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Difficulty: MediumHCF and LCM

An interior designer is planning to cover the lengths of three different hallways with square marble tiles. The lengths of the three hallways are 154\frac{15}{4} meters, 256\frac{25}{6} meters, and 358\frac{35}{8} meters. To avoid cutting any marble tiles, the designer wants to use the largest possible square tiles that can perfectly measure these lengths.

What is the maximum possible side length of each square tile?

  1. 524\frac{5}{24} metersAnswer
  2. B
    5252\frac{525}{2} meters
  3. C
    52\frac{5}{2} meters
  4. D
    245\frac{24}{5} meters

Answer

524\frac{5}{24} meters
The maximum possible side length that perfectly measures all three hallway lengths is found by calculating the Highest Common Factor (HCF) of the three fractions. According to the mathematical rule for fractions, this is the HCF of the numerators (15,25,3515, 25, 35) divided by the LCM of the denominators (4,6,84, 6, 8), which yields 524\frac{5}{24} meters.

Step-by-Step Solution

1
Identify the mathematical operation required.
Calculate the Highest Common Factor (HCF) of the three fractions to find the maximum possible side length.
The tiles must perfectly measure the lengths without being cut, meaning the tile length must be a common factor. 'Largest possible' indicates the Highest Common Factor.
2
Recall the formula for finding the HCF of fractions.
HCF of fractions = HCF of numeratorsLCM of denominators\frac{\text{HCF of numerators}}{\text{LCM of denominators}}
This is the standard rule for determining the highest common divisor among fractional values.
3
Calculate the HCF of the numerators.
Numerators are 1515, 2525, and 3535. Their HCF is 55.
The largest integer that perfectly divides 1515, 2525, and 3535 is 55.
4
Calculate the LCM of the denominators.
Denominators are 44, 66, and 88. Their LCM is 2424.
The smallest integer that is a multiple of 44, 66, and 88 is 2424.
5
Apply the results to the formula.
524\frac{5}{24} meters.
Substituting the computed values into the fraction formula provides the final answer.

Key Concept

HCF of fractions
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Topics

HCF and LCM

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