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Ducky Promenade - posted May 27, 2002
Three big ducks went out one day
With two ducklings behind.
Fourteen kilograms was their weight
When all five were combined.
Add one duck and one duckling
And their weight increased.
Nineteen kilograms was the total
Of the seven with webbed feet.
If all ducklings have one weight,
and ducks all have another,
Then you should find the total weight
of three - one child, two mothers.
The idea for this problem was suggested by this week's
mentors.
Meet the mentors of this puzzle:
Western Oregon University Spring 2002
Comments
There were many different methods that you could use to solve this
problem. Most people either set up a set of "simultaneous equations"
and solved them in different ways using algebra or used guess-and-
check. There were 132 submissions this week, 66 (50%) received credit.
Of those submissions that did not receive credit, the majority (63.6%)
of them had the correct answer, but did not include an explanation that
had enough detail.
Michael Levitin from St. Margaret's Episcopal School solved this in two
ways. The first method that he used involved simultaneous equations.
"Simultaneous equations" is a fancy name for finding two unknowns (or
variables) in two equations. Michael used the addition/subtraction
method of solving these equations. First, he used the information
given to write two equations. He knew that three mother ducks and two
ducklings weigh 14 kg and that one mother and one baby weigh 5 kg. By
subtracting these two equations, you are left with the weight of two
mother ducks and one duckling, which is what the problem asks for. He
was able to answer the question without actually finding out how much
an individual mother duck and baby duck weigh. What a creative way to
solve this problem! Julia Xu also used a method similar to this.
Michael also shows a guess-and-check method. First he wrote down the
information that he knew about the problem. Then he reasoned that he
only had two choices for the answer. He tried both of them, one
worked, and that was his answer. This is a great example of how to use
guess-and-check efficiently.
Andrew Ko from Palmyra Area Middle School used simultaneous equations,
but used the substitution method of solving them. Andrew wrote two
equations that described the information in the problem. He solved one
of them for x (the weight of a mother duck), and plugged his answer
into the second equation. This gave him a numerical value for y (the
weight of a baby duckling), then all he had to do was to plug his
answer for y into the first equation to get a numerical value for x.
This was how he was able to find the weights of the ducks and
ducklings. Ning-Jiun Jan used simultaneous equations also, but solved
them in a different way, using the addition/subtraction method. Using
algebra to solve this problem, there were many methods you could use,
the most common being Andrew’s method.
Highlighted solutions:
| From: |
Michael Levitin, age 8 |
| School: |
St. Margaret's Episcopal School, San Juan Capistrano, CA |
|
The total weight of three – one child, two mothers is 9 kg.
1. Let B be weight of a big duck and D weight of the duckling. Then
combined weight of “three big ducks with two ducklings” is 3 * B + 2
* D = 14 kg.
2. If we “add one duck and one ducklings” weight of duck’s group will
increase to a “nineteen kilograms … total”. That is an increase of
19 – 14 = 5 kg, so B + D = 5 kg.
3. If we subtract the second equation from the first we will find the
total weight of three – one child, two mothers:
(3 * B + 2 * D) – (B + D) = 14 – 5 or
3 * B + 2 * D – B + D = 9 or
2 * B + D = 9 kg
There is a way to solve the problem without using algebra.
1. Weights in the riddle are in whole kilograms, so it is logical to
assume that Mother Duck and her Duckling weights are in whole
kilograms too.
2. When Mother Duck and her Duckling joined the group the total
weight increased to 19 kg from 14 kg or by 19 – 14 = 5 kg. Because
Mother Duck normally weights more then the Duckling Mother Duck +
Duckling weight is either 3 kg + 2 kg = 5 kg or 4 kg + 1 kg = 5 kg.
3. If Mother Duck weight is 3 kg and Duckling weight is 2 kg
then “Three big ducks … with two ducklings” will weight 3 * 3 + 2 * 2
= 13 kg. But there weight must be 14 kg so our assumption is wrong.
4. If Mother Duck weight is 4 kg and Duckling weight is 1 kg
then “Three big ducks … with tow ducklings” will weight 3 * 4 + 2 * 1
= 14 kg. Bingo!
5. So the “total weight of three – one child, two mothers” is 1 * 1 +
2 * 4 = 9 kg.
| From: |
Andrew Ko, age 13 |
| School: |
Palmyra Area Middle School, Palmyra, PA |
|
Solution:
The total weight of two mother ducks and one duckling is 9 kilograms.
