考题5-6 2015 AMC 10A

AMC10数学竞赛是美国高中数学竞赛中的一项，是针对高中一年级及初中三年级学生的数学测试，该竞赛开始于2000年，分A赛和B赛，于每年的2月初和2月中举行，学生可任选参加一项即可。不管是对高校申请还是今后在数学领域的发展都极其有利！那么接下来跟随小编来看一下AMC10数学竞赛真题以及官方解答吧：

Problem 5

Mr. Patrick teaches math to  students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was . After he graded Payton's test, the test average became . What was Payton's score on the test?

Solution

If the average of the first  peoples' scores was , then the sum of all of their tests is . When Payton's score was added, the sum of all of the scores became . So, Payton's score must be 

Alternate Solution

The average of a set of numbers is the value we get if we evenly distribute the total across all entries. So assume that the first students each scored . If Payton also scored an , the average would still be . In order to increase the overall average to , we need to add one more point to all of the scores, including Payton's. This means we need to add a total of  more points, so Payton needs 

Problem 6

The sum of two positive numbers is  times their difference. What is the ratio of the larger number to the smaller number?

Solution

Let  be the bigger number and  be the smaller.

.

Solving gives , so the answer is .

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