Find the Least Common Denominator (LCD) of \(\frac{2}{3},\frac{5}{7}\). In other words, find the Least Common Multiple (LCM) of \(3,7\).
Method 1: By Listing Multiples
List the multiples of each number.
Multiples of 3
: 3, 6, 9, 12, 15, 18, 21, ...
Multiples of 7
: 7, 14, 21, ...
Find the smallest number that is shared by all rows above. This is the LCM.
Method 2: By Prime Factors
List the prime factors of each number.
Prime Factors of 3
: 3
Prime Factors of 7
: 7
Find the union of these primes.
\[3,7\]
Multiply these numbers: \(3\times 7=21\). This is the LCM.
Make the denominators the same as the LCD.
\[-\frac{2\times 7}{3\times 7}+\frac{5\times 3}{7\times 3}\]
Simplify. Denominators are now the same.
\[-\frac{14}{21}+\frac{15}{21}\]
Join the denominators.
\[\frac{-14+15}{21}\]
Simplify.
\[\frac{1}{21}\]
Decimal Form: 0.047619
1/21
