Dimes and Quarters Puzzle

Date: Sat, 05 Nov 1994 13:38:42 -0600 (MDT)
From: Catherine Baca
Subject: math problem

Dear Dr. Math,

Please help with this problem:

Jessica has 16 dimes and quarters.  Whitney has twice as many dimes 
and 1/3 as many quarters as Jessica has.  They both have the same 
amount of money. What coins does each have?

                     David

Date: Sat, 5 Nov 1994 15:59:11 -0500
From: Gabe Farmboy Cavallari
Subject: Re: math problem

Nice question, Dave.  This is Gabe, math doc.  

Ok, to start out with, you can express the amount of dimes and 
quarters which Jessica has algebraically.  Say

# of quarters Jessica has = x
# of dimes Jessica has = 16 -x

Using these, you then express the number of coins that Whitney has:

# of quarters Whitney has = x/3
# of dimes Whitney has = 2(16 - x) = 32 - 2x

Now, if they have the same amount of money, then multiplying each 
expression for the NUMBER of each coin by the value of each coin, 
and then setting up the expression so that the amount of money that 
Jessica has equals that of Whitney should give a linear equation for x.  
Solve for x, and then you can plug that value in to the expressions for 
each amount of coins for both girls:

Here's the expression you need:

Value of quarters for Jessica = 25x
Value of dimes for Jessica = 10(16 - x) = 160 -10x

Value of quarters for Whitney = 25x/3
Value of dimes for Whitney = 10(32 - 2x) = 320 - 20x

So the equation becomes:

25x + 160 - 10x = 25x/3 + 320 - 20x

If you have any questions or just want to verify your answer, just write 
back.  Hope this helps, and thanx for asking Dr. Math!!

-gabe, math doc

Date: Sat, 5 Nov 1994 16:10:20 -0500
From: Vanessa Motto) (Math Doctors
Subject: Re: math problem

Hi David,

This question will involve solving a system of two equations in order to
find out the number of quarters and dimes Jessica and Whitney have. You
know that in order to solve for two different variables you need two
different equations involving the two variables. We know that Jessica has
16 dimes and quarters, so the number of dimes plus the number of quarters
for Jessica will equal 16. So one of our equations will be:

        q + d = 16

Now let's use the second part of the problem to get the second equation. We
know that a dime is worth 10 cents and a quarter is worth 25 cents. Let's
say that Jessica has "d" number of dimes and "q" number of quarters. So the
amount of money that Jessica has is 10 times the number of dimes she has,
plus 25 times the number of quarters that she has, or 10d + 25q. Whitney has 
twice as many dimes as Jessica which is 2d, and 1/3 as many quarters
which is 1/3q. So the amount of money that Whitney has is 10(2d) + 25(1/3q).

Since both girls have the same amount of money, we can set these two
equations equal to each other. So our second equations is:

        10d + 25q = 10(2d) + 25(1/3q)
        which simplifies to
        10d + 25q = 20d + 25/3q
        30d + 75q = 60d + 25q     (multiply by 3 to get rid of the fraction)
       -30d + 50q =  0

We now have two equations in two variables so we can now solve for d and q.

        d +   q = 16
     -30d + 50q =  0

Let's multiply the first equation by 30 (we eventually want to get rid of "d")

        30d  +  30q  = 480
       -30d  +  50q  =   0

Now add the two equations and solve for q

                80q = 480
                  q = 6

Now plug this value of q back into one of the equations to find the value of d.

        d + q = 16
        d + 6 = 16
        d     = 10

So Jessica has 10 dimes and 6 quarters, which is equal to $2.50.
Since Whitney has twice as many dimes and 1/3 as many quarters, she has 20
dimes and 2 quarters, which is also equal to $2.50.

If you have any more questions, or this explanation was unclear to you,
please feel free to write back anytime.

Hope this helps,
Vanessa, M.D.