Day 176
Jack and Jill collect shells. At the start of their vacation, Jack had 380 shells and Jill had 29. Each day of their vacation, Jack found 21 shells, and Jill found 48. At the end of their vacation, Jack and Jill each had the same number of shells. How long was their vacation?

Answer:

13 days.........Let X = the number of days of their vacation. At the end of their vacation, Jack had 380 + 21X shells, and Jill had 29 + 48X shells. Since they ended up with the same number of shells, these two expressions equal each other. To solve the problem without algebra: 380 - 29 = 351......the difference in the number of shells between Jack and Jill. 48 - 21 = 27........the difference in the number of shells collected each day. 351 divided by 27 = 13.

DAY 177

14 x 82 equals 41 x 28---1,148. Likewise, 39 x 62 = 93 x 26---2,418. Why does this work?

Answer:

In each case, you get the same number of ones, tens, and hundreds. (1x8=4x2 and 3x6=9x2)

DAY 178
What patterns do you notice in the difference of the squares of two consecutive numbers?
4 squared - 3 squared = 16 - 9 = 7
5 squared - 4 squared = 25 - 16 = 9
6 squared - 5 squared = 39 - 25 = 11
7 squared - 6 squared = 49 - 36 = 13

Answer:

If you go in order, the differences are the consecutive odd numbers, and, regardless of order, the difference of the squares of two consecutive numbers is equal to the sum of those two numbers.