What is the shortest strip of fiber from which five pieces of the following lengths can be cut: 9 1/2 inches, 10 11/16 inches, 6 9/16 inches, 7 1/2 inches, and 1 6/8 inches? no allowance is made for waste.
To find the shortest strip of fiber from which five pieces can be cut without any waste, we need to find the common divisor of the lengths provided.
First, let's convert all the lengths to their equivalent improper fractions:
9 1/2 inches = 19/2 inches
10 11/16 inches = 221/16 inches
6 9/16 inches = 135/16 inches
7 1/2 inches = 15/2 inches
1 6/8 inches = 14/8 inches
Now, let's find the common denominator of these fractions, which is 16.
Multiplying the numerator and denominator of each fraction to have a common denominator, we get:
19/2 inches = 19/2 * 8/8 = 152/16 inches
221/16 inches remains unchanged
135/16 inches remains unchanged
15/2 inches = 15/2 * 8/8 = 120/16 inches
14/8 inches = 14/8 * 2/2 = 28/16 inches
Now, we can find the greatest common divisor (GCD) of these fractions:
GCD(152, 221, 135, 120, 28) = 1
Therefore, the shortest strip of fiber from which five pieces of the given lengths can be cut with no waste is 1 inch.
To find the shortest strip of fiber from which five pieces of different lengths can be cut, we need to determine the common factor or multiple of all the given lengths. Here's how we can solve this:
Step 1: Convert all the lengths to their equivalent improper fractions:
- 9 1/2 inches = (9 * 2 + 1) / 2 = 19/2
- 10 11/16 inches = (10 * 16 + 11) / 16 = 171/16
- 6 9/16 inches = (6 * 16 + 9) / 16 = 105/16
- 7 1/2 inches = (7 * 2 + 1) / 2 = 15/2
- 1 6/8 inches = (1 * 8 + 6) / 8 = 14/8 = 7/4
Step 2: Find the least common multiple (LCM) of the denominators of these fractions:
The denominators are: 2, 16, 16, 2, 4
The prime factorization of these denominators can be represented as:
2 = 2^1
16 = 2^4
16 = 2^4
2 = 2^1
4 = 2^2
To find the LCM, we take the highest power of each prime factor that appears in the list:
2^4 = 16
Step 3: Multiply the LCM found in Step 2 by the numerators of each fraction:
19/2 * 16 = 19 * 8 = 152
171/16 * 1 = 171
105/16 * 16 = 105
15/2 * 16 = 15 * 8 = 120
7/4 * 16 = 7 * 4 = 28
Step 4: Add up the results obtained in Step 3:
152 + 171 + 105 + 120 + 28 = 576
Therefore, the shortest strip of fiber from which all five pieces can be cut is 576 inches long.