180 oranges are distributed among 70 boys and girls such that each boy gets 2 and each girl gets 3 oranges. The number of boys are

## Concept & Logic This is a classic linear equations problem that can be solved using a system of two variables or by applying the method of alligation to find the ratio of groups. Let the number of boys and girls represent the two components of the total population. $$ x + y = Total $$ $$ 2x + 3y = Total\ Oranges $$ ## Step-by-Step Solution * **Given:** Total children = 70. Total oranges = 180. Each boy gets 2, each girl gets 3. * **Deduction:** Let the number of boys be $x$ and the number of girls be $y$. * **Calculation:** We can form two equations: 1) $x + y = 70$ (Equation for total children) 2) $2x + 3y = 180$ (Equation for total oranges) * **Calculation:** From equation 1, we can express $y$ as $70 - x$. * **Substitution:** Substitute $y$ into equation 2: $2x + 3(70 - x) = 180$ $2x + 210 - 3x = 180$ $-x = 180 - 210$ $-x = -30$ $x = 30$ * **Conclusion:** The number of boys is 30. ## Exam Strategy & Shortcut Use the "Assume All" shortcut. Assume ALL 70 children are girls. Then they would have received $70 \times 3 = 210$ oranges. However, we only distributed 180 oranges. The difference is $210 - 180 = 30$ oranges. This difference exists because boys receive 1 orange less than girls ($3 - 2 = 1$). Therefore, the number of boys is exactly the difference divided by the unit difference: $\frac{30}{1} = 30$. ## Common Pitfall Students often mix up the variables, accidentally solving for $y$ (girls = 40) and selecting option (c) in a rush. Always double-check what the question is asking for—in this case, the number of boys. ## Final Answer Therefore, the correct answer is **30**.