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Building Bouquets - posted March 3, 2003

This is a list of most submissions receiving Gold status this week. Comments, highlighted solutions, and a list of people on the Gold and Silver lists are also available.

Showing 16 to 25 of 25 answers:

From:  Jess Hun, age 10
School:  Abraham Lincoln Elementary School, Medford, Oregon

My answer is 51 flowers in 1 bouquet or corsage.

Building Bouquets

The problem is to determine how many flowers she might have that
would meet the situations that are given.

The question is what is the smallest possible number that Jana has to
work with.  Another question is with the number of flowers determined
above, how many bouquets or corsages and of what size can Jana make.

I know that if I can make a table I could find out how many bouquets
or corsages Jana can make.

I need to find out what the smallest possible number will come up in
the three parts of the table.

(4x)+3  7   11  15  19  23  27  31  35
    39  43  47  51
(7x)+2  9   16  23  30  37  44  51  58
    65  72  79  86
(8x)+3  11  19  27  35  43  51  59  67
    75  85  93  101
#   1   2   3   4   5   6   7   8
    9   10  11  12

Now I finished my table.  The reason I did my table is that I could
see what the smallest bouquet Jana could make in all 4, 7, and 8.

3/17=21 2/51=25.5

Jana can make 51 corsages or 17 bouquets of 3 flowers each with no
left over to waste.

In order to have no waste with 7, 8, or 9 bouquets Jana would have to
have times 7, 8, and 9.

Jana would have 504 left over flowers.

My answer is 504 left over flowers and 3 or 17.

From:  McCall A., age 9
School:  Abraham Lincoln Elementary School, Medford, Oregon
The smallest number to work with is 51.
The number of bouquets would be 3 with 17 flowers or 17 bouquets with
3 flowers.

The question is what are the smallest number possible number of
flowers that Jana has to work with. The problem is to determine how
many flowers she might have that would meet the Situations that are
given.   1  2  3  4  5  6  7  8  9  10  11  12  13

(4x)+3   7  11 15 19 23 27 31 35 39 43  47  51  55

(7x)+2   9  16 23 30 37 44 51 58 65 72  79  86  93

(8x)+3   11 19 27 35 43 51 59 67 75 83  91  99  107

My answer is she has 51 flowers left over from bigger orders of
bouquets.


The question is with the number of flowers determined above, how many
bouquets or corsages and of what size can Jana make. The problem is
what number can she use to make some bouquets and not have any left
over.
I’m going to start with the number 3and 15and multiply then and see
if they make 51. 15x3=45. That is to low so now I am going to try 3
and16.16x3=48. that is to low so now I am going to try 3 and 17.
17x3=51.
My answer is she could make 17 bouquets with 3 in each or 3 bouquets
with 17 in each.

From:  Ben Lee, age 8
School:  McDonogh School, Owings Mills, Maryland
The first answer is 51, the second answer is 51 bouquets with 1 flower
in each bouquet or 1 bouquet with 51 flowers or 17 bouquets with 3
flowers in each or 3 bouquets with 17 flowers in each and the extra
answer is 504.

I got the first answer by trying every number
after 8 unless it was a multiple of 4,7 or 8.
I kept going higher and higher and then saw that
51 worked. 51 works because 4 x 12 with a remainder
of 3 equals 51 and 7 x 7 with a remainder of 2
equals 51 and 8 x 6 with a remainder of 3 also
equals 51.  I got the second answer by taking
numbers you could multiply together to get 51
with none left over so I got the answer 51 and 1 or
1 and 51. I knew that the numbers you multiply
could not be even numbers. Then I took all the
odd numbers and knew 5 could not be one of the
numbers because anything times 5 either ends
in a 5 or a 0. Then I knew it could not be 15
because it's the same rule as with the 5. I knew
it could not be 25 or higher than 25
because 25 is almost half of 51. Then I
tried 3 and 3 worked. I kept adding by 3's
and got to 51. I needed 17 3's to get to 51
so I knew that 17 x 3= 51. So the second answer
could also be 17 and 3 or 3 and 17.
Then I got the extra answer by multiplying 7 x 8 x 9
and got (7 x 8) x 9 which is 56 x 9. 56 x 9 is
54 + 450 = 504. So that's the number of leftovers
Jana needs and that's the answer!  Jana would not have
504 flowers left over unless she saved all of her extra
flowers in her business.

