Sunday, March 11, 2012

Problems and Solutions for Kingston Math Circle 3/10/12

Solutions

1. In the correctly worked out addition problem below, different letters represent different digits. What digit does A represent?

A4+4A=BCB

First, look at how big the sum can be, and deduce that B must be 1, since the sum of two 2-digit numbers can’t be more than 198 (which is 99+99). Second, consider the units digits. Since A + 4 ends with B = 1, A must be 7.

Check: 47 + 74 = 121.

2. What is the first year in the 21st century (2001 through 2100) that is divisible by 11?

Divide: 2000 ÷ 11 = 181 with remainder 9. Add 2 years to get the next multiple of 11, which is 2002.

Alternate solution: Find a nearby multiple of 11, for example 2200, and count back by 99. Counting backwards, 2200, 2200 - 99 and 2200 - 99 - 99 are all multiples of 11. The last one is 2200 - 99 - 99 = 2002.

3. 9 apes weigh as much as 4 bears. 8 bears weigh as much as 15 cougars. 10 cougars weigh as much as 27 deer. How many deer weigh the same as 4 apes?

Let’s scale up! If 9 apes weigh as much as 4 bears, then doubling, 18 apes weigh as much as 8 bears, and so 18 apes weigh as much as 15 cougars. Let’s double that to find that 36 apes weigh as much as 30 cougars, and so 36 apes weigh as much as 3 times 27, or 81 deer. Now scale down by a factor of 9 to find that 4 apes weigh as much as 9 deer.