Lesson Oranges and grapefruits
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<H2>Oranges and grapefruits</H2> <H3>Problem 1</H3>On Monday you bought 11 oranges and 4 grapefruit for a total of $7.40. Later that day, you bought 3 oranges and 5 grapefruits for a total of $4.95. Find the price of each orange and each grapefruit. <B>Solution</B> <pre> Let "n" be the number of oranges, and let "g" be the number of grapefruits. So, you have this system of two equations in two unknowns 11n + 4g = 740, (1) 3n + 5g = 495. (2) For this system the <U>Elimination</U> method suits very well, so I will apply it. Multiply first equation by 5 (both sides) and multiply second equation by 4. In this way you get the system of <U>EQUIVALENT</U> equations 55n + 20g = 3700, (3) 12n + 20g = 1980. (4) Now the coefficients at "g" are the same in both equations. It was my goal to do it, and I reached this goal. Now subtract equation (4) from equation (3). The terms "20g" will cancel each other, and you will get SINGLE equation for ONLY ONE unknown "n": 55n - 12n = 3700 - 1980, or 43n = 1720. From the last equation you easily get n = {{{1720/43}}} = 40. Thus we just found the price of one ORANGE. It is 40 cents. Now substitute the found value of n= 40 into the either equation (1) or (2) to find "g". I will substitute into eq(2): 3*40 + 5g = 495. It gives you 120 + 5g = 495 ====> 5g = 495 - 120 = 375 ====> g = {{{375/5}}} = 75 cents. <U>Answer</U>. One orange costs 40 cents. One grapefruit costs 75 cents. </pre> Solved. <U>Lessons to learn from this solution</U> <pre> 1. The key step is to write correctly the system of equations for the given problem. It is not difficult. The major precaution you should make for this step is to read the condition carefully. 2. The second step is to look into the system and to decide which method of solution to choose. There are different methods: Elimination or/and Substitution. Actually, there are other methods too (using determinants, or the Cramer's rule). But if you only started Algebra-I, you, presumably, know only Elimination or Substitution methods. It is totally enough to solve any problem of your curriculum. 3. And the last step is to implement the method and the solution ACCURATELY. 4. Did I say "the last" ? - No, it is not the last step. The last step is to CHECK the solution. For it, simply substitute the found values into your original equations: 11*40 + 4*75 = 740. ! Correct !, and 3*40 + 5*75 = 495. ! Correct ! 5. 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