><p>1+\frac{83}{48}1+4883</p><p></p><p></p><p>6</p><p></p><p> </p><p></p><p>Convert \frac{83}{48}4883 to mixed fraction.</p><p></p><p>1+1\frac{35}{48}1+14835</p><p></p><p></p><p>7</p><p></p><p> </p><p></p><p>Simplify.</p><p></p><p>2\frac{35}{48}24835</p><p></p><p></p><p>$

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