Lesson Solving mentally word problems on two equations in two unknowns
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<H2>Solving mentally word problems on two equations in two unknowns</H2> <H3>Problem 1</H3>On Monday, Revlien paid 66 pesos for 3 doughnuts and 2 cups of coffee. On Tuesday, she paid 69 pesos for 2 doughnuts and 3 cups of coffee. How much did he pay for 1 doughnut and 1 cup of coffee on Wednesday? <B>Solution</B> This problem is intended for mental solution. <pre> Add her purchases on Monday and Tuesday: she bought 3+2 = 5 doughnuts and 2+3 = 5 cups of coffee, in total, spending 66 + 69 = 135 pesos. HENCE, one doughnut and one cup of coffee cost 1/5 of it, i.e. 135/5 = 27 pesos. <U>ANSWER</U> <U>ANSWER</U>. On Wednesday, she must pay 27 pesos. </pre> <H3>Problem 2</H3>Four times a certain number increased by three times a second number is 25. Four times the first number decreased by three times the second number is 7. Find the two numbers. <B>Solution</B> <pre> 4x + 3y = 25, (1) 4x - 3y = 7. (2) ---------------------------Subtract eq(2) from eq(1). You will get 6y = 25 - 7 = 18 ===> y = {{{18/6}}} = 3. Then from (1), 4x = 25 - 3y = 25 - 3*3 = 25 - 9 = 16; Hence, x = {{{16/4}}} = 4. <U>Answer</U>. First number is 4; second number is 3. </pre> The good style in solving such problems is to notice/(to observe) immediately from the condition that the difference between two combinations is 6 times the second number, which, in turn, is the difference 25-7 = 18, and to deduce from it <U>MENTALLY</U> that y= 3, without writing any equations. This problem is for <U>mental</U> solution, and it would be ideally if you see the solution immediately/instantly as you completed reading its condition. <H3>Problem 3</H3>Ben and Greg go to the movies and purchase snacks. Ben purchases four cookies and two bottles of water for a total cost of $9.00. Greg purchases two cookies and two bottles of water for a total cost of $7.00. What is the cost of a bottle of water? <B>Solution</B> <pre> Good style in solving such problems is to notice/(to observe) immediately from the condition that the difference between two combinations is the cost of 2 cookies, and it is equal to $9 - $7 = $2. So 2 cookies cost $2. Then subtracting $2 dollars for 2 cookies from the Greg purchase, you got the cost of two bottles - it is equal to $7 - $2 = $5. <U>Answer</U>. A bottle of water costs $2.50. </pre> Same as the previous, this problem is for mental solution, and it would be ideally if you see the solution immediately/instantly as you completed reading its condition. <H3>Problem 4</H3>An event organizer is reserving rooms for two company-wide events. For the quarterly meeting this month, she reserved 3 conference rooms and 5 ballrooms, which can seat a total of 379 attendees. For safety training next month, she reserved 3 conference rooms and 4 ballrooms, which can seat 314 attendees. How many attendees can each room accommodate? <B>Solution</B> This problem is for mental solution. <pre> 1. After reading the text, you should see that the difference of 379 - 314 = 65 is exactly the capacity of one ballroom. 2. Then the capacity of each conference room is one third of 314-4*65, or {{{(1/3)*(314-4*65)}}} = 18. <U>Answer</U>. 18 persons for each conference room and 65 persons for each ballroom. </pre> <H3>Problem 5</H3>Kira wanted to stock up on drinks for an upcoming party. First she spent $53 on 14 cases of juice and 11 cases of soda, which is all the store had in stock. A few days later, she returned to the store and purchased an additional 14 cases of juice and 8 cases of soda, spending a total of $50. What is the price of each drink? <B>Solution</B> This problem is for mental solution. <pre> 1. After reading the condition, you can easily see that the difference $53 - $50 = $3 is exactly the cost of 11-8 = 3 cases of soda, so that one case of soda costs 1 dollars. 