Lesson Sometimes one equation is enough to find two unknowns in a unique way
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<H2>Sometimes one equation is enough to find two unknowns in a unique way</H2> From the theory, we know that finding two unknowns require two equations. Generally speaking, it is true; but there are cases when one equation is enough. In this lesson, you will find examples of such problems. <H3>Problem 1</H3>A restaurant offered an eat-all-you-can menu last November. They charged 250 pesos for kids below 3 feet and 350 pesos for adults or kids beyond 3 feet. If the restaurant earned 12,450 on one day, what is the minimum number of kids below 3 feet, if there are more customers who paid 350 pesos? <B>Solution</B> <pre> In this problem, we have only one equation for two unknowns, which are "a" and "k", the number of adults and kids, respectively a*350 + k*250 = 12450. (1) So, it seems, that there are many solutions. But we have two additional restrictions. First, the solutions "a" and "k" must be integer numbers. Second, we are looking for the solution in integer numbers, where "k" should be minimal. It provides the UNIQUENESS of the solution. So, we write from equation (1) k = {{{(12450 - a*350)/250}}}, and we are looking for the first positive integer k, which we can obtain from this formula, running a = 1, 2, 3, . . . Fortunately, we do not need run very far, because just the value of k = 5 provides positive integer a = 32. So, the <U>ANSWER</U> to the problem's question is: +------------------------------------------------------------+ | the minimum number of kids under given condition is 5. | +------------------------------------------------------------+ </pre> <H3>Problem 2</H3>Lyza is selling kakanin before she goes to school. She sells putubumbong and bibingka at 1 0.50 pesos and 12.75 pesos per slice, respectively. Her total earnings today is 389.25 pesos. If she sold more putubumbong than bibingka, what is the total number of kakanin sold? <B>Solution</B> <pre> In this problem, we have only one equation for two unknowns, which are "p" and "b", the number of putubumbong and bibingka, respectively p*10.50 + b*12.75 = 389.25. (1) So, it seems, that there are many solutions. But we have two additional restrictions. First, the solutions "p" and "b" must be integer numbers. Second, we are looking for the solution in non-negative integer numbers, where "p" is greater than "b". It provides the UNIQUENESS of the solution. So, we write from equation (1) b = {{{(389.25 - p*1050)/12.75}}}, We run p = 1, 2, 3, . . . , and we are looking for positive integer values of b, which we obtain from this formula. The solutions in integer positive numbers are (p,b) = (14,19) and /or (31,5) Of these two pairs, only the pair (p,b) = (31,5) has the value of p (31) greater than b (5). So, p + b = 31 + 5 = 36 is the solution to the problem. <U>ANSWER</U> </pre>----------- <pre> The way to facilitate the required calculations is to use Excel software, if you have it in your computer. Then you can make your calculations pleasant, instead having them torturous. </pre> <H3>Problem 3</H3>A factory sorts pencils into bags such that 19 large bags plus 3 small bags contain a total of 224 pencils. Find the number of pencils in a large bag and the number of pencils in a small bag. <B>Solution</B> <pre> Let x be the number of pencils in a large bag and y be the number of pencils in a small box. Then you have this equation 19x + 3y = 224 (1) and this inequality x > y (2) Equation (1) should be solved in whole numbers. Inequality (2) is the math translation of the terms "large box" and "small box". Using inequality (2), we get the following inequality from equation (1) 19x + 3x > 224, or 22x > 224, It gives x > {{{224/22}}} = 10.1818..., and since x should be whole number, it implies x >= 11. So, we need find the solution to equation (1) in whole numbers x, y with x >= 11. There are not so many such possibilities: 19*11 = 209 and 19*12 = 228, so the only real candidate is x = 11. It gives y = {{{(224-19*11)/3}}} = {{{(224-209)/3}}} = {{{15/3}}} = 5, which is a whole number. So, the problem has a UNIQUE solution x= 11, y= 5. <U>ANSWER</U>. The number of pencils in a large bag is 11. 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