Questions on Word Problems: Coins answered by real tutors!

Case 1:  If we don't use any quarters, then we must make the whole
48 cents with dimes, nickels and pennies.

A. If we use 0 dimes we can use 0 up to 9 nickels and the rest pennies.
   That's 10 ways.

B. If we use 1 dime we can use 0 up to 7 nickels and the rest pennies.
   That's 8 ways.

C. If we use 2 dimes we can use 0 up to 5 nickels and the rest pennies.
   That's 6 ways.
 
D. If we use 3 dimes we can use 0 up to 3 nickels and the rest pennies.
   That's 4 ways.

D. If we use 4 dimes we can use 0 or 1 nickel and the rest pennies.
   That's 2 ways.

So with no quarters there are 10+8+6+4+2 or 30 ways 

--------------------

Case 2:  If we use 1 quarter, we have to make the remaining 23 cents with
dimes, nickels, and pennies 

A. If we use 0 dimes we can use 0 up to 4 nickels and the rest pennies.
   That's 5 ways.

B. If we use 1 dime we can use 0 up to 2 nickels and the rest pennies.
   That's 3 ways.

C. If we use 2 dimes we can only use 3 pennies.
   That's 1 way.

So with 1 quarter there are 5+3+1 or 9 ways to make the remaining 23 cents.

----------

              30 ways using 0 quarters
               9 ways using 1 quarter
              -----------------------
      total:  39 ways to have 48 cents in coin. 

Edwin

If we don't use any half dollars, then since 3.85/.25 = 15.4, the
maximum number of quarters is 15.

If we use a total of q quarters then we have to make $3.85-$.25q with 
only 22-q coins.  The most amount of money 22-q coins can be is the 
amount when they are all dimes, so

.10(22-q) >= 385-.25q
       
Solving this gives

        q >= 11   

So the number of quarters is between 11 and 15

But if we use 15 quarters, that's $3.75 and we would have to make 
the remaining $.10 with 7 coins which is not possible.

If we use 14 quarters, that's $3.50 and we would have to make 
the remaining $.35 with 8 coins smaller than a quarter, which 
can only be done with 3 dimes and 5 pennies.  That's 1 way.

If we use 13 quarters, that's $3.25 and we would have to make 
the remaining $.60 with 9 coins smaller than a quarter, which 
can only be done with 3 dimes and 6 nickels. That's 1 way. 
 
If we use 12 quarters, that's $3.00 and we would have to make 
the remaining $.85 with 10 coins smaller than a quarter, which 
can only be done with 7 dimes and 3 nickels. That's 1 way.

If we use 11 quarters, that's $2.75 and we would have to make 
the remaining $1.10 with 11 coins smaller than a quarter, which
 can only be done with 11 dimes. That's 1 way.

That's 4 ways using no half dollars.

It's a similar problem if we allow 1,2,3,4,5,6 or 7 half dollars.

If we use 1 half dollar then we have to make $3.35 with 21 coins.
Proceding as above we get 4 ways for this too.

If we use 2 half dollar then we have to make $2.85 with 20 coins.
Proceding as above we get 6 ways for this.

If we use 3 half dollar then we have to make $2.35 with 19 coins.
Proceding as above we get 9 ways for this.

If we use 4 half dollar then we have to make $1.85 with 18 coins.
Proceding as above we get 6 ways for this.

If we use 5 half dollar then we have to make $1.35 with 17 coins.
Proceding as above we get 7 ways for this.

If we use 6 half dollar then we have to make $.85 with 16 coins.
Proceding as above we get 5 ways for this.

If we use 7 half dollar then we have to make $.35 with 15 coins.
Proceding as above there is only 1 ways for this, 5 nickels and 10
pennies.

Total: 4+4+6+9+6+7+5+1 = 42

Edwin

There is no easy way to do this problem, for there are too many cases.
I used a computer to count them.

Answers:

1580 if you can use half dollars.
1022 if you cannot use half dollars.
 140 if you can use half dollars but no pennies.
  76 if you cannot use either half dollars or pennies.

Edwin