Lesson Virtual exchange between two persons
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<H2>Virtual exchange between two persons</H2> <H3>Problem 1</H3>John and Marie went to collect mangoes. At the end of the trip John said “Marie, give me 7 of yours and I will have two times what you have”. Marie replied, “No John, you give me seven of yours and we will have the same amount”. Find the number of mangoes that each one collected. <B>Solution</B> Let <B>x</B> be the number of mangoes that John collected, and let <B>y</B> be the number of mangoes that Marie collected. If Marie gives 7 mangoes to John, then Jon will have (x+7) mangoes and Marie will have (y-7) mangoes. According to the condition, x + 7 = 2(y-7). In opposite, if John gives 7 mangoes to Marie, then Marie will have (y+7) mangoes and John will have (x-7) mangoes. Again, according to the second condition, y + 7 = x - 7. Thus you get the system of two linear equations in two unknowns {{{system(x + 7 = 2(y - 7), y + 7 = x - 7)}}}. To solve it, express <B>y</B> from the second equation y = x - 14 and substitute it to the first equation x + 7 = 2((x-14) - 7), x + 7 = 2x - 28 - 14, x = 7 + 14 + 28 = 49. Thus John collected 49 mangoes. Marie collected y = x - 14 = 49 - 14 = 35 mangoes. <B>Answer</B>. John collected 49 mangoes. Marie collected 35 mangoes. <H3>Problem 2</H3>There is some money with Arun and some money with Kiran. If Arun gives Rs 60 to Kiran, then the amounts with them would be equal. If instead Kiran gives Rs 20 to Arun, then Arun would have Rs 160 less than twice the amount with Kiran. Find the amount of Kiran. <B>Solution</B> <pre> The given part translates to these two equations A - 60 = K + 60, (1) 2*(K - 20) = A + 20 + 160 (2) Simplify and write in standard form A = K + 120 2K - 40 = A + 180, or A = K + 120 (3) 2K = A + 220, (4) Next substitute (3) into (4) to get 2K = (K+120) + 220 ====> K = 340. <U>Answer</U>. Kiran amount is Rs 340. <U>Check</U>. Then Arun's amount is A = K + 120 = 340 + 120 = 460. Then (1) takes the form 460-60 = 400 = 340+60 ! Correct ! and (2) takes the form 2*(340-20) = 640 = 460+180 ! Correct ! </pre> <H3>Problem 3</H3>Julian and Patrick had some marbles. If Julian gave Patrick 20 marbles, both of them would have an equal number of marbles. If Patrick gave Julian 20 marbles, Julian would have three times as many marbles as Patrick. Find the number of marbles that each of them had. <B>Solution</B> <pre> From the condition, you have these two equations (1) J - 20 = P + 20 ("If Julian gave Patrick 20 marbles, both of them would have an equal number of marbles") (2) 3*(P - 20) = J + 20 ("If Patrick gave Julian 20 marbles, Julian would have three times as many marbles as Patrick") From eq(1), express J = P+40 and substitute it into eq(2). You will get 3*(P-20) = (P+40) + 20, or 3P - 60 = P + 60, 3P - P = 60 + 60 2P = 120 ====> P = 120/2 = 60. <U>Answer</U>. Patrick had initially 60 marbles; Julian had P+40 = 60+40 = 100 marbles. Check on your own that the equations (1) and (2) are hold in this case. </pre> Solve yourself next problem: <H3>Problem 4</H3>Ian and Amanda were counting the number of comic books in their collections. Ian said to Amanda, “Give me one of your comics and we’ll have the same number.” Amanda thought for a moment and then replied, “Give me one of your comics and I’ll have exactly twice as many as you do.” How many comics did each of them have? 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