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Zorluk: OrtaHCF and LCM

A botanist is preparing nutrient solutions. She has three different liquid nutrient extracts measuring 245\frac{24}{5} liters, 323\frac{32}{3} liters, and 407\frac{40}{7} liters. She wants to distribute them entirely into identical small sample vials of maximum possible capacity, such that no extract is left over and the extracts are not mixed. What should be the maximum capacity of each sample vial?

  1. 8105\frac{8}{105} litersCevap
  2. B
    1480\frac{1}{480} liters
  3. C
    480480 liters
  4. D
    88 liters

Cevap

The correct maximum capacity is 8105\frac{8}{105} liters.
To find the maximum identical capacity that can exactly measure out each of the given quantities without mixing, we must calculate the Highest Common Factor (HCF) of the given fractions. By applying the formula for the HCF of fractions, we divide the HCF of the numerators (8) by the Least Common Multiple (LCM) of the denominators (105), resulting in 8105\frac{8}{105} liters.

Adım Adım Çözüm

1
Identify the mathematical operation required.
We need to find the Highest Common Factor (HCF) of the three fractions to determine the maximum equal capacity.
The vials must have the maximum possible capacity and perfectly measure out all three volumes without mixing them or leaving any remainder.
2
Recall the formula for the HCF of fractions.
HCF of fractions = (HCF of numerators) / (LCM of denominators).
This is the standard algebraic rule for determining the greatest common divisor of fractional values.
3
Calculate the HCF of the numerators: 24, 32, and 40.
The prime factorizations are 24=23×324 = 2^3 \times 3, 32=2532 = 2^5, and 40=23×540 = 2^3 \times 5. The highest common factor is 23=82^3 = 8.
The HCF of the numerators forms the numerator of our final answer.
4
Calculate the LCM of the denominators: 5, 3, and 7.
Since 5, 3, and 7 are all prime numbers, their LCM is their product: 5×3×7=1055 \times 3 \times 7 = 105.
The LCM of the denominators forms the denominator of our final answer.
5
Construct the final fraction.
The capacity is 8105\frac{8}{105} liters.
Dividing the HCF of the numerators by the LCM of the denominators gives the final HCF of the original fractions.

Anahtar Kavram

HCF and LCM of fractions
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Konular

HCF and LCM

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