Question 390691: Amy is 3 years older than Laura. The product of their ages is equal to 2 times the sum of their ages. Could someone please tell me how to solve this without writing two independent equations? More importantly, how does a 5th grader with no concept of independent variables and equations solve this other than tedious trial and error and at the end, what has he learned?
Found 2 solutions by Edwin McCravy, solver91311:
Answer by Edwin McCravy(20056)   (Show Source):
You can put this solution on YOUR website!
Try Laura being 1.
Then Amy is 4.
The product of their ages is 4
The sum of their ages is 5.
4 is not 2 times 5
Try Laura being 2.
Then Amy is 5.
The product of their ages is 10
The sum of their ages is 7.
10 is not 2 times 7
Try Laura being 3.
Then Amy is 6.
The product of their ages is 18
The sum of their ages is 9.
Hooray! 18 is 2 times 9
That wasn't too tedious.
Edwin
Answer by solver91311(24713)  (Show Source):
You can put this solution on YOUR website!
Let's begin with your last question because the answer is fundamental to the entirety of K-12 mathematics education. The answer is simple: You learn how to solve problems. In fact, learning how to solve problems is the goal of ALL elementary and secondary education with the exception of the study of the English language. The study of our language provides us with the ability to communicate the problems and their solutions to others.
You don't need two independent equations for this problem because one of the unknown ages can be expressed in terms of the other. Let  represent Laura's age. Then, because we know that Amy is three years older than Laura, we can represent Amy's age by  .
Then we can say that the product of their ages is ) and two times the sum of their ages is \right))
Now, if your 5th grader knows how to use the distributive property, collect terms, put a quadratic equation into standard form, determine the factors of a quadratic trinomial with integer factors, and apply the Zero Product Rule, then your student is home free. If the student is unable to do any of these things, then one of the following is true: 1. The student has fallen behind the rest of the class, or 2. The problem is inappropriate for this student's ability level.
My recommendation is that you consult with your student's teacher regarding that teacher's expectations for your student's current ability level and to develop a strategy to correct any deficiencies before the student begins to fail completely.
John
My calculator said it, I believe it, that settles it
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