Barbara has a roll of material that measures $14 \frac{1}{2}$ feet long. She cuts the material into pieces, each of which measures $\frac{3}{4}$ foot. How many of these pieces of material does she have?
A. 16 pieces
B. 17 pieces
C. 18 pieces
D. 19 pieces
Answer (1)
Convert the mixed number 14 2 1 to an improper fraction: 2 29 .
Divide the total length by the length of each piece: 2 29 ÷ 4 3 = 2 29 × 3 4 .
Simplify the resulting fraction: 6 116 = 3 58 .
Convert the improper fraction to a mixed number and take the whole number part: 19 3 1 , so Barbara has 19 pieces.
Explanation
Understanding the Problem Barbara has a roll of material that measures 14 2 1 feet long. She cuts the material into pieces, each of which measures 4 3 foot. We need to find out how many pieces of material she has.
Converting to Improper Fraction First, convert the mixed number 14 2 1 to an improper fraction. To do this, multiply the whole number (14) by the denominator (2) and add the numerator (1). Place the result over the original denominator.
Calculating Improper Fraction 14 2 1 = 2 ( 14 × 2 ) + 1 = 2 28 + 1 = 2 29
Dividing Total Length by Piece Length Now, we need to divide the total length of the material ( 2 29 feet) by the length of each piece ( 4 3 foot) to find the number of pieces.
Multiplying by the Reciprocal To divide fractions, we multiply by the reciprocal of the divisor. The reciprocal of 4 3 is 3 4 . So, we have:
Performing the Multiplication 2 29 ÷ 4 3 = 2 29 × 3 4
Calculating the Result Multiply the numerators and the denominators:
Simplifying the Fraction 2 × 3 29 × 4 = 6 116
Reducing the Fraction Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
Simplified Fraction 6 116 = 6 ÷ 2 116 ÷ 2 = 3 58
Converting to Mixed Number Now, convert the improper fraction 3 58 to a mixed number to determine how many whole pieces Barbara has. Divide 58 by 3:
Finding the Whole Number 58 ÷ 3 = 19 with a remainder of 1
Determining the Number of Pieces So, 3 58 = 19 3 1 . Since Barbara can only have whole pieces, she has 19 pieces of material.
Final Answer Therefore, Barbara has 19 pieces of material, each measuring 4 3 foot.
Examples
Understanding how to divide materials into equal parts is useful in many real-life situations. For example, if you are a baker and have a certain amount of dough and need to make equally sized cookies, you would use division to determine how many cookies you can make. Similarly, if you are a construction worker and need to cut a beam into equal lengths for a project, you would use division to find the number of pieces you can obtain. This concept is also applicable in scenarios such as dividing a pizza among friends or splitting a bill evenly at a restaurant. In essence, division helps ensure fairness and efficiency in distributing resources or materials.
