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Difficulty: HardFraction Operations

A carpenter is cutting a wooden board that is initially 121212 \frac{1}{2} feet long. First, he cuts off a piece that is 3343 \frac{3}{4} feet long and sets it aside. Then, he divides the remaining board into 55 equal sections. What is the length of each of these 55 sections?

  1. A
    113411 \frac{3}{4} feet
  2. B
    11 foot
  3. 1341 \frac{3}{4} feetAnswer
  4. D
    21202 \frac{1}{20} feet

Answer

Each of the 55 sections has a length of 1341 \frac{3}{4} feet.
The correct sequence of operations is to first determine the remaining length by subtracting 3343 \frac{3}{4} from 121212 \frac{1}{2}, yielding 8348 \frac{3}{4} feet. Then, dividing this remaining length by 55 results in 1341 \frac{3}{4} feet per section.

Step-by-Step Solution

1
Calculate the length of the remaining board by subtracting the cut piece from the total length.
1212−33412 \frac{1}{2} - 3 \frac{3}{4}
The carpenter first removes a piece, leaving a smaller board to be divided.
2
Find a common denominator to perform the subtraction.
1224−33412 \frac{2}{4} - 3 \frac{3}{4}
Fractions must have the same denominator to be subtracted.
3
Borrow from the whole number to subtract the fractions.
1164−334=83411 \frac{6}{4} - 3 \frac{3}{4} = 8 \frac{3}{4} feet
Since 24\frac{2}{4} is less than 34\frac{3}{4}, one whole unit must be converted into 44\frac{4}{4}.
4
Divide the remaining length by 5 to find the length of each section.
834÷58 \frac{3}{4} \div 5
The remaining board is split into 5 equal parts.
5
Convert the mixed number to an improper fraction and multiply by the reciprocal of 5.
354×15=74=134\frac{35}{4} \times \frac{1}{5} = \frac{7}{4} = 1 \frac{3}{4} feet
Division by a whole number is equivalent to multiplying by its reciprocal.

Key Concept

Solving multi-step problems involving subtraction and division of mixed numbers with unlike denominators.
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Topics

Fraction Operations

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