Explanation:
Let x kilograms be the weight of a mother duck and y kilograms be the
weight of a duckling. Since the total weight of three mother ducks
and two ducklings is 14 kilograms, I will set up the first equation.
3x + 2y = 14 (1)
By adding one mother duck and one duckling, the total weight of the
seven is 19 kilograms. Therefore, the weight of the extra mother
duck and the extra duckling must be the difference between the total
weights, i.e. 19 - 14 or 5 kilograms. So, the second equation is:
x + y = 5 (2)
I will solve for x in equation (2).
x = 5 - y I subtract y from each side.
Now I substitute 5 - y for x in equation (1).
3(5 - y) + 2y = 14 I use Distributive Property.
15 - 3y + 2y = 14 I combine like terms on the left side.
15 - y = 14 I subtract 15 from each side.
-y = -1 I divide each side by -1.
y = 1
Next I substitute 1 for y into equation (2) to solve for x.
x = 5 - y
x = 5 - 1
x = 4
So, the weight of a mother duck is 4 kilograms and that of a ducking
is 1 kilogram.
Total weight of two mother ducks and one duckling
= 2 * 4 + 1 * 1
= 8 + 1
= 9
Therefore, the total weight of two mother ducks and one duckling is 9
kilograms.
| From: |
Julia Xu, age 12 |
| School: |
Southern Hills Middle School, Boulder, CO |
|
The total weight of two mother ducks and one duckling is 9 kilograms.
x = mother ducks
y = ducklings
3x + 2y = 14 kg or, 3 mother ducks + 2 ducklings = 14kg
Add one x, mother duck, and add one y, duckling, to get:
4x + 3y = 19 kg or, 4 mother ducks and 3 ducklings = 19 kg
This means that the mother duck and a duckling added
together weigh 5 kg. (19 kg - 14 kg)
3x + 2y 3 mothers and 2 ducklings(weight)
- x + y - 1 mother and 1 duckling (weight)
________ ___________________________
2x + y = weight of 2 mother ducks and one duckling.
Now, plug in the numbers from the previous information into the
problem.
(3x + 2y = 14 kg)
(x + y = 5 kg)
14 kg weight of 3 mothers and 2 ducklings
- 5 kg weight of 1 mother and 1 duckling
______ _________________________________________
9 kg = weight of 2 mother ducks and one duckling.
| From: |
Ning-Jiun Jan, age 13 |
| School: |
Howell Township Middle School North, Howell, NJ |
|
The weight of one child and two mothers is 9 kgs.
1st - I write out two equations from the information given. (D = big
duck and d = duckling)
3D + 2d = 14
4D + 3d = 19
2nd – I form a 3rd equation by subtracting to get what D + d =
4D + 3d = 19 - 3D + 2d = 14 = D + d = 5
3rd – Solve for d in one equation to get D in another equation.
D + d = 5
d = 5-D
3D + 2(5-D) = 14
3D + 10-2D = 14
D + 10 = 14
D = 4
4th – Use this information to solve for d
3(4) + 2d = 14
12 + 2d = 14
2d = 2
d = 1
5th – Use what you solved for to find what 1 d (child) and 2 big
ducks (mothers).
If d = 1 and D = 4, then
1d + 2D = x
1(1) + 2(4) = x
1 + 8 = x
9 = x
I have solved the problem. I child and two mothers weigh 9 kilograms.
68 students received credit this week.
