Problem 87 - College Algebra

Question

In a film, the actor Charles Coburn plays an elderly "uncle" character criticized for marrying a woman when he is 3 times her age. He wittily replies, "Ah, but in 20 years time I shall only be twice her age." How old is the "uncle" and the woman?

Step-by-Step Solution

Answer

The uncle is 60 years old and the woman is 20 years old.

Step 1: Translate the Word Problem into Equations

Let's assume, \(U\) to be the Uncle's current age and \(W\) should be the Woman's age. Now, using the information provided, two equations can be formed. The first equation is \(U = 3W\), from 'The uncle is three times the woman's age'. The second equation, after 20 years, is \(U+20 = 2(W+20)\), from 'The uncle will be twice the woman's age in 20 years'.

Step 2: Solve the System of Equations

Since the two equations are formed, now solve them simultaneously. The steps are as follows: Firstly, substitute \(U = 3W\) into the second equation to get \(3W+20 = 2(W+20)\). After simplifying it, the equation is \(3W + 20 = 2W + 40\), which simplifies directly to \(W=20\). Use this value in \(U = 3W\) to find that \(U = 60\).

Step 3: Interpret the Solution

Finally, place these values back into the context of the problem. Hence, the Uncle is 60 years old and the woman is 20 years old.