equivalent fractions for 5/6,5/7,and

8/9 using the least common denominator.

To find equivalent fractions for 5/6, 5/7, and 8/9 using the least common denominator, follow these steps:

Step 1: Find the least common denominator (LCD) of the fractions.
The LCD is the smallest multiple that the denominators have in common. In this case, the denominators are 6, 7, and 9. The multiples of these numbers are:
6: 6, 12, 18, 24, 30, ...
7: 7, 14, 21, 28, 35, ...
9: 9, 18, 27, 36, ...
The least common multiple of 6, 7, and 9 is 63.

Step 2: Rewrite each fraction using the LCD.
To rewrite each fraction using the LCD of 63, you need to find the equivalent fraction where the denominator is 63.

For 5/6: Multiply both the numerator and denominator by 63/6:
5/6 * (63/6) = (5*63)/(6*6) = 315/36

For 5/7: Multiply both the numerator and denominator by 63/7:
5/7 * (63/7) = (5*63)/(7*7) = 315/49

For 8/9: Multiply both the numerator and denominator by 63/9:
8/9 * (63/9) = (8*63)/(9*9) = 504/81

So, the equivalent fractions for 5/6, 5/7, and 8/9 using the least common denominator of 63 are:
5/6 = 315/36
5/7 = 315/49
8/9 = 504/81

To find equivalent fractions using the least common denominator, we need to find the smallest common multiple of the denominators of the given fractions. In this case, the denominators are 6, 7, and 9.

Step 1: Find the prime-factorization of each denominator.
For 6: 2 x 3
For 7: 7
For 9: 3 x 3

Step 2: Identify the highest power of each prime factor among the denominators.
For 2: 2
For 3: 3 x 3 = 9
For 7: 7

Step 3: Multiply the highest powers of the prime factors.
2 x 9 x 7 = 126

Step 4: Divide the least common denominator (126) by each original denominator and multiply the result by the numerator to get the equivalent fractions.

For 5/6:
Multiply 5 by (126 ÷ 6) = 5 x 21 = 105.
Equivalent fraction: 105/126

For 5/7:
Multiply 5 by (126 ÷ 7) = 5 x 18 = 90.
Equivalent fraction: 90/126

For 8/9:
Multiply 8 by (126 ÷ 9) = 8 x 14 = 112.
Equivalent fraction: 112/126

Therefore, the equivalent fractions for 5/6, 5/7, and 8/9 using the least common denominator of 6, 7, and 9 are 105/126, 90/126, and 112/126, respectively.

5/6 = 105/126, 5/7 = 90/126, 8/9 = 112/126.