Question 13. Three sisters whose average age is 10 all have different ages. When they get together in pairs, the average ages of two such pairs are 11 and 12. What is the age of the eldest sister? To start, we know that the average age of all three sisters is 10. Let's label each of the sisters ages as A, B, and C. So when we do A plus B plus C divided by 3, we get 10. We also know when they get together in pairs, the average ages are 11 and 12. So we can write A plus B divided by 2 equal to 11, and A plus C divided by 2 equal to 12. We can multiply these out so they look like so. A plus B plus C equal to 30, A plus B equal to 22, and A plus C equal to 24. Now we can subtract these two formulas from each other. A plus B plus C equals 30, and A plus B equals 22. We can subtract out A, B, and 22, leaving us with C equal to 8. Now we can replace all the C's in our equations with 8's, like so. Now we notice the bottom equation can be easily solved. A plus 8 equals 24, so we know that A is equal to 16. Again, we can replace all A's with 16, and with this we can find out the age of the last sister, and we get B equal to 6. So the ages of the sisters is 6, 8, and 16. So the age of the eldest sister will be answer E, 16.

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age calculation