Lesson Entertainment problems on finding three unknowns using only one equation

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Entertainment problems on finding three unknowns using only one equation

In joking form, this lesson teaches you to find three unknowns using only one single equation
(for some problems, where such approach works).

Problem 1

A school cafeteria cashier has collected $243 in one-dollar, five-dollar,
and ten-dollar bills. The number of one-dollar bills is eight more than 20 times the number of ten-dollar bills.
The cashier also has seven more than twice the number of ten-dollar bills in five-dollar bills. How many bills of each value does the cashier have?

Solution

Let x be the number of the ten-dollar bills.

Then the number of one-dollar bills is (8+20x)

and  the number of five-dollar bills is  (7+2x).


Now write the total money equation

    243 = 10x + (8+20x) + 5*(7+2x).


Simplify and find x

    243 = 10x + 8 + 20x + 35 + 10x

    243 = 40x + 43

    40x = 243 - 43

    40x =    200

      x =    200/40 = 5


ANSWER.  5 ten-dollar bills;  (8+20*5) = 108 one-dollar bills  and  (7+2*5) = 17 five-dollar bills.


CHECK.  5*10 + 108*1 + 17*5 = 243  total money.    ! Correct !

In joking form, this problem is about finding three unknown using only one equation . . .

Problem 2

You are at the store, and are trying to remember how much bread, meat, and cheese you were supposed to buy.
The bread costs $3 per loaf, the meat costs $5 per pound, and the cheese costs $2 per pound.
You know that you are to buy fifteen items, that the total cost will be $51,
and that you are supposed to purchase one more meat than cheese. How many meat do you need to purchase?

Solution

x pounds of meat,

(x-1) pounds of cheese and

(15 - x - (x-1)) = 16-2x loaves of bread.


The total money equation is

    3*(16-2x) + 5x + 2*(x-1) = 51.   (1)


Simplify and find x

    48 - 6x + 5x + 2x - 2 = 51

    46 + x = 51

         x = 51 - 46


ANSWER.  5 pounds of meat.


CHECK.  Then the left side of equation (1) is  3*(16-2*5) + 5*5 + 2*(5-1) = 51,

        which is precisely correct.

My other additional lessons on solving single linear equations and word problems in one unknown in this site (section 2) are
    - One more lesson on solving problems by the backward method
    - Choose an unknown variable in a rational way for your problem
    - Upper level word problems to solve using a single linear equation
    - OVERVIEW of my additional lessons on solving single linear equations and word problems in one unknown

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