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AMC8每日一题(2001年真题#02)

2018-05-28
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2001 AMC 8 竞赛试题/第02题

Problem

I'm thinking of two whole numbers. Their product is 24 and their sum is 11. What is the larger number?

$\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 12$

Solution 1

Let the numbers be $x$ and $y$ . Then we have $x+y=11$ and $xy=24$ . Solving for $x$ in the first equation yields $x=11-y$ , and substituting this into the second equation gives $(11-y)(y)=24$ . Simplifying this gives $-y^2+11y=24$ , or $y^2-11y+24=0$ . This factors as $(y-3)(y-8)=0$ , so $y=3$ or $y=8$ , and the corresponding $x$ values are $x=8$ and $x=3$ . These are essentially the same answer: one number is $3$ and one number is $8$ , so the largest number is $8, \boxed{\text{D}}$ .

Solution 2

Use the answers to attempt to "reverse engineer" an appropriate pair of numbers. Looking at option $A$ , guess that one of the numbers is $3$ . If the sum of two numbers is $11$ and one is $3$ , then other must be $11 - 3 = 8$ . The product of those numbers is $3\cdot 8 = 24$ , which is the second condition of the problem, so our number are $3$ and $8$ .