Out of three numbers, the first is twice the second and is half of the third. If the average of the three numbers is 56, then difference of first and third numbers is

Concept & Logic Establish a common ratio between the numbers to find their total sum in terms of parts. Use the given average to find the actual value of one part, then calculate the required difference directly from the ratio parts. Step-by-Step Solution * Let the three numbers be $A$, $B$, and $C$. * Based on the conditions: $A = 2B \implies B = \frac{A}{2}$ $A = \frac{C}{2} \implies C = 2A$ * To avoid fractions, let's assume a base variable for $B$. * Let Second number $B = x$. * First number $A = 2x$. * Third number $C = 2A = 2(2x) = 4x$. * The sum of the numbers is $2x + x + 4x = 7x$. * The average of the three numbers is given as 56. $$\text{Average} = \frac{\text{Sum}}{3} = 56 \implies \text{Sum} = 56 \times 3 = 168$$ * Equate the sum expressions: $$7x = 168$$ $$x = 24$$ * The First number $A = 2(24) = 48$. * The Third number $C = 4(24) = 96$. * Difference between the third and first number = $96 - 48 = 48$. Exam Strategy & Shortcut Write the ratio directly: $A:B:C = 2:1:4$. The average of these parts is $\frac{2+1+4}{3} = \frac{7}{3}$ units. We know $\frac{7}{3}$ units = 56. Therefore, $1$ unit = $56 \times \frac{3}{7} = 24$. The question asks for the difference between the first and third numbers ($4 - 2 = 2$ units). Difference = $2 \times 24 = 48$. This ratio method skips solving for every individual number. Common Pitfall Misinterpreting the phrasing "and is half of the third". Students sometimes mistakenly think the *second* number is half of the third ($B = C/2$), instead of the *first* number ($A = C/2$). Always link the conjunction "and" back to the primary subject of the sentence. Final Answer Therefore, the correct answer is 48.