Ratio and sum — Two numbers are in the ratio 3:4 and their sum is 420. Compute the greater number using the parts method.

Introduction / Context:
Ratio problems are efficiently solved by interpreting ratios as parts of a whole. Given both the ratio and the total sum, you can find the value of one part and then scale each term to obtain the actual numbers.

Given Data / Assumptions:

• Ratio of two numbers: 3:4.
• Sum of the numbers: 420.
• We seek the greater number corresponding to the 4-part share.

Concept / Approach:
If the numbers are 3k and 4k, then their sum is 7k. Solving for k from the sum and multiplying yields each number. This method generalizes to any ratio-sum combination and avoids trial and error.

Step-by-Step Solution:
Let numbers be 3k and 4k.Sum: 3k + 4k = 7k = 420.Solve for k: k = 420 / 7 = 60.Greater number: 4k = 4 * 60 = 240.

Verification / Alternative check:
Add the two computed values: 3*60 + 4*60 = 180 + 240 = 420, matching the given sum; the ratio 180:240 simplifies to 3:4, confirming correctness.

Why Other Options Are Wrong:

• 175/200/315/180: These either do not align with a 3:4 split of 420 or correspond to the smaller number, not the greater one.

Common Pitfalls:
Dividing 420 by 3 or 4 prematurely; forgetting to divide by the total number of parts (7); mistakenly reporting the smaller number instead of the greater one.

Final Answer:
240