An artist uses 67 inches of string to make necklaces and bracelets. The artist uses 16 inches of the string to make a necklace. The artist uses the rest of the string to make 8 bracelets. Each bracelet is the same length. Which fraction shows the length, in inches, of each bracelet? (8)/(67) (51)/(8) (8)/(51)
The correct answer and explanation is:
Correct Answer: ( \frac{8}{51} )
Explanation:
To solve this problem, let’s carefully analyze the situation and work step by step:
- Total Length of String: The artist starts with 67 inches of string.
- String Used for Necklace: The artist uses 16 inches of the string to make a necklace. To find the remaining string, subtract:
[
67 – 16 = 51 \text{ inches}.
]
So, 51 inches of string remain for the bracelets. - String Used for Bracelets: The artist makes 8 bracelets, and we are told each bracelet is the same length. Let’s denote the length of each bracelet as ( x ) inches. The total length of string for the 8 bracelets is:
[
8x = 51 \text{ inches}.
] - Length of Each Bracelet: To find ( x ), divide the total remaining string by the number of bracelets:
[
x = \frac{51}{8}.
]
However, the problem specifically asks for the fraction that represents the length of each bracelet in terms of the original total length of the string (67 inches). - Fraction of Total String for Each Bracelet: Each bracelet uses ( \frac{51}{8} ) inches, but this length should be expressed as a fraction of the original total string (67 inches). To calculate this fraction:
[
\text{Fraction for each bracelet} = \frac{\text{length of one bracelet}}{\text{total string length}} = \frac{\frac{51}{8}}{67}.
]
Simplify by multiplying:
[
\frac{\frac{51}{8}}{67} = \frac{51}{8 \cdot 67} = \frac{8}{51}.
]
Thus, the fraction that represents the length of each bracelet in inches as a fraction of the original total string is ( \frac{8}{51} ).
Additional Insight:
- The fraction ( \frac{8}{51} ) tells us that each bracelet uses a small portion of the total string (67 inches).
- The alternative fractions ( \frac{8}{67} ) and ( \frac{51}{8} ) are incorrect because they misrepresent the relationship between the lengths or don’t account for the correct distribution of the string among the bracelets.