From:  Jessie Oehrlein, age 7
School:  St. John's Episcopal School, Oklahoma City, Oklahoma
The smallest possible number of flowers that Jana has to work with is
51.

She can make 3 bouquets of 17 flowers or 17 bouquets of 3 flowers.

No, I don't think she would ever have enough leftovers.

#1
Four is a factor of 8 and when she tries either 4 or 8 she has 3
flowers left. So, I ignored the 4.

When she tried 7, she had 2 flowers left. So the multiple of 7 had to
be one more than the multiple of 8.

I counted by 8s until I got to 48. I stopped at 48 because 49 was a
multiple of 7 and 48 was one less than 49. Then, I added 3 to 48 and
got 51. So I know she had 51 flowers to work with.

#2
I made a list of numbers and divided 51 by each number. When, I got
to 3 and I divided, I got 17. So, I made the list of numbers to 17.
When I finished divided, 3 and 17 were the only ones that went into
51 evenly. Since the results were that 3, 17,and 51 were a fact
family, Jana could have 3 bouquets of 17 flowers or 17 bouquets of 3
flowers.

Extra
I think this because the lowest number that is a multiple of 7, 8,
and 9 is 504. I know that 8 x 7  = 56. Then I multiplied 56 x 9. I
came up with 504. To check it, I did 9 x 8 = 72. Then I multiplied
72 x 7. It's answer was 504. Then I did 63 x 8. The answer was 504
again.

504 was the lowest multiple of 7, 8, and 9. I do not think that Jana
will ever have that many leftovers.

From:  Michael Do, age 10
School:  Gymea North Public School, Sydney, Australia
1.The smallest possible number of flowers that Jana has to work with
is 51
2. Jana can make 3 bouquets of 17 flowers, 17 bouquets of 3 flowers.
In theory, there can be 1 bouquet of 51 flowers or 51 corsages of 1
flower. However, in reality, she would not make these arrangements.

Let n be the number of bouquets that Jana tried with 8 flowers in
each. The number of all the leftover flowers is 8*n + 3. I have to
find the smallest n so that 8*n + 3 flowers can be arranged into some
bouquets of seven flowers with a remainder of two flowers.

The 8*n + 3 flowers can be arranged into some bouquets of seven
flowers with a remainder of three flowers, because 8*n + 3 = 4*(2n) +
3. This means that the number of bouquets of four is 2*n.

If I take 2 flowers away from 8*n + 3 flowers, I will have 8*n + 3 -
2 = 8*n + 1. These 8*n + 1 flowers must be arranged into a number of
bouquets of seven flowers without any remaining flowers. In other
words, 8*n + 1 must be divisible by 7. 8*n + 1 is the same as 7*n +
(n + 1), so n + 1 must be divisible by 7, because 7*n is already
divisible by 7. The smallest n to make n + 1 divisible by 7 must be
7 - 1 = 6. The smallest number of flowers is

8*6 + 3 = 48 + 3 = 51 flowers

Obviously this number of flowers can be arranged into 12 bouquets of
4 flowers with remaining 3 flowers, because

8*6 + 3 = 4*12 + 3

I check by subtracting 2 from this and seeing if that is divisible by
7.

51 - 2 = 49
49 = 7*7

Therefore I must be correct.

51 = 3*17 and 1*51 so Jana can make 3 bouquets of 17 flowers, 17
bouquets of 3 flowers, 1 bouquet of 51 flowers or 51 corsages of 1
flower. In theory, there can be 1 bouquet of 51 flowers or 51 corsages
of 1 flower. However, in reality, she would not make these arrangements.

Extra:

In theory, it is possible, for example she can have 7*8*9 left-over
flowers. In reality, it is unlikely that she will have so many
left-over flowers.

From:  Laura Herrle, age 8
School:  Wexford Elementary School, Wexford, Pennsylvania
1. The smallest possible number of flowers that Jana has to work
with is 51 flowers.
2. With 51 extra flowers, Jana would probably make 17 bouquets with
3 flowers each.  She could also make 3 larger bouquets with 17
flowers each.

Extra:  I do not think that Jana would ever have enough leftovers to
make bouquets of 7, 8 or 9 flowers because she would have to have
504 flowers left over.

