Lesson More complicated problems on finding number of elements in finite subsets
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<H2>More complicated problems on finding number of elements in finite subsets</H2> Introductory problems on finding the number of elements of subsets of finite sets are placed in the lesson - <A HREF=http://www.algebra.com/algebra/homework/word/misc/Counting-elements-in-sub-sets-of-a-given-finite-set.lesson>Counting elements in sub-sets of a given finite set</A> Here in this lesson, you will find more complicated problems of this type. <H3>Problem 1</H3>All 30 students in a class study at least one of the two subjects History and Geography. Twice as many study History as Geography. 8 students study only Geography. Find the number of students who study both History and Geography. <B>Solution</B> <pre> Let x students study only History; y students study only Geography; z students study both subjects. Then we have these three equations x + y + z = 30 (1) x + z = 2*(y+z) (2) y = 8 (3) Substitute y= 8 from equation (3) into equations (1) and (2) x + 8 + z = 30 (1') x + z = 2*(8+z) (2') Simplify x + z = 22 (1'') x - z = 16 (2'') From equations (1'') and (2'') you get by adding 2x = 22+16 = 38, x = 38/2 = 19. From equations (1'') and (2'') you get by subtracting 2z = 22-16 = 6, z = 6/2 = 3. Thus 19 students study only History; 8 students study only Geography, and 3 students study both subjects. <U>ANSWER</U>. 3 students study both subjects. </pre> <H3>Problem 2</H3>A class of 30 students took two quizzes. Sixteen passed the first quiz and 20 passed the second. If 4 students failed both quizzes, how many passed both? <B>Solution</B> <pre> From the condition, 30-16 = 14 failed the first quiz, and 30-20 = 10 failed the second quiz, while 4 failed both quizzes. Hence, the number of those who failed at least one quiz is 14 + 10 - 4 = 20. It means that 30-20 = 10 students passed both quizzes. <U>ANSWER</U> </pre> <H3>Problem 3</H3>In a city school, 60% of students have blue eyes, 55% have dark hair, and 20% have neither blue eyes nor dark hair. How many students have blue eyes and dark hair? <B>Solution</B> <pre> Since 20% have neither blue eyes nor dark hair, we conclude that 100 -20% = 80% are those who either has blue eyes OR dark hair. So, now we have 60% of students have blue eyes n(B) = 0.6, 55% have dark hair n(D) = 0.55, and their union (B U D) is 80%: n(B U D) = 0.8. Having it, we write 0.6 + 0.55 - n(B and D) = 0.8, 0.6 + 0.55 - 0.8 = n(B and D), n(B and D) = 0.35. Therefore, the probability that a randomly selected student will have blue eyes and dark hair is P = {{{0.35/1}}} = 0.35 = 35%. <U>ANSWER</U> </pre> My other additional lessons on Miscellaneous word problems (section 3) in this site are - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Solving-problems-by-the-Backward-method.lesson>Solving problems by the Backward method</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Minimax-problems-that-to-be-solved-MENTALLY-based-on-common-sense.lesson>Minimax linear problems to solve MENTALLY based on common sense</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Solving-linear-optimization-problems-by-reduction-to-linear-function.lesson>Solving linear optimization problems without LP-method by reduction to linear function</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Solving-one-special-linear-minimax-problem-in-100-D-space.lesson>Solving one special linear minimax problem in 100-D space by the Linear Programming method</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Miscellaneous-logical-problems.lesson>Miscellaneous logical problems</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Upper-class-entertainment-Math-problems-for-all-ages.lesson>Upper class entertainment Math problems for all ages</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/OVERVIEW-of-my-additional-lessons-on-Miscellaneous-word-problems-section-3.lesson>OVERVIEW of my additional lessons on Miscellaneous word problems, section 3</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.
