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Zorluk: OrtaHCF and LCM

A gourmet coffee roaster has three batches of rare coffee beans weighing 152\frac{15}{2} kg, 254\frac{25}{4} kg, and 358\frac{35}{8} kg respectively. The roaster wants to package all the beans into smaller, equal-sized bags such that each bag contains the maximum possible weight of coffee beans without mixing the batches, and no beans are left over. What should be the weight of each bag?

  1. A
    5252\frac{525}{2} kg
  2. B
    5258\frac{525}{8} kg
  3. 58\frac{5}{8} kgCevap
  4. D
    52\frac{5}{2} kg

Cevap

The weight of each bag should be 58\frac{5}{8} kg.
The problem asks for the maximum possible equal capacity for the bags, which requires calculating the Highest Common Factor (HCF) of the given fractional weights. The HCF of a set of fractions is found by calculating the HCF of their numerators (1515, 2525, 3535) and dividing it by the LCM of their denominators (22, 44, 88). The HCF of the numerators is 55, and the LCM of the denominators is 88. Therefore, the correct weight is 58\frac{5}{8} kg.

Adım Adım Çözüm

1
Determine the mathematical operation required.
We need to find the Highest Common Factor (HCF) of the given fractional weights.
The bags must be of equal size, hold the maximum possible weight, and leave no remainder, which matches the definition of HCF.
2
Recall the formula for finding the HCF of fractions.
The formula is: (HCF of numerators) / (LCM of denominators).
This formula ensures the resulting fraction correctly divides all the given fractions into integers.
3
Calculate the HCF of the numerators.
The numerators are 1515, 2525, and 3535. Their HCF is 55.
55 is the largest integer that divides 1515, 2525, and 3535 without leaving a remainder.
4
Calculate the LCM of the denominators.
The denominators are 22, 44, and 88. Their LCM is 88.
88 is the smallest integer that is a multiple of 22, 44, and 88.
5
Combine the results to find the HCF of the fractions.
The final HCF is 58\frac{5}{8}.
Substituting the calculated HCF and LCM into the fraction formula yields 58\frac{5}{8}.

Anahtar Kavram

Finding the Highest Common Factor (HCF) of fractions.
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Konular

HCF and LCM

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