Three storage vessels in a chemical laboratory contain \frac{12}{35} litres, \frac{18}{25}… — State PSC Exam

Three storage vessels in a chemical laboratory contain $\frac{12}{35}$ litres, $\frac{18}{25}$ litres, and $\frac{24}{49}$ litres of different reagents. What is the maximum capacity (in litres) of a single measuring container that can be used to measure the exact quantity of reagent from any of the three vessels without leaving any remainder?

$\frac{6}{1225}$ litresAnswer
B
$\frac{72}{1225}$ litres
C
$\frac{1225}{6}$ litres
D
$72$ litres

Answer

The maximum capacity of the measuring container is

\frac{6}{1225}

litres.

To find the largest measuring unit that can measure fractional quantities exactly, we need the HCF of those fractions. The HCF of fractions is defined as

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

. Calculating

\text{HCF}(12, 18, 24) = 6

and

\text{LCM}(35, 25, 49) = 1225

yields

\frac{6}{1225}

litres.

Step-by-Step Solution

Identify the mathematical rule required for measuring fractional quantities exactly.

To find the maximum capacity of a container that measures given fractional quantities an exact number of times, compute the HCF of the fractions:

\text{HCF}\left(\frac{a}{b}, \frac{c}{d}, \frac{e}{f}\right) = \frac{\text{HCF}(a, c, e)}{\text{LCM}(b, d, f)}

The measuring container's capacity must be a common divisor of all three fractional volumes.

Calculate the HCF of the numerators

12

18

, and

24

\text{HCF}(12, 18, 24) = 6

The prime factorizations are

12 = 2^2 \times 3

18 = 2 \times 3^2

, and

24 = 2^3 \times 3

. The highest common power of prime factors shared by all three numbers is

2^1 \times 3^1 = 6

Calculate the LCM of the denominators

35

25

, and

49

\text{LCM}(35, 25, 49) = 1225

The prime factorizations are

35 = 5 \times 7

25 = 5^2

, and

49 = 7^2

. The LCM takes the highest power of each prime:

5^2 \times 7^2 = 25 \times 49 = 1225

Substitute the calculated HCF of numerators and LCM of denominators into the fraction HCF formula.

\text{HCF} = \frac{6}{1225}

litres.

Dividing the HCF of the numerators by the LCM of the denominators gives the exact maximum volume.

Key Concept

HCF of Fractions

Estimated Time:2m 0s

Question

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Step-by-Step Solution

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