Joan, Kylie, Lillian, and Miriam all celebrate their birthdays today. Joan is 2 years younger than Kylie, Kylie is 3 years older than Lillian, and Miriam is one year older than Joan. Which of the following could be the combined age of all four people today?
So it took me a long time to realize that everyone's age was consecutive. The order goes:
L -> J -> M -> K
So L is the youngest person and everyone's age can be expressed as a function of his age. L + L + 1 + L + 2 + L + 3 = 4L+ 6. So their combined age must be 6 greater than a multiple of 4.
the choices were:
- 51
- 52
- 53
- 54
- 55
So the answer is 54.
Is there a way to solve this problem using algebra and not coming to a realization that the ages were consecutive? Why not? Or Why?
Let $J,K,L,M$ have their obvious meanings. Then, $K=J+2$, $L=K−3=J−1$, and $M=J+1$. And so $$ J+K+L+M=J+(J+2)+(J−1)+J+1=4J+2\equiv 2\pmod{4}, $$ leading to $54$ as the answer.