1. Math Problem -- ages Making your own variable equation to solve a math riddle

2. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them?

3. What do we know from this?

4. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? What do we know from this? What information is given?

5. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? What do we know from this? What information is given? We know Joe is twice as old as Bill, and that together they are three times as old as Antwone. We also know that if you add up the ages of Joe and Bill and Antwone, the total age of the three of them would be 80 years old.

6. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? What do we know from this? What information is given? How can we write it out as variables and numbers? We can use variables for the age of each person. In other words, we can use the variable J to represent Joe’s age, B to represent Bill’s age and A to represent Antwone’s age. If we do, we get the following:

7. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? How can we write it out as variables and numbers? J = Joe’s age B = Bill’s age A = Antwone’s age

8. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? How can we write it out as variables and numbers? J = Joe’s age B = Bill’s age A = Antwone’s age Now, let’s replace what we said we knew with the appropriate variable.

9. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? How can we write it out as variables and numbers? J = Joe’s age B = Bill’s age A = Antwone’s age Now, let’s replace what we said we knew with the appropriate variable. J is twice B, and together they are 3 times A. We also know J + B + A = 80.

10. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? How can we write it out as variables and numbers? J = Joe’s age B = Bill’s age A = Antwone’s age Now, let’s replace what we said we knew with the appropriate variable. J is twice B, and together they are 3 times A. We also know J + B + A = 80. Now, let’s rewrite the information line right above this sentence, and rewrite it completely as math statements.

11. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? J = Joe’s age B = Bill’s age A = Antwone’s age J is twice B, and together they are 3 times A. We also know J + B + A = 80. Now, let’s rewrite the information line right above this sentence, and rewrite it completely as math statements. For J is twice B, you rewrite it as J = 2B

12. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? J = Joe’s age B = Bill’s age A = Antwone’s age J is twice B, and together they are 2 times A. We also know J + B + A = 80. Now, let’s rewrite the information line right above this sentence, and rewrite it completely as math statements. For “J is twice B”, you rewrite it as J = 2B For “together they are 3 times A”, you rewrite it as J + B = 3A

13. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? J = Joe’s age B = Bill’s age A = Antwone’s age J is twice B, and together they are 2 times A. We also know J + B + A = 80. Now, let’s rewrite the information line right above this sentence, and rewrite it completely as math statements. For “J is twice B”, you rewrite it as J = 2B For “together they are 3 times A”, you rewrite it as J + B = 3A For “J + B + A = 80,” you don’t have to change anything: J + B + A = 80

14. So here is what you now have: Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? J = Joe’s age B = Bill’s age A = Antwone’s age J = 2B J + B = 3A J + B + A = 80

15. So here is what you now have: Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? J = Joe’s age B = Bill’s age A = Antwone’s age J = 2B J + B = 3A J + B + A = 80 Now what?

16. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? J = Joe’s age B = Bill’s age A = Antwone’s age J = 2B J + B = 3A J + B + A = 80 You can’t solve variable equations with more than one variable…. So now what?

17. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? J = Joe’s age B = Bill’s age A = Antwone’s age J = 2B J + B = 3A J + B + A = 80 So now what? Is there any way to rewrite things so you are working with just one variable?

18. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? J = Joe’s age B = Bill’s age A = Antwone’s age J = 2B J + B = 3A J + B + A = 80 Can you replace any of the variables with one of the other variables so we have fewer variables in the equation?

19. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? J = Joe’s age B = Bill’s age A = Antwone’s age J = 2B J + B = 3A J + B + A = 80 How about J? Can we replace it with 2B? If we could, we would then have two different variables instead of three… How would that work?

20. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? J = Joe’s age B = Bill’s age A = Antwone’s age J = 2B J + B = 3A J + B + A = 80 Are these two equations the same? J + B +A = 80 and 2B + B + A = 80

21. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? J = Joe’s age B = Bill’s age A = Antwone’s age J = 2B J + B = 3A J + B + A = 80 Are these two equations the same? J + B +A = 80 and 2B + B + A = 80 If J = 2B then J + B + A = 80 is the very same thing as 2B + B + A = 80. This is because J = 2B; they are the same thing, so you can replace one of them with the other.

22. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? J = Joe’s age B = Bill’s age A = Antwone’s age J = 2B J + B = 3A J + B + A = 80 2B + B + A = 80 So we have added 2B + B + A = 80 to the list of things we know above…. But that still gives us an equation with two different variables, and you only know the value of a variable when you get your equation down to a single variable.

23. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? J = Joe’s age B = Bill’s age A = Antwone’s age J = 2B J + B = 3A J + B + A = 80 2B + B + A = 80 Now what? Can we rewrite 2B + B + A = 80 somehow so it only has one variable instead of two variables (B and A are two variables)?

24. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? J = Joe’s age B = Bill’s age A = Antwone’s age J = 2B J + B = 3A J + B + A = 80 2B + B + A = 80 Can we rewrite 2B + B + A = 80 ? Let’s look back at what we know. J + B = 3A and J = 2B. We have already seen that this means 2B + B = 3A. That means that 3B = 3A. So let’s add that to the list of things we know: 3B = 3A.

25. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? J = Joe’s age B = Bill’s age A = Antwone’s age J = 2B J + B = 3A J + B + A = 80 2B + B + A = 80 3B = 3A If 3B = 3A, doesn’t that mean that B = A?

26. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? J = Joe’s age B = Bill’s age A = Antwone’s age J = 2B J + B = 3A J + B + A = 80 2B + B + A = 80 3B = 3A If 3B = 3A, doesn’t that mean B = A? Let’s add that to the list of things we know: B = A.

27. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? J = Joe’s age B = Bill’s age A = Antwone’s age J = 2B J + B = 3A J + B + A = 80 2B + B + A = 80 3B = 3A B = A If B = A, then can’t we go back to the equation 2B + B + A = 80, and replace A with B?

28. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? J = Joe’s age B = Bill’s age A = Antwone’s age J = 2B J + B = 3A J + B + A = 80 2B + B + A = 80 3B = 3A B = A Now, if we replace A with B in the equation 2B+ B + A = 80, what we get is 2B + B + B = 80.

29. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? J = Joe’s age B = Bill’s age A = Antwone’s age J = 2B J + B = 3A J + B + A = 80 2B + B + A = 80 3B = 3A B = A So let’s add that to the list of things we know: 2B + B + B = 80.

30. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? J = Joe’s age B = Bill’s age A = Antwone’s age J = 2B J + B = 3A J + B + A = 80 2B + B + A = 80 3B = 3A B = A 2B + B + B = 80 Now, if 2B + B + B = 80, we have reduced the problem to one having just one variable: B.

31. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? J = Joe’s age B = Bill’s age A = Antwone’s age J = 2B J + B = 3A J + B + A = 80 2B + B + A = 80 3B = 3A B = A 2B + B + B = 80 If we combine all of the B’s on the lefthand side of the equation, we get 4B = 80.

32. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? J = Joe’s age B = Bill’s age A = Antwone’s age J = 2B J + B = 3A J + B + A = 80 2B + B + A = 80 3B = 3A B = A 2B + B + B = 80 If 4B = 80, we can divide each side by 4 and end up with B = 20. So let’s add that to what we know.

33. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? J = Joe’s age B = Bill’s age A = Antwone’s age J = 2B J + B = 3A J + B + A = 80 2B + B + A = 80 3B = 3A B = A 2B + B + B = 80 B = 20 If B = 20, let’s look back at what we know to see how that helps us tell what the other variables are.

34. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? J = Joe’s age B = Bill’s age A = Antwone’s age J = 2B J + B = 3A J + B + A = 80 2B + B + A = 80 3B = 3A B = A 2B + B + B = 80 B = 20 J = 2B, and if B = 20, then J = 2(20), meaning J = 40. We also know that B + A, so A is also = 20. So let’s add this to what we know.

35. Joe is twice as old as Bill, and together they are three times as old as Antwone. All together the three of them are 80 years old. How old is each of them? J = Joe’s age B = Bill’s age A = Antwone’s age J = 2B J + B = 3A J + B + A = 80 2B + B + A = 80 3B = 3A B = A 2B + B + B = 80 B = 20 J = 40 A = 20 And since J = Joe’s age, B = Bill’s age, and A = Antwone’s age, then Joe is 40, Bill is 20 and Antwone is 20. Does this work out?

36. Now let’s try it again, only a little different. Juan is five times as old as Mary. Mary is twice as old as Rajin. All of them together are 39 years old. How old are they?

37. Is there any reason to think you would work this problem any differently than you did the last one?

38. Juan is five times as old as Mary. Mary is twice as old as Rajin. All of them together are 39 years old. How old are they? So how did you start working on the last problem?