Class Summary - 10/23/16

Instructors:	Mr. Rameel Rizvi
Attendance:	Ayyan, Aiden, Omar, Areej, AbdulRafay,
Homework:	Read: Chapter 4.9 (this is a review of all main concepts from the chapter--like last time, make sure to do this slowly and thoroughly! While memorizing these concepts is important, make sure you also understand each and every one of them so that you can actually apply them in the future (which is, after all, the goal of education)); Problems to solve:  4.55 (parts (b),(d),(f),(h),(j),(l)), 4.56, 4.58, 4.66 (all parts), 4.67, 4.74, 4.82 (all parts), 4.86, 4.87, 4.88 (all parts), 4.89, 4.90 (part (f) only) Note: Again, start early. These shouldn't be too difficult but there is a lot of computational effort involved.
Challenge Problem:	Jar A contains four liters of a solution that is 45% acid. Jar B contains five liters of a solution that is 48% acid. Jar C contains one liter of a solution that is k% acid. From jar C, m/n liters of the solution is added to jar A, and the remainder of the solution in jar C is added to jar B. At the end both jar A and jar B contain solutions that are 50% acid. Given that m and n are relatively prime positive integers, find k + m + n. (Source: AIME I 2011)
Class Activity:	Finished Chapter 4.
Concepts:	Introduced fractions, values of the form "a over b" where "over" indicates a horizontal bar with numerator a immediately above and denominator b immediately below. Said that fractions are the same values we obtain when we divide a by b, and we already covered the concept of division, all of whose rules apply to fractions by definition. For example, reciprocals and exponents behave exactly the same (as they should!) when dealing with fractions as they do when dealing with division.  Showed how the concepts of GCD and LCM we learned previously apply to fractions: GCD negation is used in obtaining simplest form fractions (i.e. coprime, or relatively prime, numerals as the numerator and denominator), and LCM is used in adding/subtracting fractions with unequal denominators (by multiplying each such fraction by some other fraction k/k (unique for each individual fraction) such that the denominators of the resulting fractions are all the LCM of the denominators of the original set of fractions). Also covered mixed numbers, which are, basically, the division-with-remainder representation of a fraction with numerator greater than its denominator. Hence the main left-hand-side numeral of a mixed number corresponds to how many "whole" times the denominator of the fraction in question goes into the numerator, and the fraction on the right of this larger numeral corresponds to the remainder. Recall, for integers m,n with m >= n we can write m = q*n + r for some integers q,r where r < n. In this representation q corresponds to the main numeral of the mixed fraction, and r corresponds to the remainder (in the mixed number this is the right-hand-side fraction with numerator r and denominator n). Arithmetic with mixed vs. non-mixed numbers is, of course, equivalent; it's just a matter of notation.
Student Difficulties:	Glad to see the obvious increase in effort placed in homework! Everyone should keep up this level of effort. In particular, ensure each week that you remember and (more importantly) understand all the concepts previously covered in the course. Everything builds on things we've already covered, and this mirrors the development of mathematics in general. Failure to grasp something now will only begin a buildup of confusion that will surely grow as we progress. Oh, and one more thing: make sure for multi-part homework problems that you do every part that has been assigned. For some reason many of you randomly skipped a part of some multi-part question in the previous assignment--I'm assuming this was by accident, so just remember to double-check that you've done all the assigned parts in the future.
Notes	Please hand-in your homework solutions, written in a consistent notebook, in the next class.