1) A chime clock strikes 1 chime at one o'clock, 2 chimes at two o'clock, 3 chimes at three o'clock, and so forth. What is the total number of chimes the clock will strike in a twelvehour period?
2) A boy has the following seven coins in his pocket: 2 pennies, 2 nickels, 2 dimes, and 1 quarter. He takes out two coins, records the sum of their values, and then puts them back with the other coins. He continues to take out two coins, record the sum of their values, and put them back. How many different sums can he record at most? 3) The product of two numbers is 144 and their difference is 10. What is the sum of the two numbers? Solutions: 1) Method 1: T, the total number of chimes, equals 1 + 2 + 3 + 4 + ... + 10 + 11 + 12. The sum of this series is 78. Method 2: T = 1 + 2 + 3 + 4 + ... + 10 + 11 + 12 T = 12 + 11 + 10 + 9 + ... + 3 + 2 + 1 2T = 13 + 13 +13 +13 + ... + 13 + 13 + 13 Notice that the series on the second line is the reverse of the series on the first line. Each time we add a term and the corresponding term above, the sum is 13. then 2T is equal to 12 x 13 or 156. But 2T is twice the sum of the first series. Therefore, T = 78. 2) The following pairs of numbers represent the values of the two coins the boy could take from his pocket: (1,1) (1,5) (1,10) (1,25) (5,5) (5,10) (5,25) (10,10) (10,25) Each of the above pairs has a sum that is different from the sum of each of the other pairs. Therefore there are 9 different sums. 3) Examine the pairs of whole number whose product if 144. The only pair that has a difference of 10 is 18 and 8. Their sum is 26.
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