Franz's Dilemma (A study in increasing fractions and decreasing numbers)
David W. Hansen (2011)
Franz had a very wonderful mother who delighted in doing nice things for him, but who, I'm afraid, also liked to tease him at times.
One morning he awoke to the delicious odor of fresh-baked cookies which his mother was just taking out of the oven.
"Oh, Mother", cried Franz as he leaped out of bed and ran downstairs to the kitchen, "how good those cookies smell! I could eat all of them right now and still have room for more. Please may I have one?"
His mother laughed at him and said, "Franz, I have made 30 cookies for you to eat over the next few days. If you eat them all now, you will get nothing but a big tummy ache! But I have a plan. Today, the 1st day, you may eat 1/10th (10%) of the cookies; tomorrow, the 2nd day, you may eat 1/9th (about 11%) of the remaining cookies; the 3rd day, you may eat 1/8th (12.5%) of the remaining cookies; the 4th day, you may eay 1/7th (about 14%) of the remaining cookies; and so on, continuing in the same manner, until the next to the last day, when you may eat 1/2 (50%) of the remaining cookies, and on the last day you may eat all (100%) of the cookies that are left! That way, you will never eat too many cookies and get a tummy ache, and each day you will get to eat a higher percentage of the remaining cookies than you did the previous day."
"Oh, thank you, Mother," said Franz as he hugged her. "That sounds like a fine plan, and I can look forward to eating more and more cookies each day!"
"Don't be too sure of that, my son", said his mother as she gave him a sweet smile.
"Can I have my cookies for today now?" asked Franz.
"Of course you can, but you must first figure out how many cookies you get for this 1st day," she said.
"OK, Mom. Let's see now. On the 1st day, I get 1/10th of the cookies, and since there are 30 cookies, I get 1/10th of 30 which is (1/10)30 = 30/10, or 3."
"That's right, Franz, and here they are. Hope you enjoy them!"
The next day (the 2nd day), Franz jumped out of bed early and ran to his mother, saying, "Mom, today I get more cookies than yesterday because today I get 1/9th, which is about 11% and bigger than the 1/10 or 10% of yesterday."
"Yes, my son, but remember that you get 1/9th of the remaining cookies. How many cookies are remaining?"
"Well, you baked 30 cookies and I ate 3 cookies yesterday so that means there are 27 cookies remaining today."
"That's right, Franz, so how many cookies do you get today?"
"I get 1/9th of the 27 remaining cookies, so I get 1/9th of 27 which is (1/9)27 = 27/9 = 3. So, I guess I get 3 cookies just as I did yesterday. But, Mom, I thought I would get more today than yesterday!"
His mother smiled. "Well, I guess that 1/9 is not enough bigger than 1/10 to get you any extra cookies yet. Maybe tomorrow will be better when you get 1/8 of the remaining cookies. Let's hope so, dear."
"OK, Mom, but doesn't it seem strange that I get exactly the same number of cookies today as yesterday? It would seem that I should get at least a teeny bit more than 3 cookies today if only just 1/10 or 1/3 of a cookie more."
"It would seem so, Franz, but your calculations show that you're to get exactly 3 cookies today, neither more nor less, and here they are" she said with a small laugh.
On the third day when Franz got up, he thought to himself, "Today I am to get 1/8 of the remaining cookies. Since I ate 3 cookies on the first day and three on the second day, there are now 24 cookies left, and I will get 1/8 of those remaining 24 cookies, which will be 24/8 or 3 cookies. But, wait! That's just the same number of cookies that I got on the 1st and 2nd days!" This surprised him so much that he decided to construct a table showing the number of cookies he would receive each day until they were all eaten. And here is the table he made.
Table 1. 30 cookies made
Day Number of cookies Fraction of cookies Number of cookies Total number
remaining to be eaten that day eaten that day of cookies eaten
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1 30 1/10 10% (1/10)30 = 3 3
2 30 - 3 = 27 1/9 11.1% (1/9)27 = 3 6
3 30 - 6 = 24 1/8 12.5% (1/8)24 = 3 9
4 30 - 9 = 21 1/7 14.3% (1/7)21 = 3 12
5 30 - 12 =18 1/6 16.7% (1/6)18 = 3 15
6 30 - 15 =15 1/5 20% (1/5)15 = 3 18
7 30 - 18 =12 1/4 25% (1/4)12 = 3 21
8 30 - 21 = 9 1/3 33.3% (1/3)9 = 3 24
9 30 - 24 = 6 1/2 50% (1/2)6 = 3 27
10 30 - 27 = 3 1/1 100% (1/1)3 = 3 30
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Franz could clearly see from his table that he would always get just 3 cookies a day to eat even though the fraction of cookies he ate each day increased. Of course, what he had forgotten was that the number of cookies remaing to be eaten were decreasing each day and apparently in just such a manner as to cancel out the increase in the fractional amount of cookies which he would be allowed to eat for that day. "How very interesting," he thought to himself, "I wonder why that is true?"
Later that day, he showed his table to his mother, and they both laughed at this amazing result!
"Did I fool you a little bit, Franz?" asked his mother.
"Yes, you certainly did, Mother, but it was all a lot of fun and has taught me to be very careful about assuming too much!"
