Examkin

Soru

Zorluk: OrtaFractions and Decimals

A botanical research institute is preparing identical standardized nutrient doses for a plant growth experiment. They have three concentrated liquid nutrient solutions measuring 143\frac{14}{3} liters, 356\frac{35}{6} liters, and 214\frac{21}{4} liters respectively. The researchers need to distribute these solutions into small testing vials such that every vial contains the exact same maximum possible volume of solution, without mixing the different types of nutrients, and leaving no solution behind in the original containers. What should be the volume capacity of each testing vial?

  1. 712\frac{7}{12} litersCevap
  2. B
    1210\frac{1}{210} liters
  3. C
    210210 liters
  4. D
    77 liters

Cevap

712\frac{7}{12} liters
To find the maximum identical volume that can evenly divide all three solutions without remainders, we must calculate the Highest Common Factor (HCF) of the fractions 143\frac{14}{3}, 356\frac{35}{6}, and 214\frac{21}{4}. Using the rule that the HCF of fractions equals the HCF of their numerators divided by the LCM of their denominators, we evaluate HCF(14, 35, 21) = 7 and LCM(3, 6, 4) = 12. This results in exactly 712\frac{7}{12} liters per vial.

Adım Adım Çözüm

1
Identify the mathematical operation required based on the scenario.
The problem asks for the maximum identical volume to evenly distribute three different quantities without remainders. This requires finding the Highest Common Factor (HCF) of the given fractional volumes.
HCF is used when dividing varying quantities into the largest possible equal segments.
2
Recall the formula for calculating the HCF of fractions.
HCF=HCF of numeratorsLCM of denominatorsHCF = \frac{\text{HCF of numerators}}{\text{LCM of denominators}}
This is the standard rule for finding the greatest common divisor of fractional values.
3
Calculate the HCF of the numerators (14,35,2114, 35, 21).
The prime factors are: 14=2×714 = 2 \times 7; 35=5×735 = 5 \times 7; 21=3×721 = 3 \times 7. The highest common factor is 77.
To form the numerator of our final fraction answer.
4
Calculate the LCM of the denominators (3,6,43, 6, 4).
The multiples of the denominators meet at 1212. (Since 1212 is divisible by 33, 66, and 44).
To form the denominator of our final fraction answer.
5
Substitute the calculated values back into the fraction HCF formula.
712\frac{7}{12} liters.
Provides the final vial capacity.

Anahtar Kavram

Highest Common Factor (HCF) of Fractions in Word Problems
Bu soruyu puanla

Konular

Fractions and Decimals

Bu konudan daha fazla soru