1. From the problem, I know that if Jana makes bouquets of 4, she
has 3 extra flowers left.  If she makes bouquets of 7, she has 2
flowers left.  And if she makes bouquets of 8, she has 3 flowers
left.  I decided to make a table, solving these equations for the
different number of bouquets and see where the answers were the same.

I started solving equations for bouquets of 4.  I decided to solve 5
equations.  So, I got the answers 7, 11, 15, 19, and 23.  Then I
solved equations for bouquets of 7.  I got the answers of 9, 16, and
23.  When I got 23, and it was the same, I went right to equations
for bouquets of 8.  However, I got answers of 11, 19, and 27.  Since
it didn’t match, I went back to bouquets of 4.  I solved the next
equation for bouquets of 4 and got 27.  Even though it matched the
answer I got for bouquets of 8, I realized that since the last
answer for bouquets of 7 was 23, and all the answers increase by 7,
27 would not match for bouquets of 7 because 23 + 7 = 30.  So, I
continued and solved 5 more equations for bouquets of 4 and got the
answers 31, 35, 39, 43, and 47.  Solving for bouquets of 7, I got
30, 37, 44 and 51 for answers.  Since 51 is 4 more than 47, I solved
one more equation for bouquets of 4 and got 51 as the answer.  Then
I solved more equations for bouquets of 8 and got 35, 43 and 51 for
answers. The table is listed below.

Bouquets of 4       Bouquets of 7       Bouquets of 8
4 x 1 + 3 = 7       7 x 1 + 2 = 9       8 x 1 + 3 = 11
4 x 2 + 3 = 11      7 x 2 + 2 = 16      8 x 2 + 3 = 19
4 x 3 + 3 = 15      7 x 3 + 2 = 23      8 x 3 + 3 = 27
4 x 4 + 3 = 19      7 x 4 + 2 = 30      8 x 4 + 3 = 35
4 x 5 + 3 = 23      7 x 5 + 2 = 37      8 x 5 + 3 = 43
4 x 6 + 3 = 27      7 x 6 + 2 = 44      8 x 6 + 3 = 51
4 x 7 + 3 = 31      7 x 7 + 2 = 51
4 x 8 + 3 = 35
4 x 9 + 3 = 39
4 x 10 + 3 = 43
4 x 11 + 3 = 47
4 x 12 + 3 = 51

I found that all the equations could be solved with an answer of
51.  And 51 was the smallest number that all matched for all three
of the equations.  I double checked all my equations to make sure I
didn’t make a mistake.  Since I was positive I didn’t make a mistake
and all the answers matched, I knew that 51 was the correct answer.

2. To find out how many bouquets she could make with the leftovers,
I needed to find out what numbers 51 is divisible by.  Since it is
odd, both divisors would have to be odd, so 2 wouldn’t work.

Then I tried 3, and found that 51 / 3 = 17.

So, I thought that she would make 17 bouquets of 3 flowers each.  To
double check, I multiplied 3 x 17 = 51.

Since both 3 and 17 only have 1 and themselves as factors, they are
prime numbers, there are no other ways to group the flowers to make
bouquets.  This is the only possible solution, because the flowers
can not be grouped any other way.  There is no way to divide either
3 or 17 evenly into any smaller groups.

She could make 3 bigger bouquets of 17 flowers, but she would
probably make 17 smaller bouquets because she could probably make
more money that way by charging more per flower and still having
cheaper bouquets.

Extra:  I knew that for her to make bouquets of 7, 8 or 9 flowers,
the number of leftovers would have to be a number divisible by 7, 8
and 9.  To make that possible, 7, 8 and 9 have to all be factors.

Multiplying, 7 x 8 x 9 = 504 flowers.

This number seems too big for Jana to have as leftover flowers.  So
I don’t think that would ever happen.

I think Jana should count her leftover flowers first so she could
figure out what that number is divisible by.  Even though it might
take some time to do that, I think it would save her time because
she wouldn’t just keep making wrong guesses.

From:  Zachery Davis, age 8
School:  Gates Mills Elementary School, Gates Mills, Ohio
#1. The smallest possible number of flowers that Jana has to work with is 51.
#2. With the number of flowers determined above, Jana can make 17
bouquets of 3 or 3 bouquets of 17 each.

