Solving the Steel Bar Cutting Problem

This problem asks us to find the decimal fraction of a steel bar that is left over after cutting multiple pieces of a specific length. We start with a total length of the steel bar and need to determine how many pieces can be cut and what length remains.

Step-by-Step Calculation for Leftover Steel Bar

Here’s how we can solve this steel bar cutting problem:

1. Calculate the maximum number of pieces: We need to find out how many pieces of 5.25 m length can be cut from the 50 m long bar. This is done by dividing the total length by the length of each piece. The number of pieces must be a whole number, as you cannot cut a fraction of a piece.

   Total length = \(50 \, \text{m}\)

   Length per piece = \(5.25 \, \text{m}\)

   Number of pieces = \(\frac{\text{Total length}}{\text{Length per piece}} = \frac{50}{5.25}\)

   Let's perform the division:

   \(\frac{50}{5.25} = \frac{50 \times 100}{5.25 \times 100} = \frac{5000}{525}\)

   Now, simplify the fraction:

   \(\frac{5000}{525} = \frac{1000 \times 5}{105 \times 5} = \frac{1000}{105}\)

   \(\frac{1000}{105} = \frac{200 \times 5}{21 \times 5} = \frac{200}{21}\)

   Dividing 200 by 21:

   \(200 \div 21 \approx 9.52\)

   Since only whole pieces can be cut, the workman can cut 9 pieces.

2. Calculate the total length used: Multiply the number of pieces cut by the length of each piece.

   Number of pieces = 9

   Length per piece = \(5.25 \, \text{m}\)

   Total length used = \(9 \times 5.25 \, \text{m}\)

   \(9 \times 5.25 = 9 \times (5 + 0.25) = (9 \times 5) + (9 \times 0.25) = 45 + 2.25 = 47.25 \, \text{m}\)

   The total length used is 47.25 m.

3. Calculate the length left: Subtract the total length used from the original total length of the bar.

   Original total length = \(50 \, \text{m}\)

   Total length used = \(47.25 \, \text{m}\)

   Length left = \(50 \, \text{m} - 47.25 \, \text{m} = 2.75 \, \text{m}\)

   The length of the bar left is 2.75 m.

4. Calculate the decimal fraction of the whole left: Divide the length left by the original total length of the bar.

   Length left = \(2.75 \, \text{m}\)

   Original total length = \(50 \, \text{m}\)

   Decimal fraction left = \(\frac{\text{Length left}}{\text{Original total length}} = \frac{2.75}{50}\)

   To convert this fraction to a decimal, we can multiply the numerator and denominator by 2 to make the denominator 100:

   \(\frac{2.75}{50} = \frac{2.75 \times 2}{50 \times 2} = \frac{5.5}{100} = 0.055\)

   The decimal fraction of the whole bar left is 0.055.

Conclusion on the Decimal Fraction

After cutting as many 5.25 m pieces as possible from the 50 m steel bar, the length remaining is 2.75 m. When expressed as a decimal fraction of the original length, this leftover piece is 0.055 of the whole bar.

Revision Table: Steel Bar Problem

Additional Information: Understanding Decimal Fractions

A decimal fraction is a fraction where the denominator is a power of ten (like 10, 100, 1000, etc.). However, the term is also commonly used to refer to any number written in decimal form, especially when it represents a part of a whole. In this problem, the decimal fraction represents the proportion of the original steel bar that remains after cutting. Calculating this involves division and understanding how to represent quantities as parts of a whole.

When dealing with real-world problems like steel bar cutting, it's important to consider practical constraints, such as only being able to cut whole pieces. This often means using integer division (finding the quotient) to determine the number of items that can be made or cut, before calculating the remainder or leftover quantity.