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Now we’re done talking about general word problem strategies and we can get into the particular question types. So the first question type we’ll talk about are age questions. This category concerns people of different ages.
What’s tricky about these questions is that mathematical relationships can be specified at two different times.
Right now here is one relationship they have and then so many years in the past or so many years in the future. Here is another relationship they had. For example, right now, Steve’s age is half of Tom’s age.
In eight years, twice Tom’s age will be more than five will be five more than three times Steve’s age.
How old is Tom right now? So, what I’m gonna say about this is, the big mistake associated with this problem is thinking of the ages as a fixed number. People get very sloppy, and they say, okay well, S is Steve’s age and T is Tom’s age.
Well then that first equation we can certainly say Steve’s age is half of Tom’s, S equals one half T, we can say that.
But for the second equation we cannot just say, well Tom’s age will be five more than 30 times Steve’s age, 2T equals 3S plus five, we can’t say that because in eight years, the ages will be different, and we need to take that into account.
It’s not enough to say that T is Tom’s age, we have to specify Tom’s age when. So, I’m gonna say in most age problems, it usually makes sense to pick the variables to represent the age right now.
Pick the age in the present moment that usually is the best strategy. So for example, it’s not that F is gonna equal Frieda’s age. F is gonna equal Frieda’s age right now, and then we use addition or subtraction to create expressions for ages at other times.
So for example, F minus 5 would be Frieda’s age five years ago. F plus 7 would be Frieda’s age 7 years from now.
We can create other ages based on the age now. We will use these expressions to create equations that correspond to statements about age relationships in the past or the future.
So we have in seven years Edgar will be twice as old as Frieda. Well suppose we picked E as Edgar’s age now, and F as Frieda’s age now. Well in seven years Edgar will be E plus 7, and Frieda will be F plus 7.
So E plus 7 will be equals twice 2 times F plus 7, and that’s our actual equation. So having seen that, think about this practice problem. You can pause the video and then we’ll talk about this. Okay.
Right now, Steve’s age is half of Tom’s age. In eight years, twice Tom’s age will be five more than three times Steve’s age. How old is Tom right now?
Well, first thing I’m gonna say T is gonna be Tom’s age now, S is gonna be Steve’s age now, and I could write it as S equals one-half T, but I think it’d make more sense to write it as T equals 2S that way I don’t have to think about fractions.
Now for the other ages I’m gonna have to use T plus 8 and S plus eight. So twice Tom’s age, 2 times T plus 8, will be, equals, five more than. So we’re gonna add 5, 3 times Steve’s age, 3 is times Steve’s age is S plus 8.
So we get this equation, 2 times T plus 8 equals 3 times S plus 8 and that quantity plus 5. Well since that I don’t have to worry about fractions at all I’m just gonna plugin for T.
I realize that I’m not solving for my target value. I’ll have to go back and find T later on but that’s fine cuz I can avoid fractions by doing it this way.
So I just plug that in for T now I’ve got an equation in S, multiply everything out, get everything all collected and we get s equals 13. So that is Steve’s age right now, and that’s half of Tom’s age, so Tom must be 26.
In summary age questions can be tricky when different mathematical relationships among the ages are specified at different times and so we have to very clear in defining the variables.
It’s not enough to say that F is Frank’s age we have to specify when F is Frank’s age now. We have to be very clear on that.
Choose variables to represent the ages now and then use addition or subtraction to create expressions for ages at other times.
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