To solve this problem, I first had to find a number that was divisible by 4
with 3
left over and was also divisible by 7 with 2 left over and also was divisible
by
8 with 3 left over.  To do that, I made three columns. They were multiples of
4
plus 3, multiples of 7 plus 2 and multiples of 8 plus 3.  I kept doing this
until I
found  the same number in each column, and that number was 51.

4+3=7          7+2=9          8+3=11
8+3=11       14+2=16     16+3=19
12+3=15     21+2=23     24+3=27

These are just a few examples until I got to

(12x4) +3=51                (7x7)+2=51                    (6x8)+3=51
48+3=51                        49+2=51                        48+3=51


To figure out how many bouquets or corsages and of what size can Jana
make with none left over, I found out what 51 could be divided into evenly.I
knew it was not 4, 7 or 8 because:

51  4 = 12 with 3 left
51  7= 7 with 2 left
51  8 = 6 with 3 left

I tried 51  3 = 17

So she could make 3 bouquets of 17 or 17 bouquets of 3.

I had another idea of how she can make bouquets with none left over. If she
made
5 bouquets of 7,  she could then make 4 bouquets of 4 or 2 bouquets of 8 with
what's left.

5 x 7 =35       51 - 35 = 16         16  4 = 4   or     16  8 = 2


extra:  For it to not matter if she makes bouquets of 7, 8 or 9, she would
have
to have a number of flowers that was evenly divisible by 7, 8 and 9. That
number is 504. I multiplied 7x8x9.

7x8=56  56x9=504

I tried some smaller multiples of 504, but I couldn't find a smaller number
that
was divisible evenly by all three.

From:  Rachel Pay, age 10
School:  Abraham Lincoln Elementary School, Medford, Oregon
1. Jana has 51 leftovers.
2. She will be able to build 17 corsages with 3 flowers on each,
(or,3 bouquets with 17 flowers on each).

Explanation(s):
1. I had to find a common number between 8+3 and 7+2(These are the
amount of leftovers she tried in her “Leftover Bouquets” plus
the “double leftovers” that had after that),  and found 51.  I found
51 by making a chart:
    1   2   3   4   5   6   7
7+2 9   16  23  30  37  44  51 
8+3 11  19  27  35  43  51 59
Here is the process in which the chart was used: It times for
example, 7 times 3 plus 2 equals 23.
That is how the chart was used.
That is the answer to the first question.
2. On the second Question, I decided to divide to get the answer. So
first I tried dividing 4 into 51, and it didn’t work. Next I tried
three and it went into it 17 times. So there would be 17 corsages
with 3 flowers on each.

From:  Kevin Weber, age 11
School:  Howell Township Middle School North, Farmingdale, New Jersey
The smallest possible number of flowers Jana has to work with is 51
flowers. With 51 flowers, Jana can make 17 corsages with three
flowers in each or three bouquets with 17 flowers in each.

After reading this problem, I believed that it would involve many
steps.  To solve this problem I set it up as a ratio.  The top number
would be the number of bouquets or corsages.  The bottom number is
the number of flowers in each bouquet or corsage.

The first ratio was for four flowers in each bouquet or corsage.
The first bouquet or corsage would have seven flowers because Jana
made them with four flowers in each and had three flowers left over.
For the second bouquet or corsage, she would have 11 flowers. Two
bouquets or corsages with four flowers in each and 3 extra.  For
three bouquets or corsages, she would need fifteen flowers; three
bouquets or corsages with four flowers in each and 3 flowers left
over. I then noticed a pattern, add four flowers each time you add
another bouquet or corsage.  I figured out how many flowers would be
in each bouquet or corsage until I reached ten bouquets or corsages.

The second ration was in the same setup, except for seven flowers in
each bouquet or corsage.  The first would have 9 flowers; 7 for the
bouquet or corsage and then 2 left over. When two bouquets or
corsages are made, you need 16 flowers; 2 bouquets equal 14 flowers
with two left over.  To make 3 bouquets or corsages, 23 flowers are
needed; 7 flowers equals 21 plus 2 left over.  I noticed a pattern in
this ratio also; add seven flowers every time you add another bouquet
or corsage.  I then went up to ten bouquets or corsages to see how
many flowers were needed.

