If Keynes was three times as old as he is now, he would be 10 years younger than Hayek. If Hayek is 50 years older than Keynes, how old is Keynes?
This word problem tests our ability to assign variables to unknown quantities and translate the relationships described between these variable into equations. Our first step is to identify the unknown quantities and assign a variable to each. Generally, for age problems such as this one, our unknowns will be the current ages of the people involved. In this case, let K represent Keyne’s current age and let H represent Hayek’s current age. Now we must determine what relationships are described between these two people’s ages and create an equation for each relationship.
Our problem states, “If Keynes was three times as old as he is now, he would be 10 years younger than Hayek.” In other words, if we multiply Keyne’s age by 3, we would get the same number as if we subtracted 10 from Hayek’s age, or:
Next, we are told that Hayek is 50 years older than Keynes, or that if we take Keyne’s age and add 50, we get Hayek’s age:
This gives us a system of equations. Of the two most common methods for solving a system of equations (substitution and elimination), substitution seems to be the easiest method for this system. As such we will substitute K + 50 in for the H in our first equation, since we know these two quantities to be equal. This gives us the following equation:
We then solve algebraically for K:
This gives us our final answer: Keynes is currently 20 years old.
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