My name is Kathy, I am a parent trying to help my son. Please help with the following questions: Middle school 7-8

1. Zachary wanted to run the shore-to shore marathon next month, a 26 mile race from one side of the island where he lived to the other. He decided to work up to 26 miles by running five days a week for the next five weeks. Each day he'd run one mile longer than he did the day before, except for the first day of the week, when he'd run the same number of miles he ran on the last day of the previous week. He wanted his last two practice runs to be 26 and 27 miles long.
   
   How many miles would he have to run on his first practice run?
   
   
2. A school just bought a new set of encyclopedias. The front and back covers are 1/8 inch thick, and the pages are 1/100 inch thick (in other words, there are 100 pages per inch). the 26 volume set has a total of 5,250 pages. If you assume there is no extra space left between books, then .........
   
   What length of bookshelf will the encyclopedias take up?

Thanks so much

Mother in need

1. Zach is going to run 5 days a week for 5 weeks, so 25 days in total. Each day the number of miles he runs is 1 more than the the previous day, and the last day he runs 27 miles. If he started with 1 mile on the first day, 2 miles aon the second day, 3 miles on the third day and so on, he would run 27 miles on the twenty-seventh day. This is 2 days too long! You can fix this problem by ignoring the first 2 days and starting with 3 miles on the first day, 4 miles on the second day ... and then you end with 27 miles on the twenty-fifth day.
   
   
2. Imagine that you have the books unassembled, just the covers and the pages.

   Put the covers in a stack. There are 52 covers (a front and back cover for each of the 26 volumes) and each cover is  ¹/₈ inch thick. Hence the stack of covers is
   
   

    ¹/₈ x 52 = 6 ¹/₂ inches thick.

   Take the 5,250 pages and put them in stacks of 100 pages each. You get 52 stacks and 50 pages left over. Each 100 page stack is 1 inch high and the 50 page stack is  ¹/₂  inches high. Hence you have
   
   

   52 ¹/₂ inches of pages.
   Together you need
   
   6 ¹/₂ + 52 ¹/₂ = 59 inches.