The sum of two positive numbers is 55 and the sum of their squares is 1525. What is the smaller number?
Answer and explanation
Correct answer: 25
Let one number be x; because the sum is 55, the other is 55 − x. The sum-of-squares condition becomes x² + (55 − x)² = 1525. Expanding gives 2x² − 110x + 3025 = 1525, hence 2x² − 110x + 1500 = 0. Dividing by 2 gives x² − 55x + 750 = 0, which factors as (x − 25)(x − 30) = 0. Thus the two numbers are 25 and 30. Both are positive, and the smaller is 25, so option C is correct. Substitution confirms 25 + 30 = 55 and 25² + 30² = 625 + 900 = 1525. Option D is the larger number, while 20 and 22 fail the square-sum condition.
Frequently asked questions
What is the correct answer to this question?
25
Why is this the correct answer?
Let one number be x; because the sum is 55, the other is 55 − x. The sum-of-squares condition becomes x² + (55 − x)² = 1525. Expanding gives 2x² − 110x + 3025 = 1525, hence 2x² − 110x + 1500 = 0. Dividing by 2 gives x² − 55x + 750 = 0, which factors as (x − 25)(x − 30) = 0. Thus the two numbers are 25 and 30. Both are positive, and the smaller is 25, so option C is correct. Substitution confirms 25 + 30 = 55 and 25² + 30² = 625 + 900 = 1525. Option D is the larger number, while 20 and 22 fail the square-sum condition.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Word Problems and Applications.
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