Birthday Line Up [Problem #4040]



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Seth and his mother share the same birthday. On his fourteenth birthday, his mother turned 42 and Seth noticed that her age was exactly three times his age.

He realized that when his mother is 60, he will be 32 and she won't even be twice as old as Seth at that point.

He wondered when his mother's age will be twice his age, and when her age will be five times his age.

Help Seth figure out when these situations occur. Make sure to write an explanation for how to solve the problem.

Extra: When Seth's age is q years, what expression would give his mother's age?

Comments and Sample Solutions

Have any of you ever tried to figure out how old you'll be when your mom is twice your age? It's a fun little exercise. And you'll probably notice that you'll be a lot older than you are now!

There were three methods that most people used to solve the problem. The most common was to make a chart, like Najma and Madeleine did. Najma started by figuring out that Seth's mother is 28 years older than he is. She started her chart at 1 (for Seth) and 29 (for Seth's mom) and kept going until she found both answers.

Madeleine started her chart at 14 and 42 and kept going up until she found when Seth's mom was twice is age. In doing this, Madeleine didn't have to first figure out the difference in their ages. She only had to use the fact that each year they are each one year older. (She did figure it out later to find the answer to the second question.)

Annemarie used a combination of reasoning, algebra, and guess and check. She first reasoned that if they're 28 years apart, Seth will be 28 when his mom is twice his age (since her age is 28 + 28). Then she uses some algebraic notation to express how their ages need to be related when Mom is five times Seth's age. Then she uses guess and check to find the actual answer.

Elizabeth used algebra to solve the problem. She does a nice job of explaining her equations and telling the steps she did to solve them. If you are just learning how to use algebra, I encourage you to read her solution.

As we said, a lot of students used a chart to solve the problem. But some people forgot to tell us how they knew what to put in the chart! Instead of saying, "I started my chart with 0 and 28", say how you knew to use those two numbers. Why not 0 and 32?

Other students carefully showed why 28 and 56, and 7 and 35, work. But they did not say anything about where the numbers came from. What did you do to find the numbers that you used? Remember that each step you take is important, and you should try to help someone else understand why you did the things you did.

A big Happy Birthday to everyone who had a birthday while this problem was running!

--Annie and Suzanne, for the Pre-Algebra PoW

From: Najma T

My answer to the birthday problem was when Seth was 7, his mom was 35. So the mom was 5 times older. And when Seth will be 28, his mom will be 56 years old. That means that she will be twice as old.

The answer to the birthday problem is when Seth was 7, his mom was 35 years old. She was 5 times older than him. I got this by making a chart. The mom will be 28 years older forever. So when Seth was 1 years old, his mom was 29. When Seth was 2 his mom was 30. And it goes on. Then I was making a chart and I made it up to 20 years for Seth and 48 years for his mom. Then I checked to see which age will 5 times older. I got 7 and 35. Then I checked to see which number is 2 times older. I couldn't find any. So I countinued the chart. I stopped at 28 years and 56 years. I stopped there because I found out that 28 years multipied by 2 equals 56. So this is my answer, 7 years and 35 years for checking which age is 5 times older. And I also got 28 years and 56 years to see which age is twice as old.  

From: Madeleine G

Seth's mother will be twice his age when Seth is 28 and his mother is 56. Seth's mother was five times his age when Seth was 7 and his mother was 35. Extra: q+28

I solved this problem by setting up a chart.  The first row was Seth's age, and the second row was Mom's age.  I started by putting 14 in Seth's row, and 42 in Mom's.  Then I counted up from those ages and continued to write down each age in the appropriate row.  My chart looked somewhat like this:

Seth    14  15  16  17  18  19  20

Mom    42  43 44  45  46 47  48     etc.

I quickly scanned over Seth's row, mentally multiplying each number by two and checking to see if it matched the age in Mom's row.  Eventually, I got to the number 28 in Seth's row, multiplied it by two, and my product was 56, the same number in Mom's row.  So, when Seth is 28, his mother will be twice as old as him.

Next, I had to find how old Seth would be when his mother was five times his age.  From the last step, I knew that as they both aged, the amount of times his mother was his age went gradually down.  So, I made another chart counting the years before Seth's fourteenth birthday...

Seth    5   6   7   8    9    10   11

Mom    33 34 35  36  37   38  39   etc.

And I continued my process, only this time multiplying each number in Seth's row by five instead of two.  Soon, I came to 7 in Seth's row, and since 7 x 5 = 35, and 35 was the number in Mom's row, I knew that when Seth was 7, his mother was five times his age.

Extra:  To make an expression to show Seth's mother's age, I simply found the difference between their two ages.  42 - 14 = 28, so q + 28 shows how to find Seth's mother's age.

From: Elizabeth A

So Seth was 7 and mom was 35 yrs. old when they are 5 times the age. So Seth was 28 and mom was 56 yrs. old when they are half the age.

X=seth's age We are given X+28 equals moms age because she is 42 when he is 14. We want X when mom's age , or X+28, equals 5 time Seth's age, or 5X. Now I need to solve 5X=X+28. Subtracting X from both sides leaves 4x=28. Dividing both sides by 4 leaves me with X=7. So Seth was 7 and mom was 35 yrs. old.

For Seth being half mom's age, we have X=1/2(X+28). multiplying both sides by 2 we have 2X=X+28. Subtract X from both sides leaves me with 1X=28. Divide each side by one which leaves me with X=28. So Seth was 28 and mom was 56 yrs. old.

From: Annemarie C

When Seth is 28, his mother will be 56, and that is twice his age. When Seth is 7, his mom will be 35, and that is 5 times as much. Extra: If Seth=q, then his mom=m, so his mom's age is q+28=m.

 First, I realized that Seth was his mom's age-28, so that means his mom's age is Seth's+28. I figured this by looking at what the relationship was between Seth's age and his mom's age. 42-14=28, and 60-32=28. This means they are 28 years apart. Once you figure this out, the rest of the problem is easier.

Sine they are 28 years apart, when Seth is 28 that is 1/2 of his mom's age which is 56. Since m=s+28, then 28+28=56, so I know it works.

To figure out when she will be 5 times his age, I started out with the original sentence, m=s+28. Then I realized that s=m/5, so that means m=m/5+28. Whatever my answer was it had to be divisible by five and I had to be able to add 28 and get the original number that I divided by 5.

Then I tried different numbers for m. First I tried 30, but that didn't work. Then I tried 35, and it worked. 35=35/5+28. 35/5=7, and 7+28=35. 

Extra: I already figured out that m=s+28, so instead of Seth=s, he equals q. Then it is just m=s+28.

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