The ratio of two numbers is 3 : 4 and their LCM is 120. The sum of numbers is

## Concept & Formula When the ratio of two numbers is given as $a:b$, we can express the actual numbers as $ax$ and $bx$, where $x$ is their Highest Common Factor (HCF). The LCM of these two numbers is simply the product of the ratio components and the HCF: $\text{LCM} = a \times b \times x$. ## Step-by-Step Solution * **Given:** Ratio of numbers = $3:4$. LCM = 120. * **Step 1:** Let the common multiplier (HCF) be $x$. The numbers are $3x$ and $4x$. * **Step 2:** Express the LCM in terms of $x$. Since 3 and 4 are co-prime, their LCM is $3 \times 4 = 12$. The LCM of $3x$ and $4x$ is $12x$. * **Step 3:** Equate to the given LCM and solve for $x$. $$12x = 120$$ $$x = 10$$ * **Step 4:** Find the individual numbers and their sum. * First number = $3(10) = 30$ * Second number = $4(10) = 40$ * $\text{Sum} = 30 + 40 = 70$ ## Exam Strategy & Shortcut If the numbers are in the ratio $3:4$, their sum is a multiple of $(3 + 4) = 7$. Look at the options: (a) 70 (Multiple of 7) (b) 140 (Multiple of 7) (c) 35 (Multiple of 7) (d) 105 (Multiple of 7) Since all options are multiples of 7, elimination doesn't immediately narrow it down to one, but you can quickly calculate the multiplier: $x = \frac{120}{3 \times 4} = 10$. Then directly multiply by the sum of ratios: $(3+4) \times 10 = 70$. This takes 5 seconds! ## Common Pitfall A common mistake is assuming the LCM is just the product of the numbers. Students might set up $3x \times 4x = 120$, leading to $12x^2 = 120$, which incorrectly yields $x = \sqrt{10}$. Remember that $\text{LCM} \times \text{HCF} = \text{Product of numbers}$. ## Final Answer Therefore, the correct answer is 70.