All the next week as Franz munched on his 3 cookies each day, he pondered this strange occurence and wondered what would have happened if his mother had made him 24 cookies instead of 30. Of course, she couldn't have started him out on the first day with eating 1/10th of the cookies because 1/10 of 24 cookies is not a whole number of cookies. But she could start him out with eating 1/6th of the cookies on the first day, 1/5th of the remaining cookies on the second day, and so forth. He decided to make another table to show what would happen if his mother had made him 24 cookies instead of 30. And here is what he got.
Table 2. 24 cookies made
Day Fraction of cookies Number of cookies Number of cookies Total number
to be eaten that day remaining eaten that day of cookies eaten
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1 1/6 16.7% 24 (1/6)24 = 4 4
2 1/5 20% 24 - 4 = 20 (1/5)20 = 4 8
3 1/4 25% 24 - 8 = 16 (1/4)16 = 4 12
4 1/3 33% 24 - 12 = 12 (1/3)12 = 4 16
5 1/2 50% 24 - 16 = 8 (1/2) 8 = 4 20
6 1/1 100% 24 - 20 = 4 (1/1) 4 = 4 24
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As happened before, Franz noticed that again he would always get the same number of cookies to eat each day; in this case, 4 cookies each day for 6 days. To try to explain this phenomenom, he decided to use a little algebra. He let A represent the total amount of cookies made by his mother, he let 1/d represent the fraction of cookies to be eaten on the 1st day, 1/(d-1) to represent the fraction of cookies to be eaten on the 2nd day, 1/(d-2) to be the fraction of cookies eaten on the 3rd day, etc., and he let n represent the number of cookies eaten on the 1st day. Then, he reasoned as follows.
On the 1st day, there were A cookies made, he could eat 1/d of those A cookies, so he ate (1/d)A or A/d cookies. Since n represents the number of cookies eaten on the 1st day, Franz knew that
A/d = n, or A = dn (1)
On the 2nd day, there were A - n cookies remaining, he could eat 1(d-1) of these A - n cookies, so he ate
1 A - n
------- (A - n) = --------- . Replacing A by dn from (1) above, Franz got
d - 1 d - 1
dn - n n(d - 1)
= ---------- = ------------- = n, proving that he ate exactly n cookies on
d - 1 d - 1 the 2nd day.
On the 3rd day, there were A - 2n cookies remaining, he could eat 1(d-2) of these A - 2n cookies, so he ate
1 A - 2n
------- (A - 2n) = --------- . Replacing A by dn from (1) above, Franz got
d - 2 d - 2
dn - 2n n(d - 2)
= ---------- = ------------- = n, proving that he ate exactly n cookies on
d - 2 d - 2 the 3rd day.
On any day, say the kth day, there will be A - (k-1)n cookies remaining, and he can eat 1/[d-(k-1)] of these A - (k-1)n cookies, so he will eat
1 A - (k-1)n
----------- [A - (k-1)n] = ------------- cookies . Replacing A by dn from (1) above, Franz gets
d - (k-1) d - (k-1)
dn - (k-1)n n[d - (k-1)]
= --------------- = ------------- = n, proving that he will eat exactly n cookies on
d - (k-1) d - (k-1) any day, say the kth day.
Franz was very pleased to have found the answer to his dilemma, but somehow he never got around to sharing this proof with his esteemed mother. However, he did eat as many of her cookies as he could possibly get!
This story is about reducing fractions to equivalent fractions whose numerators and denominators are both smaller than the original given fraction. However, this reduction is not accomplished as it usually is done by dividing the numerator and denominator by the same number, such as
12 12/2 6
-------- = -------- = -------- ,
6 6/2 3
but by subtracting different numbers from the numerator and denominator of the fraction to so as to end up with an equivalent fraction, such as
12 12 - 8 4
-------- = -------- = -------- .
6 6 - 4 2
Note that the numerator of the fraction, 12, is twice as large as the denominator, 6, of the fraction, so it would seem reasonable to subtract twice as much from the numerator as from the denominator if we wished to reduce this fraction to an equivalent fraction by subtraction. Accordingly, we have subtracted 4 from the denominator, and thus subtracted 8 from the numerator, and did, indeed, get an equivalent fraction for 12/6 by subtraction.
In general, if a / b = n, then a = bn, (2), so the numerator of the fraction a/b is n times as large as its denominator. Thus, if we were to subtract k from the denominator of a fraction, it seems reasonable that we should subtract nk (a number n times as large as k) from the numerator of the fraction if we wished to get an equivalent fraction in reduced form by subtraction. Let's see if this is so. Subtracting k from the denominator and nk from the numerator of a fraction a/b whose value is n, we get
a - nk bn - nk n(b - k)
--------- . Replacing a by bn from (2), we get ----------- = ------------ = n, which is the value of a/b.
b - k b - k b - k
Thus, a a - nk
--- = ---------- . (3)
b b - k
Example. Since 52 / 13 = 4, then, by using (3) where a = 52, b = 13, and n = 4, we can write
52 52 - 4(1) 52 - 4(2) 52 - 4(3) 52 - 4(4)
---- = ----------- = ------------ = ------------- = ------------ = . . . ,
13 13 - 1 13 - 2 13 - 3 13 - 4
48 44 40 36
= ------- = -------- = ------ = ------- = . . .
12 11 10 9