Grades 9-10 Video Solutions 2014-Levels 9&10 Video Solutions 2014 problem11

Video Transcription

Question number 11. This year, in 2014, a grandmother, her daughter, and her granddaughter can say that the sum of their ages is equal to 100. In what year  was the granddaughter born if each of their ages is a power of 2? So let's try to add up powers of 2. We have 2 to the power 0, which is 1, then 2 to the power  1, 2 to the power 2, 2 to the power 3, and then 4, 5 gives us 32, 64 is next, and  then 128, but that's too much. So then we see that one of those ages has to be 64. Why is that? Because if we remove 64 from the list, the next three lowest  numbers sum up to less than 100. So one woman, probably the grandmother, is exactly 64, so we'll say that's the grandmother. Then 100 minus 64 gives us  36, and the only way to obtain the sum of 36 with 2 of the remaining numbers is to add a 32 and a 4. So that is 32 plus 4. So here we see that the daughter has  to be 32, and that leaves us with the granddaughter as the 4-year-old. And if she is 4 years old this year, meaning in the contest year 2014, she was born  exactly 4 years ago in 2010. And that is then our answer, C, 2010.

Video Summary

In 2014, a grandmother, her daughter, and her granddaughter together are 100 years old, with each age being a power of 2. The grandmother is identified as 64 years old. Subtracting 64 from 100 leaves 36, which matches the ages of the daughter and granddaughter being 32 and 4, respectively. Thus, the granddaughter was born in 2010.