Five years ago, John’s age was half of the age he will be in 8 years. How old is he now?
Solution:
Let x = John's current age
x - 5 = John's age 5 years ago
Let x + 8 = John's age in 8 years
My Equation:
x - 5 = (1/2)(x + 8)
Correct?
Like those Joseph? Try this one:
Jim has three daughters.
His friend Joe wants to know the ages of his daughters.
Jim gives him the first hint: The product of their ages is 72.
Joe says this is not enough information, so Jim gives him a second hint:
The sum of their ages is equal to my house number.
Joe goes out and look at the house number and says:
“I still do not have enough information to determine the ages”.
Jim then gives Joe the third hint:
The oldest of the girls likes strawberry ice-cream.
"OK, got it!" says Joe.
What are the ages?
If you were wondering: the whole point of "the oldest of the girls likes strawberry ice cream" is tell us that there is an oldest girl. We also have to assume that we are giving the ages of the girls as integers, not as, for example, "7 and 2/3 years".
"The product of the ages of the girls is 72". How can we factor 72 into 3 ages? The prime factorization of 72 is $\displaystyle 2^3(3^2)$ so their ages can be any of
2, 2, and 18 which sum to 22
2, 3, and 12 which sum to 17
2, 4, and 9 which sum to 15
2, 6, and 6 which sum to 14
3, 3, and 8 which sum to 14
3, 4, and 6 which sum to 13
We are also told that their ages sum to the house number. We do not know what that house number is but the neighbor does. So why could he not look at that list and tell which is the right one? Because the house number was 14 for which there are two possible sets of ages! Knowing that there is an "oldest" child tells the neighbor that it cannot be "2, 6, and 6" and so must be "3, 3, and 8"