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Difficulty: Very hardFraction Addition and Subtraction in Context

A machinist has a steel rod of an unknown initial length. She cuts off a piece measuring 5585 \frac{5}{8} inches for a fitting and another piece measuring 3563 \frac{5}{6} inches for a bracket. She then welds on a spare segment that is 2342 \frac{3}{4} inches long. After these modifications, she trims exactly 712\frac{7}{12} of an inch off the end to square it up. If the final length of the rod is 141214 \frac{1}{2} inches, what was its original length?

  1. 21192421 \frac{19}{24} inchesAnswer
  2. B
    205820 \frac{5}{8} inches
  3. C
    75247 \frac{5}{24} inches
  4. D
    2772427 \frac{7}{24} inches

Answer

The original length of the steel rod was 21192421 \frac{19}{24} inches.
The correct answer is found by working backward from the final length. This requires reversing all operations: adding back the pieces that were cut and trimmed, and subtracting the piece that was welded. Using the least common denominator of 24, the calculation becomes 141224+51524+32024−21824+142414 \frac{12}{24} + 5 \frac{15}{24} + 3 \frac{20}{24} - 2 \frac{18}{24} + \frac{14}{24}, which accurately sums to 21192421 \frac{19}{24} inches.

Step-by-Step Solution

1
Set up an equation representing the physical changes to the rod from its initial length.
Initial−558−356+234−712=1412\text{Initial} - 5 \frac{5}{8} - 3 \frac{5}{6} + 2 \frac{3}{4} - \frac{7}{12} = 14 \frac{1}{2}
Cutting and trimming decrease the overall length (subtraction), while welding increases the length (addition).
2
Rearrange the equation to solve for the initial length by applying inverse operations.
Initial=1412+558+356−234+712\text{Initial} = 14 \frac{1}{2} + 5 \frac{5}{8} + 3 \frac{5}{6} - 2 \frac{3}{4} + \frac{7}{12}
To work backward from the final length to the initial length, we must reverse all operations (add what was removed, subtract what was added).
3
Determine the least common denominator (LCD) for the fractions 12\frac{1}{2}, 58\frac{5}{8}, 56\frac{5}{6}, 34\frac{3}{4}, and 712\frac{7}{12}.
The LCD of 2, 8, 6, 4, and 12 is 24.
Fractions must share a common denominator before they can be added or subtracted.
4
Convert each mixed number to an equivalent fraction with a denominator of 24.
141224+51524+32024−21824+142414 \frac{12}{24} + 5 \frac{15}{24} + 3 \frac{20}{24} - 2 \frac{18}{24} + \frac{14}{24}
Equivalent fractions allow for direct arithmetic on the numerators.
5
Perform the addition and subtraction for the whole numbers and the fractions.
Whole numbers: 14+5+3−2=2014 + 5 + 3 - 2 = 20. Fractions: 12+15+20−18+1424=4324=11924\frac{12 + 15 + 20 - 18 + 14}{24} = \frac{43}{24} = 1 \frac{19}{24}. Total length = 20+11924=21192420 + 1 \frac{19}{24} = 21 \frac{19}{24} inches.
Combining the whole number and fractional sums yields the fully simplified initial length of the rod.

Key Concept

Fraction Addition and Subtraction in Context
Estimated Time:4m 0s
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Fraction Addition and Subtraction in Context

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