A wholesale merchant has three batches of specialty syrups measuring \frac{63}{5} liters, … — State PSC Exam

A wholesale merchant has three batches of specialty syrups measuring $\frac{63}{5}$ liters, $\frac{72}{25}$ liters, and $\frac{81}{10}$ liters in volume. To distribute them in uniform sample sizes, the merchant wants to repackage all the syrup into identical smaller bottles filled exactly to the brim, ensuring no syrup is left over from any of the original batches. What is the maximum possible capacity (in liters) of each sample bottle?

Cevap: 0.18 liters

Cevap

The maximum possible capacity of each sample bottle is 0.18 liters.

To find the maximum capacity of uniform bottles that can hold the syrup from any batch without leftover, we must find the Highest Common Factor (HCF) of the respective volumes. For fractions, the rule is

\text{HCF} = \frac{\text{HCF of numerators}}{\text{LCM of denominators}}

. The HCF of the numerators (

63, 72, 81

) is

9

, and the LCM of the denominators (

5, 25, 10

) is

50

. The resulting maximum capacity is

\frac{9}{50}

, which equals

0.18

liters.

Adım Adım Çözüm

Determine the mathematical operation required.

Calculate the HCF of

\frac{63}{5}

\frac{72}{25}

, and

\frac{81}{10}

Finding the largest equal container size that leaves no remainder requires computing the Highest Common Factor.

Apply the rule for finding the HCF of fractions.

\text{HCF of fractions} = \frac{\text{HCF of numerators}}{\text{LCM of denominators}}

This standard formula correctly identifies the greatest common divisor for a set of rational numbers.

Find the HCF of the numerators.

\text{HCF}(63, 72, 81) = 9

The largest integer that perfectly divides 63, 72, and 81 is 9.

Find the LCM of the denominators.

\text{LCM}(5, 25, 10) = 50

The smallest integer that is a multiple of 5, 25, and 10 is 50.

Compute the final fraction and convert it to a decimal.

\frac{9}{50} = 0.18

Dividing the HCF of the numerators by the LCM of the denominators yields the exact capacity in liters.

Anahtar Kavram

HCF and LCM of fractions

Soru

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