How Old are Ben and Kris?Date: 10/17/1999 at 17:44:56 From: Emily Subject: Age word problems I have been given several word problems on age like: A man is 13 times as old as his son is. In 10 years he will be 3 times as old as his son will be. How old are they now? The book does show how to solve that particular problem, but gives a new twist on the problems they solve! Here's one. Please give me a step-by-step on how to solve: Five years ago Ben was 2/3 as old as Kris. Ten years from now he will be 5/6 as old as Kris. How old are they now? Thanks. Date: 11/28/1999 at 10:21:29 From: Doctor Aileen Subject: Re: Age word problems Hi Emily, One way to solve this problem is to write a system of equations with two variables. Let x be Kris' age now and y be Ben's age. Then x - 5 is how old Kris was 5 years ago. Since Ben was two thirds of Kris' age 5 years ago, then 2/3*(Kris' age five years ago) = Ben's age five years ago. Therefore 2/3*(x - 5) = y - 5 The problem also tells you that in 10 years from the present time Ben's age will be 5/6 Kris' age. In other words 5/6 * (Kris' age in 10 years) = Ben's age in 10 years or: 5/6 * (x + 10) = y + 10 You need to find an x and y that will be true for both equations. To do this you need to solve for the variables one at a time. By subtracting the second equation from the first you can formulate a third equation that has only one variable. 2/3*(x-5) = y - 5 - 5/6*(x+10) = y + 10 -------------------------------- (2/3*(x-5) - 5/6*(x+10) = -15 The equations can be either subtracted or added, whatever works to eliminate one variable. Now you can find the value of this variable. After doing so you can substitute it back into one of the original equations. Setting up the equations is the most important aspect of the problem. Once you have the proper equations it is simply a matter of solving. Also, if you are uncomfortable with the method of eliminating a variable, you can try graphing both equations. The intersection point represents the x and y values that satisfy both equations. I hope this helps. - Doctor Aileen, The Math Forum http://mathforum.org/dr.math/ Date: 1/20/2002 at 13:01:58 From: Judy Ann Brown Subject: Re: Age word problems Here's the same problem done in two more ways: Five years ago Ben was 2/3 as old as Kris. Ten years from now he will be 5/6 as old as Kris. How old are they now? We can set up a chart: 5 years ago now in ten years Ben (2/3)x (2/3)x + 5 (2/3)x + 15 Kris x x + 5 x + 15 From the problem we know that Ben's age in 10 years = 5/6 Kris' age in 10 years (2/3)x + 15 = 5/6(x + 15) 4x + 90 = 5x + 75 (we multiplied both sides by 6 to get rid of the fractions) 15 = x If we know that Kris was 15 five years ago, he is now 20. Ben was 2/3(15) or 10 five years ago, so he is now 15. In ten years Ben will be 25 and Kris will be 30. There is also an elementary guess-and-check approach to solving this problem that does not use variables. Five years ago Kris had to be an age that was a multiple of 3. The 2/3 for Ben's age tells me so. Let's make a chart and guess: Let's start with 3, because that is the youngest Kris could have been 5 years ago. GUESS 1: Kris is 3 5 years ago now in ten years Ben 2 7 17 (more than 5/6 of 18, so 3 is too little) Kris 3 8 18 GUESS 2: Kris is 30 5 years ago now in ten years Ben 20 25 35 (less than 5/6 of 45, so 30 is too much) Kris 30 35 45 I can also tell that Kris' age in ten years must be a multiple of 6. But that's only going to happen if I start out with an odd number for his age, because I will have to add 15, and the only way to get an even number when one of the addends is odd is to have the second addend also odd. GUESS 3: Kris is 21 5 years ago now in ten years Ben 14 19 29 (less than 5/6 of 36, so 21 is too much) Kris 21 26 36 I know that I'm getting close in my guesswork, but I need a smaller number for Kris' age, and it must be an odd multiple of 3. I choose 15. GUESS 4: Kris is 15 5 years ago now in ten years Ben 10 15 25 (this is exactly 5/6 of 30) Kris 15 20 30 I know I have a correct solution. But what was the question? How old are they now...? Ben is 15, and Kris is 20. - Judy Ann Brown |
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