Three quantities are given as 0.\overline{6}, \frac{12}{25}, and 1.4. What is the exact ra… — State PSC Exam

Three quantities are given as $0.\overline{6}$ , $\frac{12}{25}$ , and $1.4$ . What is the exact ratio of the Least Common Multiple (LCM) of these three quantities to their Highest Common Factor (HCF)?

6300Answer
B
$\frac{1}{6300}$
C
420
D
84

Answer

6300

Converting the given values yields the reduced fractions

\frac{2}{3}

\frac{12}{25}

, and

\frac{7}{5}

. Applying standard fraction formulas,

\text{LCM} = \frac{\text{LCM}(2,12,7)}{\text{HCF}(3,25,5)} = \frac{84}{1} = 84

and

\text{HCF} = \frac{\text{HCF}(2,12,7)}{\text{LCM}(3,25,5)} = \frac{1}{75}

. Taking the ratio

\frac{\text{LCM}}{\text{HCF}}

gives

84 \div \frac{1}{75} = 6300

Step-by-Step Solution

Convert all terms to irreducible fractions

0.\overline{6} = \frac{6}{9} = \frac{2}{3}

\frac{12}{25} = \frac{12}{25}

, and

1.4 = \frac{14}{10} = \frac{7}{5}

All numbers must be expressed as simplified fractions

\frac{a}{b}

in lowest terms before applying fraction HCF and LCM formulas.

Calculate the Least Common Multiple (LCM) of the fractions

\text{LCM}\left(\frac{2}{3}, \frac{12}{25}, \frac{7}{5}\right) = \frac{\text{LCM}(2, 12, 7)}{\text{HCF}(3, 25, 5)} = \frac{84}{1} = 84

The LCM of a set of fractions is given by

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

Calculate the Highest Common Factor (HCF) of the fractions

\text{HCF}\left(\frac{2}{3}, \frac{12}{25}, \frac{7}{5}\right) = \frac{\text{HCF}(2, 12, 7)}{\text{LCM}(3, 25, 5)} = \frac{1}{75}

The HCF of a set of fractions is given by

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

Find the ratio of the LCM to the HCF

\text{Ratio} = \frac{84}{\frac{1}{75}} = 84 \times 75 = 6300

Dividing the computed LCM by the computed HCF yields the required ratio.

Key Concept

HCF and LCM of Fractions and Recurring Decimals

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