To figure out how many flowers Jana would need in each bouquet or
corsage, I followed the same procedure as the other two ratios.  In
each bouquet or corsage, she used 8 flowers and had three left over.
For the first bouquet or corsage, she needed 11 flowers; 8 for the
bouquet and 3 left over.  For two bouquets, she would need 19
flowers; 2 times 8 equals 16 flowers plus 3 left over. If she decided
to make 3 bouquets, Jana would need 27 flowers; 3 times 8 equals 24
plus 3 left over.  I calculated all these numbers until I reached how
many flowers she would need in ten bouquets or corsages.

After the ratios were complete, the numbers 11, 19, 23, 43, and 51
were in at least two ratios.  The only number that might work is 51
because it is the only number in which the remaining ratio has not
yet been calculated for.  The ratio missing 51 was bouquets with 4
flowers in each.  I continued that ratio to see if it would hit the
number 51 and it did.

Therefore, 51 flowers is the least amount of flowers Jana can use.
If she uses them in bouquets or corsages with four flowers in each
she will make 12 bouquets or corsages; 4 times 12 equals 48 plus 3
left over.  If Jana makes bouquets or corsages with seven flowers in
each she will make seven of them; 7 times 7 equals 49 plus 2 left
over.  If Jana uses eight flowers in each bouquet or corsage she can
make 6 of them, 8 times 6 equals 48 with 3 left over.  Jana can also
make three bouquets with 17 flowers in each or 17 corsages with 3
flowers in each.

Extra: If Jana decides she wants to make bouquets or corsages out of
I think she might be able to build bouquets or corsages out of 7, 8,
or 9 flowers. If Jana ever gets 504 leftover flowers, she can make
bouquets or corsages out of 7, 8, or 9 flowers. The only thing is,
this is alot of leftove flowers. Therefore, it may be hard for Jana t
have that many leftover flowers but if she does have 504 leftover
flowers, Jana will be able to make bouquets or corsages out of 7, 8,
or 9 flowers.If she makes bouquets or corsages out of 7 flowers, she
can make 72 of them. If she decides to make bouquets or corsages out
of 8 flowers, she can make 63 of them. If she decides to make
bouquets or corsages out of 9 flowers, she can make 56 pf them. This
is what Jana can do with enough leftovers.




From:  Sheldon Nguyen, age 11
School:  Howell Township Middle School North, Farmingdale, New Jersey
1) The smallest # of flowers Jana has to work with is 51.
2) With 51 flowers, Jana can make 17 bouquets with 3 flowers each, or
3 bouquets with 17 flowers each without wasting a single flower.

    To find the solution I first needed to figure out the equation. I
knew that there must be 3 different equations because Jana wanted to
try out different-sized bouquets: bouquets with 4 flowers each, with
7 flowers in each, and with 8 flowers in each. Thus, the equations
are the following:
4 flowers x # bouquets completed + 3 remaining flowers = X flowers
7 flowers x # bouquets completed + 2 remaining flowers = X flowers
8 flowers x # bouquets completed + 3 remaining flowers = X flowers

Next, I created 3 separate tables for each size of bouquet to find a
common number for all three categories. The tables look like this:

                   BOUQUETS WITH 4 FLOWERS EACH
# bouquets completed 1  2  3  4  5  6  7  8  9  10  11  12  13
      X              7  11 15 19 23 27 31 35 39 43  47  51  55


                   BOUQUETS WITH 7 FLOWERS EACH
# bouquets completed 1  2  3  4  5  6  7  8  9  10  11  12  13
      X              9  16 23 30 37 44 51 58 65 72  79  86  93


                   BOUQUETS WITH 8 FLOWERS EACH
# bouquets completed 1  2  3  4  5  6  7  8  9  10  11  12  13
      X              11 19 27 35 43 51 59 67 75 83  91  99 107

I knew that in order to determine the smallest possible number of
flowers that Jana has to work with, I needed to find the smallest
number that matched in all 3 tables. I kept filling in the tables
until I discovered a match, which was the number 51. This number
occurred when 12 bouquets of 4 flowers each were completed, when 7
bouquets of 7 flowers were completed,  and when 6 bouquets of 8
flowers were completed. Thus, I arrived at the above the solution,
which fit into the equations.

The above solution only applies to the equations with leftover
flowers, however, since Jana did not want to waste any flowers, the
best way is for her to create 17 bouquets with 3 flowers each (17 x 3
= 51), or  3 bouquets with 17 flowers each ( 3 x 17 = 51).




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