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25 Coins for a Dollar

Date: 02/13/2003 at 20:02:48
From: Melissa
Subject: Math

I need 25 coins for my dollar. What are they?

Date: 02/14/2003 at 02:40:45
From: Doctor Jeremiah
Subject: Re: Math

Hi Melissa,

Lets use d for dimes, n for nickels and p for pennies.

The total number of coins is d + n + p (which equals 25) so one of the 
equations is d + n + p = 25.

The total number of cents of all the dimes is 10d and the total number 
of cents of all the nickels is 5n, and since pennies are worth one 
cent the total number of cents of all the pennies is just p.  

The total number of cents of all the coins put together is 
10d + 5n + p (and all the coins add up to 100 cents) so the other 
equation is 10d + 5n + p = 100

Now if we can solve these equations we will get somewhere.

We can combine the equations to remove p.  When we do that we get:

  10d + 5n + p = 100
    d +  n + p = 25
   9d + 4n     = 75

Let's isolate one of the variables - say d for example:

    d + 4n/9   = 75/9

The left has two things added together, so we should change the right 
side so that we have two things added together.

    d + 4n/9  = 7 + 12/9

Notice how the left side and the right side match up really nicely?  
If we make d equal 7, then 4n will equal 12 when n equals 3.

So d=7 and n=3. But that doesn't make 25 coins. Remember that there 
were pennies too? If we stick d=7 and n=3 into this equation:

    d + n + p = 25
    7 + 3 + p = 25
            p = 15

Now let's check our answer by sticking d=7 and n=3 in this equation:

    10d + 5n + p = 100
     70 + 15 + p = 100
               p = 15

So our answers are right. But they are not the only answer. If you use 
quarters also, you can do it the same way and get d=3 q=1 n=6 p=15.

- Doctor Jeremiah, The Math Forum 
Associated Topics:
Middle School Algebra
Middle School Word Problems

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