--------- #1490 - What are their ages?
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- Can you figure out how old Jenn is in this case. We know Debbie is 44. Debbie is twice as old as Jenn was when Deb was as old as Jenn is now. How old is Jennifer now?
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- Do not read below this line if you want to try to solve the problem on your own. The answer and the solution are below for those who need a hint.
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- The problem with ages can be solved by realizing that the difference in the ages always remains a constant. Let’s call the constant = “k”. Debbie’s age - Jenn’s age = k
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- However, the ratio of the ages or the percentages of age comparison is constantly changing from one time to another. For example: When me and my brother were ages 2 and 4 , I was twice as old as he was, 200%.
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- However, when I was 52 and he was 50 I was only 4% older than he was, 2 / 50 = 4%
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- We have 2 time frames, now and some other time identified as “when”. Let’s start with the first time frame. Debbie is 44 now. The question is what is Jenn’s age now, in the first time frame?
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------------------- Debbie(1) = 44
------------------- Jenn (1) = ?
------------------ Debbie - Jenn = the constant = k
------------------ D1 - J1 = k
------------------ 44 - J1 = k
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-------------------- Now let’s work with the second time frame: , When Deb was as old as Jenn is now. We are at time (2) and Debbie(2) = Jenn (1)
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------------------------- D2 = J1
------------------ Debbie(2) is twice as old as Jenn(2) when …. Deb(2) = 2 * (Jenn (2))
------------------ Therefore , Jenn(2) is half as old as Debbie (1),
-------------------- Jenn(2) = 0.5 * 44 = 22
------------------------- J2 = 22
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- The constant, k, remains the same in the second time frame, and D2 = J1:
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--------------------- Debbie(2) - Jenn(2) = k
--------------------- J1 - 22 = k
-------------------- k = k , the difference between their ages remains constant regardless of time frame
-------------------- J1 - 22 = 44 - J1
------------------- 2 * J1 = 66
------------------- J1 = 33
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- When Debbie was 44 years old, Jenn was 33 years old.
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- Now that you have the secret on how to calculate ages, I will give you one more to work on:
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- When Jenn was in high school their combined ages summed to 44 years. Debbie was twice as old as Jenn was when Debbie was half as old as Jenn will be when Jenn is 3 times as old as Debbie was when Debbie was 3 times as old as Jenn. How old was Jenn and Debbie at that time Jenn was in high school?
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(1) To learn the answer, as soon as I figure out how it is done, I will be glad to share.
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- Here is the answer to the above ages calculations puzzle: Debbie is 27.5 years old and Jennifer is 16.5 years old and still in High School.
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- Split up the problem into the 4 different time frames (1) through (4):
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-When Jenn (J1) was in high school their combined ages summed to 44 years.
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---------------------- D1 + J1 = 44
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- Debbie (D1) was twice as old as Jenn (J2) was when:
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----------------------- D1 = 2 * J2
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-Debbie (D2) was half as old as Jenn (J3) will be when :
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-------------------- D2 = ½ * J3
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-Jenn J3) is 3 times as old as Debbie (D4) was when:
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---------------------- J3 = 3 * D4
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- Debbie (D4) was 3 times as old as Jenn (J4):
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----------------- D4 = 3 * J4
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- Next we define the difference in their ages as “k”, since it is always constant regardless of the time frame.
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------------------- D4 - J4 = k
------------------- J4 = (D4 - k)
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- Next work backwards through each time frame until we get back to the start where :
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------------------ D4 + J4 = 44 years.
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- Debbie (D4) was 3 times as old as Jenn (J4):
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------------------ D4 = 3 * J4 = 3 * (D4 - k)
------------------ D4 = 3 * k / 2
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- Jenn (J3) is 3 times as old as Debbie (D4) was when:
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- Next: -------- J3 = 3 * D4
------------------- J3 = 9 * k / 2
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-Debbie (D2) was half as old as Jenn (J3) will be when :
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- Next: -------- D2 = J3 / 2
------------------- D2 = 9 * k / 4
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- Debbie (D1) was twice as old as Jenn (J2) was when:
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- Next: ---------- D1 = 2 * J2
------------------- J2 = D2 - k = 9 * k / 4 - 4 * k / 4 = 5 * k / 4
----------------------D1 = 5 * k / 2
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- Next solve two simultaneous equations for “k“:
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--------- ----------- D1 = J1 + k
--------------------- J1 = D1 - k = 5 k / 2 - 2 k / 2 = 3k /2
--------------------- D1 + J1 = 44
--------------------- 5 k / 2 + 3 k / 2 = 44 = 88 / 2
---------------------- 8 k / 2 = 88 /2
----------------------- k = 11
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- The difference in Debbie and Jenn ages is 11 years. The sum of their ages is 44 years. Solve these two simultaneous equations:
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----------------- D1 - J1 = 11
---------------- D1 + J1 = 44
----------------- 2 D1 = 55
----------------- D1 = 27.5 years
----------------- J1 = 16.5 years
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- To double check their ages at the various time frames, here is how the puzzle was created:
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------------------ D4 + J4 = 44 years. ------------ 27.5 + 16.5 = 44
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- Debbie (D4) was 3 times as old as Jenn (J4):
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------------------ D4 = 3 * J4 = 3 * (D4 - k) --------- 16.5 - 5.5 = 11
------------------ D4 = 3 * k / 2
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- Jenn (J3) is 3 times as old as Debbie (D4) was when:
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- Next: -------- J3 = 3 * D4 ------------- 60.3 - 49.5 = 11
------------------- J3 = 9 * k / 2
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-Debbie (D2) was half as old as Jenn (J3) will be when :
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- Next: -------- D2 = J3 / 2 ------------ 24.75 - 13.75 = 11
------------------- D2 = 9 * k / 4
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- Debbie (D1) was twice as old as Jenn (J2) was when:
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- Next: ---------- D1 = 2 * J2 ---------------- 27.5 - 16.5 = 11
------------------- J2 = D2 - k = 9 * k / 4 - 4 * k / 4 = 5 * k / 4
----------------------D1 = 5 * k / 2
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