A number multiplied by 6 less than itself gives 112. What is the larger number?
Answer and explanation
Correct answer: 14
Let the larger number be x. The phrase “6 less than itself” refers to x − 6, so the condition becomes x(x − 6) = 112. Expanding gives x² − 6x − 112 = 0. Factoring, (x − 14)(x + 8) = 0, so x = 14 or x = −8. The pair associated with x = 14 is 14 and 8, and its product is 112; therefore the larger number is 14. Option B is correct. The negative solution represents the ordered pair −8 and −14, whose larger member is −8, but the intended positive-number interpretation and the listed choices identify 14. A check shows 14 × (14 − 6) = 14 × 8 = 112. The other positive options do not produce 112 when multiplied by six less than themselves.
Frequently asked questions
What is the correct answer to this question?
14
Why is this the correct answer?
Let the larger number be x. The phrase “6 less than itself” refers to x − 6, so the condition becomes x(x − 6) = 112. Expanding gives x² − 6x − 112 = 0. Factoring, (x − 14)(x + 8) = 0, so x = 14 or x = −8. The pair associated with x = 14 is 14 and 8, and its product is 112; therefore the larger number is 14. Option B is correct. The negative solution represents the ordered pair −8 and −14, whose larger member is −8, but the intended positive-number interpretation and the listed choices identify 14. A check shows 14 × (14 − 6) = 14 × 8 = 112. The other positive options do not produce 112 when multiplied by six less than themselves.
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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