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### Lucky Seven Fractions Puzzle

```
Date: 12/22/2001 at 16:36:11
From: Victor Hugo
Subject: Maths lucky seven problem

Put each number 1 to 9 in the X in order to make the equation correct.

XX / XXX + XX / XX = 7
```

```
Date: 12/23/2001 at 02:48:27
From: Doctor Greenie
Subject: Re: Maths lucky seven problem

Hello, Victor -

What a great problem! I got a lot of good exercise in reasoning and
mental math figuring this one out.

We have

XX   XX
--- + -- = 7
XXX   XX

where each of the X's represents a different digit from 1 through 9.

To start with, notice that the first fraction has a 2-digit numerator
and a 3-digit denominator, so that the first fraction is less than 1;
with the sum of the two fractions being 7, that means the second
fraction must be more than 6.

Then, if the fraction XX/XX, where numerator and denominator are
2-digit numbers, is greater than 6, then the denominator must be at
most 16. So the possible denominators are 12, 13, 14, 15, and 16.

If the denominator of the second fraction is 12, then, for the
fraction to have a value more than 6 but less than 7, the possible
numerators are 73 to 83. Similarly,

if the denominator is 13, the possible numerators are 79 to 90;
if the denominator is 14, the possible numerators are 85 to 97;
if the denominator is 15, the possible numerators are 91 to 98; and
if the denominator is 16, the possible numerators are 97 to 98.

Note that the possible numerators for the last two cases are limited
by the fact that they must be 2-digit numbers; and furthermore, of
course, some of the possible numerators for each possible denominator
are excluded by the requirement that all the digits in the problem be
different.

Having narrowed the possible numerator/denominator combinations by
rejecting those that have repeated digits, most of the remaining
possible numerators can quickly be rejected because of the value that
would then be required for the first fraction. For example, if the
second fraction were 73/12, then the first fraction XX/XXX would have
to have the value 11/12, using the digits 4, 5, 6, 8, and 9. The
smallest possible 3-digit denominator using these 5 digits is 456, and
you can't get a fraction whose value is 11/12 having a 2-digit
numerator and denominator 456.

We can also reject most of the possibilities with denominator 15 (in
fact, as it turns out, all of them) because, if the second fraction is
XX/15, then the first fraction XX/XXX must reduce to a fraction ??/15.
Since 0 is not one of the digits we can use, and since the digit 5 is
used in the denominator of the second fraction, the first fraction
must reduce to something that does not have denominator 5 or 15 - so
it must reduce to either 1/3 or 2/3. But 1/3 = 5/15 and 2/3 = 10/15;
then, in either case, the numerator of the first fraction would have
to be a multiple of 5.

Following is a table of all the possibilities for the second fraction,
with the resulting requirements for the first fraction. The numbers in
the last column of the table refer to notes as to whether the search
for a solution to the problem needs to be continued for the given
second fraction. The notes are as follows:

(1) these possibilities can be rejected, because it is not possible to
form a first fraction, using the remaining digits, which has the
required value
(2) these possibilities can be rejected, because the numerator of the
first fraction would have to be a multiple of 5
(3) these possibilities remain to be investigated

then the first    and would have
fraction would     to be formed
if the second      have to be        using the
fraction is     equivalent to    remaining digits      notes
--------------------------------------------------------------------
73/12            11/12            45689              (1)
74/12            10/12            35689              (1)
75/12             9/12            34689              (1)
76/12             8/12            34589              (1)
78/12             6/12            34569              (1)
79/12             5/12            34589              (1)
83/12             1/12            45679              (3)

79/13            12/13            24568              (1)
82/13             9/13            45679              (1)
84/13             7/13            25679              (1)
85/13             6/13            24679              (1)
86/13             5/13            24579              (3)
87/13             4/13            24569              (3)
89/13             2/13            24567              (3)

85/14            13/14            23679              (1)
86/14            12/14            23579              (1)
87/14            11/14            23569              (1)
89/14             9/14            23567              (1)
92/14             6/14            35678              (1)
93/14             5/14            25678              (1)
95/14             3/14            23678              (3)
96/14             2/14            23578              (3)
97/14             1/14            23568              (3)

92/15            13/15            34678              (1),(2)
93/15            12/15            24678              (1),(2)
94/15            11/15            23678              (1),(2)
96/15             9/15            23478              (1),(2)
97/15             8/15            23468              (1),(2)
98/15             7/15            23467              (1),(2)

97/16            15/16            23458              (1)
98/16            14/16            23457              (1)

So we are left with only a very few possibilities for the second
fraction:

83/12; 86/13, 87/13, 89/13; 95/14, 96/14, 97/14

Note that none of these is reducible; so the fully reduced forms for
the corresponding first fractions must be

1/12; 5/13, 4/13, 2/13; 3/14, 2/14, 1/14

From here, the solution seems to require just a lot of work trying the
different possibilities. My approach was to find the possible 3-digit
denominators that could reduce to the required denominator and hope
the remaining 2 digits can make the required numerator.

For example, with the possible second fraction of 83/12, a fraction
equivalent to 1/12 must be made using the digits 45679. Using
divisibility rules, a number divisible by 12 must be divisible by 3
and by 4; so the sum of its digits must be divisible by 3 and the last
two digits must be divisible by 4. Using the digits 45679, the
possible denominators that satisfy those requirements are 456, 756,
576, and 564. The fractions equivalent to 1/12 with these denominators
are

1    38    63    48    47
-- = --- = --- = --- = ---
12   456   756   576   564

All of the numerators of these possible first fractions repeat a digit
that is either in the second fraction 83/12 or in the denominator of
the first fraction, so there is no solution with the second fraction
83/12.

I won't go through the details for all the other possible second
fractions. (You might want to, since I am doing this late at night and
have not gone back and checked all my work.) My work revealed a single

86    95
-- + --- = 7
13   247

I hope all this helps. Write back if you have any further questions or

- Doctor Greenie, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Logic
High School Puzzles
Middle School Logic
Middle School Puzzles

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