Lesson Solving mentally word problems on two equations in two unknowns

Solving mentally word problems on two equations in two unknowns

Problem 1

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.   ANSWER


ANSWER.  On Wednesday, she must pay 27 pesos.

Problem 2

4x + 3y = 25,     (1)
4x - 3y =  7.     (2)
---------------------------Subtract eq(2) from eq(1). You will get

6y = 25 - 7 = 18  ===>  y =  = 3.


Then from (1),  4x = 25 - 3y = 25 - 3*3 = 25 - 9 = 16;


Hence, x =  = 4.


Answer.  First number is  4;  second number is  3.

Problem 3

    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. 


    Answer.  A bottle of water costs  $2.50.

Problem 4

    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   = 18.


Answer.  18 persons for each conference room and  65 persons for each ballroom.

Problem 5

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.

Problem 6

    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.

Problem 7

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  = 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.


Answer.  Airplane with 60 gallons of fuel weights 2231.25 pounds.

Problem 8

It follows immediately from the condition, that 58-22 = 36 minutes cost 21.70-16.30 = 5.40 dollars, i.e.  = 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.


Answer.  After 74 minutes of calls the remaining credit will be  13.90 dollars.

Problem 9

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   = 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.



ANSWER.  $2.50 per pound for the cinnamon tea and $3.00 per pound for the spice tea.

Problem 10

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.


From it, you get the price of one hand towel  15/3 = 5 dollars.