2. After finding this, you conclude that 14 cases of juice cost 53 - 11*1 = 42 dollars. Hence, each case of the juice costs 42/14 = 3 dollars. </pre> <H3>Problem 6</H3>One Friday night, two large groups of people called centerville taxi service. The first group requested 3 sedans and 2 minivans, which can seat a total of 21 people. The second group asked for 3 sedans and 3 minivans, which can seat a total of 27 people. How many passengers can each type of taxi seat? <B>Solution</B> This problem is for MENTAL solution, and the solution is in two lines: <pre> 1. The difference of 27-21 = 6 is exactly the number of peoples that can seat in minivan. 2. From the part related to the 2-nd group, you see that 1 sedan plus one minivan can carry 9 passengers. Then from n.1 you get that each sedan carries 9-6 = 3 passengers. </pre> <H3>Problem 7</H3>Suppose that the weight (in pounds) of an airplane is a linear function of the amount of fuel (in gallons) in its tank. When carrying 18 gallons of fuel, the airplane weighs 1999 pounds. When carrying 50 gallons of fuel, it weighs 2175 pounds. How much does the airplane weigh if it is carrying 60 gallons of fuel? <B>Solution</B> This problem is for MENTAL solution. <pre> Airplane + 50 gallons of fuel weights 2175 pounds. (1) Airplane + 18 gallons of fuel weights 1999 pounds. (2) ----------------------------------------------------------Subtract (2) from (1). You will get 32 gallons of fuel weight 2175 - 1999 = 180 pounds. ====> ====> 1 gallon of fuel weights {{{180/32}}} = 5.625 pounds. (Airplane + 60 gallons of fuel)'s weight = (Airplane + 50 gallons of fuel weight) + weight of 10 gallons of fuel = 2175 + 10*5.625 = 2231.25 pounds. <U>Answer</U>. Airplane with 60 gallons of fuel weights 2231.25 pounds. </pre> <H3>Problem 8</H3>The remaining credit on a prepaid phone card after 22 minutes of calls is $21.70 and the remaining credit after 58 minutes of calls is $16.30. What is the remaining credit after 74 minutes of calls? <B>Solution</B> This problem is for MENTAL solution. <pre> It follows immediately from the condition, that 58-22 = 36 minutes cost 21.70-16.30 = 5.40 dollars, i.e. {{{540/36}}} = 0.15 dollars per minute. Further, 74 - 58 = 16 minutes, and they cost 16*0.15 = 2.4 dollars. So, after 74 minutes of calls the remaining credit will be 16.30 - 2.40 = 13.90 dollars. <U>Answer</U>. After 74 minutes of calls the remaining credit will be 13.90 dollars. </pre> <H3>Problem 9</H3>A merchant mixed 10 lb of a cinnamon tea with 4 lb of spice tea. The 14-pound mixture cost $37. A second mixture included 16 lb of the cinnamon tea and 4 lb of the spice tea. The 20-pound mixture cost $52. Find the cost per pound of the cinnamon tea and of the spice tea. <B>Solution</B> This problem is for MENTAL solution. <pre> An attentive reader will notice momentarily from the condition, that the difference in costs $52 - $37 = $15 is exactly the cost of 16 lb - 10 lb = 6 lb of the cinnamon tea. So the price of the cinnamon tea is {{{15/6}}} = 2.50 dollars per pound. Then the cost of 4 lb of the spice tea is 37 - 10*2.5 = 37 - 25 = 12 dollars; hence, the price of the of the spice tea is 12/4 = 3 dollars per pound. <U>ANSWER</U>. $2.50 per pound for the cinnamon tea and $3.00 per pound for the spice tea. </pre> <H3>Problem 10</H3>Jayce bought 2 bath towels and 3 hand towels for $35. His sister jayna bought 3 bath towels and 3 hand towels for $45. Determine the prices of a bath towel and hand towel. <B>Solution</B> This problem is for MENTAL solution. <pre> If you read the problem attentively, you will notice, that the difference in items bought by two women, is exactly 1 bath towel. The difference in the payed cost is $45 - $35 = 10 dollars. Hence, one bath towel price is 10 dollars. (Half of the problem is just solved (!) ). Next, Jayce paid 2*10 = 20 dollars for 2 bath towel -- hence, the cost of 3 hand towels is 35 - 20 = 15 dollars. 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