The ratio of two numbers is $3 : 5$ and their least common multiple is 42 more than their highest common factor. What is the value of the larger of the two numbers?

Explanation

The ratio of two numbers is $3 : 5$. Let the numbers be (a) = $3x$ (b) = $5x$ the highest common factor be (HCF) = $x$ Least common multiple (LCM) = $x + 42$ Given that the least common multiple is 42 more than the greatest common divisor. $a \times b = HCF \times LCM$ $3x \times 5x = x \times (42 + x)$ $15x^2 = x \times (42 + x)$ $15x = 42 + x$ $x = 3$ Larger number (b) = $5x = 5 \times 3 = 15$ Smaller number (a) = $3x = 3 \times 3 = 9$