Introduction / Context:
This question combines the concept of ratio with algebraic manipulation involving squares of numbers. You are given the ratio between two positive numbers and the sum of their squares, and you are asked to determine the sum of the original numbers. This kind of problem is frequently tested to see if candidates can translate ratio information into algebraic expressions and then solve a simple equation involving squares.
Given Data / Assumptions:
- Let the two positive numbers be x and y.
- Their ratio is x : y = 3 : 4.
- The sum of their squares is x^2 + y^2 = 400.
- Both numbers are positive.
Concept / Approach:
When two numbers are in the ratio 3 : 4, we can write x = 3k and y = 4k for some positive constant k. Substituting these expressions into the condition about the sum of squares gives an equation in k. Solving this equation yields the value of k, from which we can compute both numbers and then their sum. This approach is systematic and avoids guesswork.
Step-by-Step Solution:
1) Let the two numbers be x and y with x : y = 3 : 4.
2) Express x and y in terms of k: x = 3k and y = 4k.
3) Given that x^2 + y^2 = 400.
4) Substitute: (3k)^2 + (4k)^2 = 400.
5) This gives 9k^2 + 16k^2 = 400.
6) Combine like terms: 25k^2 = 400.
7) Solve for k^2: k^2 = 400 / 25 = 16, so k = 4 (positive root because numbers are positive).
8) Then x = 3k = 3 * 4 = 12 and y = 4k = 4 * 4 = 16.
9) Sum of the numbers = x + y = 12 + 16 = 28.
Verification / Alternative check:
We can check the solution by verifying the sum of squares condition with x = 12 and y = 16. Compute x^2 + y^2 = 12^2 + 16^2 = 144 + 256 = 400, which matches the given condition exactly. This confirms that both numbers and their sum have been calculated correctly, and that no sign or arithmetic mistakes have been made.
Why Other Options Are Wrong:
Any other sum choice such as 22, 24, 26 or 30 would correspond to different values of x and y, which would not satisfy both the given ratio and the required sum of squares. For example, if the sum were 24 and the ratio is 3 : 4, the numbers would be 9 and 15, whose squares sum to 81 + 225 = 306, not 400. Similar checks show that none of the other options satisfy both conditions simultaneously.
Common Pitfalls:
Students sometimes confuse the sum of numbers with the sum of squares and try to directly assign values based on the ratio without forming the correct equation. Another common error is taking the negative root of k even though the question specifies positive numbers. Always pay attention to the word positive in such questions, and make sure you substitute back to check that your final numbers satisfy all original conditions.
Final Answer:
The sum of the two positive numbers is
28.
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