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Difficulty: MediumFractions and Decimals

An agricultural research center is preparing experimental test plots. The facility has exactly 2582 \frac{5}{8} kg, 211122 \frac{11}{12} kg, and 31163 \frac{1}{16} kg of three different premium organic soil additives. The lead researcher wants to distribute these additives into smaller, identical sample packets such that no material is left over from any of the three original quantities. To ensure efficiency, the packets must be as large as possible. What should be the exact weight capacity of each sample packet?

  1. A
    4735\frac{4}{735} kg
  2. B
    1341 \frac{3}{4} kg
  3. 748\frac{7}{48} kgAnswer
  4. D
    18334183 \frac{3}{4} kg

Answer

The correct packet capacity is 748\frac{7}{48} kg.
To find the largest possible identical packet size that leaves no remainder, we must calculate the Highest Common Factor (HCF) of the three fractional quantities. After converting the mixed numbers to improper fractions (218\frac{21}{8}, 3512\frac{35}{12}, 4916\frac{49}{16}), we apply the required formula: HCF of fractions = (HCF of numerators) / (LCM of denominators). The HCF of the numerators (21,35,49)(21, 35, 49) is 77, and the LCM of the denominators (8,12,16)(8, 12, 16) is 4848. Therefore, the required capacity is 748\frac{7}{48} kg.

Step-by-Step Solution

1
Convert the mixed numbers representing the quantities of additives into improper fractions.
258=2182 \frac{5}{8} = \frac{21}{8}, 21112=35122 \frac{11}{12} = \frac{35}{12}, and 3116=49163 \frac{1}{16} = \frac{49}{16}.
Calculations involving the HCF and LCM of fractions require the numbers to be in improper fraction format.
2
Identify the mathematical operation required to solve the problem.
Calculate the Highest Common Factor (HCF) of the three fractions.
The problem asks for the largest possible identical packet size that can divide all quantities without a remainder, which maps to finding the HCF.
3
Apply the standard formula for the HCF of fractions.
The formula is: HCF of fractions = (HCF of numerators) / (LCM of denominators).
This formula allows the determination of the greatest common divisor for fractional quantities.
4
Calculate the HCF of the numerators and the LCM of the denominators.
The HCF of the numerators (21,35,49)(21, 35, 49) is 77. The LCM of the denominators (8,12,16)(8, 12, 16) is 4848.
Finding these values provides the numerator and denominator for the final fractional answer.
5
Combine the calculated values to form the final fraction.
The maximum identical capacity is 748\frac{7}{48} kg.
Dividing the numerator HCF by the denominator LCM yields the exact required capacity.

Key Concept

Calculating the Highest Common Factor (HCF) of fractions using the formula: HCF of numerators / LCM of denominators.
Estimated Time:1m 30s
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Topics

Fractions and Decimals

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