Alin Apostolescu, age 12 - School No.12, Bucharest, Romania
Craig Ashinsky, age 12 - New Providence Middle School, New Providence, NJ
Kara B, age 13 - Nathan Hale Middle School, Norwalk, CT
Mitchell Berkowitz, age 13 - Howell Township Middle School North, Howell, NJ
Bogdan Bintu, age 12 - School nr. 8 Onesti, Onesti, Romania
Cory Bovenzi, age 13 - Howell Township Middle School North, Howell, NJ
Stephen C, age 13 - Nathan Hale Middle School, Norwalk, CT
Erica Che, age 11 - William Annin Middle School, Basking Ridge, NJ
Monica Chong, age 14 - Nanyang Girls High School, Singapore, Singapore
Vivian Chui, age 12 - Sperling Elementary School, Burnaby, British Columbia, Canada
Evan Clark, age 11 - Davis Waldorf School, Davis, CA
Kristin Cutrona, age 12 - Hopkinton Middle School, Hopkinton, MA
Marina D'Angelo, age 16 - Islands International School, Buenos Aires, Argentina
Nathan De Ruvo, age 13 - Nathan Hale Middle School, Norwalk, CT
Kyle Desjardins, age 14 - Brookhurst Junior High School, Anaheim, CA
Ashley Doucette, age 13 - Howell Township Middle School North, Howell, NJ
Angelica English, age 13 - J. S. Russell Junior High School, Lawrenceville, VA
Jason F, age 12 - Nathan Hale Middle School, Norwalk, CT
Anthony Garcia, age 14 - Chaparral Middle School, Diamond Bar, CA
Hermina Ghenu, age 14 - University of Toronto Schools, Toronto, Canada
Kaylin Gosack, age 13 - Howell Township Middle School North, Howell, NJ
Melissa H, age 13 - Nathan Hale Middle School, Norwalk, CT
Brian H., age 13 - Nathan Hale Middle School, Norwalk, CT
Ping He, age 12 - Wagner Middle School, New York, NY
Siu Ho Chung, age 13 - Queen's College, Hk Sar, China
Alyssa Imperial, age 14 - Chaparral Middle School, Diamond Bar, CA
Ning-Jiun Jan, age 13 - Howell Township Middle School North, Howell, NJ
Helen Joo, age 12 - Highland Junior High School, Toronto, Ontario, Canada
Kelsey K., age 11 - Nathan Hale Middle School, Norwalk, CT
Bill Kinslow, age 13 - West Essex Junior High School, North Caldwell, NJ
Andrew Knopp, age 13 - Nathan Hale Middle School, Norwalk, CT
Andrew Ko, age 13 - Palmyra Area Middle School, Palmyra, PA
Rusi Kolev, age 15 - Asen Zlatarov, Haskovo, Bulgaria
Elena Kwan, age 13 - King Edward Public School, Toronto, Ontario, Canada
Brad LaPoff, age 13 - West Essex Junior High School, North Caldwell, NJ
Andrei Lazanu, age 13 - School No. 205, Bucharest, Romania
Xin Yi Lee, age 14 - Nanyang Girls High School, Singapore, Singapore
Kelly Lemon, age 14 - Chaparral Middle School, Diamond Bar, CA
Michael Levitin, age 8 - St. Margaret's Episcopal School, San Juan Capistrano, CA
Bernie Liu, age 11 - Snyder Middle School, Bensalem, PA
David M, age 12 - Nathan Hale Middle School, Norwalk, CT
Justin Mar, age 13 - Chaparral Middle School, Diamond Bar, CA
Jessica Martino, age 13 - Howell Township Middle School North, Howell, NJ
Lizzy Mills, age 10 - National Cathedral School-Girls, Washington, DC
Sam N, age 13 - Nathan Hale Middle School, Norwalk, CT
K O, age 14 - District, Dundas, Canada
Chris P., age 12 - Nathan Hale Middle School, Norwalk, CT
Ronak Patel, age 13 - Chaparral Middle School, Diamond Bar, CA
Amanda Pickford, age 13 - J. S. Russell Junior High School, Lawrenceville, VA
Doug S, age 13 - Nathan Hale Middle School, Norwalk, CT
Laz S, age 13 - Nathan Hale Middle School, Norwalk, CT
Sathya Sankaran, age 11 - Whitney Elementary School, Strongsville, OH
Jessica Smith, age 13 - Chaparral Middle School, Diamond Bar, CA
Genia Sokolskaya, age 10 - Penn Valley Elementary School, Narberth, PA
Sashe Sokolsky, age 14 - Welsh Valley Middle School, Narberth, PA
Leo Sprinzen, age 13 - Far Brook School, Short Hills, NJ
Rose Swift, age 14 - Harvest Park Middle School, Pleasanton, CA
Willie Sze, age 13 - Howell Township Middle School North, Howell, NJ
Sarah Tops, age 13 - George MacDougal High School, Airdrie, Alberta, Canada
Justin V, age 13 - Nathan Hale Middle School, Norwalk, CT
Jennifer Wagner, age 14 - Palmyra Area Middle School, Palmyra, PA
Jessica Wagner, age 11 - Paringa Park Primary School, Adelaide, Australia
Trevon Williams, age 13 - Chaparral Middle School, Diamond Bar, CA
Sean Woltering, age 13 - Howell Township Middle School North, Howell, NJ
Jeffrey Wu, age 13 - Chaparral Middle School, Diamond Bar, CA
Julia Xu, age 12 - Southern Hills Middle School, Boulder, CO
Lynn Yi, age 11 - P.S. 101, New York, NY
Karla Zerehi, age 13 - Chaparral Middle School, Diamond Bar, CA
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