Expert: Josh - 11/5/2009

Question
Hello:

2/3 of Mary’s age equals Sarah’s age and 3/4 of Ruth’s age equals Sarah’s age.
If the total ages equal 46, how old is each girl?

Answer: Mary 18, Ruth 16, Sarah 12

Solution:

3/2 of Sarah’s age equals Mary’s age.
4/3 of Sarah’s age equals Ruth’s age.
6/6 of Sarah’s age equals Sarah’s age.

46 divided by 23/6 equals Sarah’s age of 12.

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Can the same method for determining the ages in the above examples be used to determine the ages of the girls in the example below?

1/2 of Mary's age and 2/5 of Ruth's age and 1/4 of Sarah's age equal 14.  
If the total ages equals 37, how old is each girl?

Answers: 10, 15, 12

Here is the complete calculation: 1/2 of Mary's age [10] and 2/5 of Ruth's age [15] and 1/4 of Sarah's age [12] equals 14.

1/2 X 10 = 5
2/5 X 15 = 6
1/4 X 12 = 3

10 + 15 + 12 = 37
-or-
5 + 6 + 3 = 14

The missing ages are in the brackets. Can you determine these ages by using the solution similar to the first example? There will be more than one possible age for each girl.

NOTE: I am interested in only a solution that matches or closely matches the one in the first example above.

I thank you for your reply.


Answer
Kenneth,

The usual way to navigate such a problem is by using linear algebra. This will tell us if the ad hoc strategy works in the second example.

Let Sarah's age be "s", Mary's age be "m", Ruth's age be "r".
From the information given, we obtain two equations:
m/2 + 2r/5 + s/4 = 14 ...[1] and  m + r + s = 37 ...[2].

whereas in example one, we had
2m/3 = s   =>  m = 3s/2  [A1]
3r/4 = s   =>  r = 4s/3  [A2]
m + r + s = 46  [A3]
unique solutions were obtained by substituting [A1] and [A2] into [A3].

Now, looking at these equations, it is obvious that the ad hoc strategy used in the first example cannot be used in the second example. Reason being: From [1], the variables "m", "r" and "s" are inter-related. We therefore cannot make a separation, expressing "m" directly in terms of "s", and expressing "r" in terms of "s" (which is what you have done in the first instance). It is not possible to then write [2] in terms of "s" only.