Chapter 0, Problem 5P

To determine

To find: The total possible ways to change a dollar

Answer to Problem 5P

Thetotal possible ways are 29

Explanation of Solution

Given:

Consider the values of dollar, quarter, dime and nickel in terms of cents as follows:

1 dollar =100 ȼ

1 quarter =25 ȼ

1 dime =10 ȼ

1 nickel =5 ȼ

Calculation:

Consider the values of dollar, quarter, dime and nickel in terms of cents as follows:

1 dollar =100 ȼ

1 quarter =25 ȼ

1 dime =10 ȼ

1 nickel =5 ȼ

Now, change for dollar could be in terms of single coin or in mix of them.

Comment

Step 2 of 3

Considering single coins,

1. 4 quarters make $\$1-1$ way

2. 10 dimes make $\$1-1$ way

3. 20 nickels make $\$1-1$ way

Now, consider mix of 2 types of coins

4. Quarters and dimes: since dime involves factor of 10 quarters must have value which is factor of 10. Only combination of quarters between 25 ȼ and 100 ȼ (not including 100,as it has been included above ) 50 ȼ.

Thus, it will have 2 quarters and 5 dimes -1 way.

5. Quarters and nickels: Here, we can have 1 quarter, 2 quarters or 3 quarters in the mix

Thus − 3 ways.

6. Dimes and Nickels: Here we can have mix ranging from 1 dime to 9 dimes (as 0 and 10 arealready covered under single coins)

Thus - 9 ways.

Now, considering mix of all three types of coins as,

Various combinations are possible here,

7. with 1 quarter. With 25 cents gone, there can be 1 to 7 dimes (as 0 dimes is already considered under 2 type mix and 8 will exceed limit of $1)

Thus - 7 ways.

8. with 2 quarters: With 50 cents gone, there can be 1 to 4 dimes (as 0 and 5 dimes is alreadyconsidered under 2 type mix

Thus - 4 ways.

9. with 3 quarters: With 75 cents gone, there can be 1 to 2 dimes (as 0 dimes is alreadyconsidered under 2 type mix and 3 will exceed limit of $1 )

Thus there are 2 ways.

Counting all ways of all three scenarios,

The total possible ways are,

  $=1+1+1+1+3+9+7+4+2$

=29

Conclusion:

Hence, the total possible ways are 29