NEW YEAR'S RESOLUTIONS

Date of the Problem
January 03, 2022

Peitlyn decided that she would like to read more in 2022, so she made a resolution to read 30 minutes each day. If Peitlyn reads an average of 1 page every 2 minutes and she finishes her 6th book at the end of her reading session on March 31, 2022, what is the average number of pages per book for these six books?

Since Peitlyn is reading 1 page every 2 minutes, she is reading 15 pages each day. There are 31 + 28 + 31 = 90 days that she will have read after reading on March 31, 2022. This is a total of 15 × 90 = 1350 pages. If we spread this evenly over her 6 books, this is an average of 1350 ÷ 6 = 225 pages per book.

Nora’s resolution is to get more exercise. She decided that starting with the first full week in 2022, she will do a 45-minute workout four times each week (four days during every seven-day period). This means she will have to exercise four of the days during the week of Jan. 1 through Jan. 7, and then four of the days during the week of Jan. 8 through Jan. 14, etc. If Nora sticks to her resolution, what is the first possible date on which she could reach a total of 25 hours of exercise for the year?

If Nora sticks to her resolution, she will do four 45-minute workouts each week, which is 4 × 45 = 180 minutes each week, which is 180 ÷ 60 = 3 hours each week. By the end of her eighth week, she will have exercised 3 × 8 = 24 hours. The start dates of these 8 weeks are January 1, 8, 15, 22, 29 and February 5, 12, 19. Therefore, during the week starting on February 26, she will need only one more hour of exercise. If she exercises on the 26th, she’ll be up to 24 hours and 45 minutes. If she exercises on the 27th, too, that will put her over the 25-hour mark. Therefore, the first possible date is February 27th if she sticks to her resolution.

Riley decided to cut back on the time that she spends watching television. During the first week of 2022 she will allow herself to watch her usual 22 hours of television. However, during each successive week of 2022, she will only allow herself to watch 90% of the time that she watched during the previous week. During which week (first, second, third, etc.) will her viewing time first be under 5 hours?

We know that during the first week of the year, Riley is watching 22 hours of television. During the second week, she is allowing herself 22 × 0.9 = 19.8 hours. During the third week, she is allowing herself 19.8 × 0.9 or (22 × 0.9) × 0.9 = 22 × 0.92 hours. During the fourth week, she will allow herself ((22 × 0.9) × 0.9) × 0.9 = 22 × 0.93 hours. We can see that during the nth week, she will allow herself to watch 22 × 0.9n–1 hours. We are trying to determine when 22 × 0.9n–1 is less than or equal to 5 hours. Continuously multiplying by 0.9 without performing any rounding along the way, we see that we have to multiply 22 by 0.9 a total of 15 times, which will put us in the n – 1 = 15 → n = 16th week.

Page 1 of the linked PDF contains PROBLEMS & SOLUTIONS.

Page 2 contains ONLY PROBLEMS. ♦

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