A rope that is $[tex]253 / 4[/tex]$ feet long is to be cut into pieces that are $[tex]1 / 2[/tex]$ feet each. What is the maximum number of pieces the rope can be cut into?
Answer (2)
The rope measuring 4 253 feet long can be cut into a maximum of 51 pieces, with each piece being 2 1 feet long. We determine this by converting the mixed number to an improper fraction, dividing by the piece length, and taking the floor of the result. Therefore, the final answer is 51 .
;
Convert the mixed fraction to an improper fraction: 25 4 3 = 4 103 .
Divide the total length by the length of each piece: 4 103 ÷ 2 1 = 2 103 = 51.5 .
Take the floor of the result to find the maximum number of whole pieces.
The maximum number of pieces is 51 .
Explanation
Understanding the Problem We are given a rope of length 253/4 feet that needs to be cut into pieces of length 1/2 feet each. We need to find the maximum number of pieces that can be cut.
Converting to Improper Fraction First, let's convert the mixed fraction 253/4 to an improper fraction. To do this, we multiply the whole number 25 by the denominator 4 and add the numerator 3. Then we put the result over the denominator 4. So, we have 25 4 3 = 4 25 × 4 + 3 = 4 100 + 3 = 4 103 Thus, the length of the rope is 4 103 feet.
Dividing the Lengths Next, we need to divide the total length of the rope by the length of each piece to find the number of pieces. So, we divide 4 103 by 2 1 :
4 103 ÷ 2 1 = 4 103 × 1 2 = 4 × 1 103 × 2 = 4 206 = 2 103 = 51.5 So, we can cut the rope into 51.5 pieces.
Finding the Maximum Number of Pieces Since we can only have whole pieces, we take the floor of the result, which is the largest integer less than or equal to 51.5. Therefore, the maximum number of pieces is 51.
Final Answer Therefore, the maximum number of pieces the rope can be cut into is 51.
Examples
Imagine you are a tailor and you have a long piece of fabric. You need to cut it into smaller pieces to make scarves. This problem helps you determine the maximum number of scarves you can make from the fabric, given the length of the fabric and the length of each scarf. Understanding how to divide fractions and find the maximum number of whole pieces is essential for efficient use of materials in various real-life scenarios, such as cooking, construction, and crafting.
