Lesson Math Olympiad level problem on caught fishes
Algebra
->
Customizable Word Problem Solvers
->
Misc
-> Lesson Math Olympiad level problem on caught fishes
Log On
Ad:
Over 600 Algebra Word Problems at edhelper.com
Word Problems: Miscellaneous Word Problems
Word
Solvers
Solvers
Lessons
Lessons
Answers archive
Answers
Source code of 'Math Olympiad level problem on caught fishes'
This Lesson (Math Olympiad level problem on caught fishes)
was created by by
ikleyn(52779)
:
View Source
,
Show
About ikleyn
:
<H2>Math Olympiad level problem on caught fishes</H2> <H3>Problem 1</H3>One day Vani caught fishes of weight 100 kg. The total weight of 3 largest fishes is 35 kg , and total weight of 3 smallest fishes is 25 kg. How many fishes Vani caught in total ? <B>Solution</B> <pre> Let the 3 smallest fishes weight a, b and c kilograms. Then a + b + c = 25 kg (1) Notice, that from equation (1), the weight at least one of the fishes {a,b,c} <U>MUST BE MORE than</U> 8 kilograms (otherwise, their total weight would be not more than 24 kg). Let call this fish M8 ("more than 8 kilograms"): M8 > 8 kg. <<<---=== note the strong inequality (!) Let the 3 largest fishes weight x, y and z kilograms. Then x + y + z = 35 kg (2) The sum (1) plus sum (2) is 25 + 35 = 60 kilograms, which is 40 kilograms less than 100 kilograms. Hence, it should be at least 4 or more other fishes, distinct from a, b, c, x, y and z, to balance this difference of 40 kilograms. Now, if the set of these distinct fishes contain 5 or more fishes, then the weight of at least one of these 5 or more distinct fishes must be <U>LESS than OR equal</U> to 8 kilograms (otherwise, the total weight of these 5 distinct fishes would be more than 40 kilograms). Let call this fish L8 ("less or equal to 8 kilograms"): L8 <= 8. Then, replacing M8 by L8 in the set {a,b,c}, we would obtain the new set of 3 fishes weighing in total LESS THAN 25 kilograms. It gives us a CONTRADICTION with the statement that the three smallest fishes weight 25 kilograms. The contradiction means that the set of distinct fishes consists of EXACTLY 4 fishes. Hence, the total number of fishes is 3 + 4 + 3 = 10. <U>ANSWER</U> </pre> My other additional lessons on Miscellaneous word problems in this site are - <A HREF=https://www.algebra.com/algebra/homework/word/misc/I-do-not-have-enough-savings-now.lesson>I do not have enough savings now</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/In-a-jar-all-but-6-are-red-marbles.lesson>In a jar, all but 6 are red marbles</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/How-many-boys-and-how-many-girls-are-there-in-a-family.lesson>How many boys and how many girls are there in a family ?</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/What-is--the-last-digit-of-the-number-a%5En-.lesson>What is the last digit of the number a^n ?</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Find-the-last-three-digits-of-these-numbers-.lesson>Find the last three digits of these numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/What-are-the-two-last-digits-of-the-number-3%5E123%2B7%5E123%2B9%5E123.lesson>What are the last two digits of the number 3^123 + 7^123 + 9^123 ?</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Really-complicated-logic-problem.lesson>Advanced logical problems</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Prove-that-if-a-b-and-c-are-the-sides-of-a-triangle-.lesson>Prove that if a, b, and c are the sides of a triangle, then so are sqrt(a), sqrt(b) and sqrt(c)</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Calculus-optimization-problems-for-shapes-in-2D-plane.lesson>Calculus optimization problems for shapes in 2D plane</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Calculus-optimization-problems.lesson>Calculus optimization problems for 3D shapes</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Solving-some-linear-minimax-problems-in-3D-space.lesson>Solving some linear minimax problems in 3D space</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Solving-one-non-linear-minimax-problem-in-3D-space.lesson>Solving one non-linear minimax problems in 3D space</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Solving-linear-minimax-problem-in-three-unknowns-by-the-simplex-method.lesson>Solving linear minimax problem in three unknowns by the simplex method</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/The-pigeonhole-principle-problems.lesson>The "pigeonhole principle" problems</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/In-the-worst-case.lesson>In the worst case</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Page-numbers-on-the-left-and-right-facing-pages-of-an-opened-book.lesson>Page numbers on the left and right facing pages of an opened book</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Selected-problems-on-counting-elements-in-subsets-of-a-given-finite-set.lesson>Selected problems on counting elements in subsets of a given finite set</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/How-many-numbers-from-1-to-300-are-divisible-by-X.lesson>How many integer numbers in the range 1-300 are divisible by at least one of the integers 4, 6 and 15 ?</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Nice-problems-to-setup-them-using-Venn-diagram.lesson>Nice problems to setup them using Venn diagram</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Wrapping-a-gift.lesson>Wrapping a gift</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/In-preparation-for-Halloween-.lesson>In preparation for Halloween</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Nice-entertaiment-problems-related-to-divisibility-numbers.lesson>Nice entertainment problems related to divisibility property</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Stars-and-bars-method-for-Combinatorics-problems.lesson>Stars and bars method for Combinatorics problems</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Olimpiad-level-Math-problem-on-caves-and-bats.lesson>Math Olympiad level problem on caves and bats</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Math-Olympiad-level-problem-on-pigeonhole-principle.lesson>Math Olympiad level problem on pigeonhole principle</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Math-Olympiad-level-problem-on-placing-books-in-bookcase.lesson>Math Olympiad level problem on placing books in bookcase</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/OVERVIEW-of-additional-lessons-on-Miscellaneous-word-problems.lesson>OVERVIEW of additional lessons on Miscellaneous word problems</A> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-I. Use this file/link <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-II.
