Age Word Problems - Problem 1

This looks like it's going to be a really easy word problem about the ages of a father and his daughter. However, it gets really tricky when you're doing Math with the passage of time.

A father is 26 years older than his daughter. In 4 years the father will be 2 more than 5 times her age. That sentence is really complicated. There's lots going on, we'll deal with that in a second. Find each of their ages.

Okay, I want to set up a system of equations. The first equation is going to come from the first sentence. A father is 26 more than the daughter. That's not right. This means the father number plus 26 gives me the daughter. That would mean the daughter was older. That's not right, but a lot of students make this mistake right from the bat because they're reading this sentence like f, 26 equals d, like I can understand why people make this mistake, but you have to think about your equations before you actually go through with them.

The daughter plus 26 equals the father because he’s older. That represents the first sentence. Keep in mind that's not going to change. He is always going to be 26 more than her.

Okay let's look at the second sentence. In four years, that means I'm going to have to add 4 to d, and also 4 to f, the father will be 2 more than 5 times her age. Okay well let's start with the father. Father plus 4 because it's in the future is equal to 2 more than 5 times her age. Here is the 2 more part because 2 more means adding 2, 5 times her age. Well her new age got 4 added to it. Let's go through that one more time because that sentence and that equation is really tricky. 4 years from now both the father has a plus 4, and the daughter has a plus 4 because 4 years have gone by. His age is equal to 2 more than 5 times her age, that's where that comes from. Also keep in mind this represents her old age plus 4, so this whole little piece inside the parentheses is her new age.

Okay, once you have those equations set up, it's a really basic system of equations. It's already set up for you to substitute because f is already isolated, so instead of f right there, I'm going to write the expression d plus 26 instead. D plus 26 plus 4 is equal to 2 plus I'm going to distribute so I have 5d plus 20. Now both of these sides need to be simplified before I combine any like terms. D plus 30, not before I combine like terms, before I solve I'm sorry. D plus 30 is equal to 5d plus 22, let's subtract d from both sides and subtract 22 from both sides, so now I have 4 times the daughter's age is equal to 8, that means the daughter oops d, daughter is equal to 2.

Currently, the daughter is 2 years old, that's her current age because my d that I defined represented her age right now. Currently the daughter is 4, excuse me is 2, daughter is 2 currently, label your answers to remember what that stands for. Daughter is 2 currently. If the daughter is 2 currently and the father is 26 more than that, that means that currently the father is equal to 28.

And I'm done, I have their current ages for both of them that's what the problem asked for. If I wanted to be a superstar, I would think about okay, 4 years from now so when he's 32, that should be equal to 2 plus 5 times her new age. Her new age would be 6, 5 times that would be 30. I add 2 more to it boom, there it is 32.

When you're checking your systems of equations problems, you don’t have to write it out. A lot of the times you can do it in your head like that and make sure you have a reasonable answer. When you see these age problems in your homework, it can be tricky to deal with the passage of time. Just make sure that when you add 4 years to one of your people in the equation, you also want to add 4 years to the other person.