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Difficulty: MediumHCF and LCM

A boutique chocolatier is preparing special gift assortments. They have three large blocks of premium cocoa weighing 212\frac{21}{2} kg, 354\frac{35}{4} kg, and 498\frac{49}{8} kg. They must mold these blocks into the largest possible identical solid chocolate bars such that no cocoa is left over from any of the three blocks. What should be the exact weight of each chocolate bar?

  1. 78\frac{7}{8} kgAnswer
  2. B
    72\frac{7}{2} kg
  3. C
    7352\frac{735}{2} kg
  4. D
    77 kg

Answer

The exact weight of each chocolate bar should be 78\frac{7}{8} kg.
To find the maximum identical weight that can perfectly divide all three cocoa blocks, we must calculate the Highest Common Factor (HCF) of the three fractions. The formula dictates finding the HCF of the numerators (21, 35, 49), which is 7, and dividing it by the LCM of the denominators (2, 4, 8), which is 8. This results in the correct weight of 7/8 kg.

Step-by-Step Solution

1
Identify the mathematical operation required.
The problem asks for the 'largest possible identical' divisions of the blocks with no remainders, meaning we must find the Highest Common Factor (HCF) of the given fractional weights.
HCF provides the maximum uniform size that can perfectly divide a given set of quantities.
2
Recall the formula for finding the HCF of fractions.
HCF of fractions = (HCF of Numerators) / (LCM of Denominators).
This is the standard mathematical rule for calculating the greatest common divisor for fractional values.
3
Calculate the HCF of the numerators: 21, 35, and 49.
The highest common factor for 21, 35, and 49 is 7.
21 = 3 × 7; 35 = 5 × 7; 49 = 7 × 7. The greatest shared prime factor is 7.
4
Calculate the LCM of the denominators: 2, 4, and 8.
The lowest common multiple for 2, 4, and 8 is 8.
8 is a multiple of both 2 and 4, making it the smallest common denominator.
5
Apply the calculated values to the fraction formula.
78\frac{7}{8} kg.
Combining the calculated numerator HCF and denominator LCM yields the final maximum weight.

Key Concept

HCF of Fractions
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Topics

HCF and LCM

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