How many coins
When i dumped the change out of my pocket, I noticed that i could pay the exact price for any item from one cent up to and through one dollar without receiving any change. What is the fewest number of coins I could have had in my pocket if i had pennies, nickels, dimes, and quarters? What are the coins?
4 pennies
1 nickle
2 dimes
3 quarters
start with
4 pennies, 2 nickels , 2 dimes , 3 quarters
covers from
1 to 9 --- 1 nickel, 4 pennies
10 to 14 -- 1 dime, 4 pennies
15 to 19 = 1 dime, 1 nickel, 4 pennies
20 to 24 = 1 dime, 2 nickels 4 pennies OR 2 dimes 4 pennies
25 to 29 --- 1 quarter, 4 pennies
30 to 34 --- 1 quarter, 1 nickel, 4 pennies
35 to 39 ---- 1 quarter, 1 dime, 4 pennies
40 to 44 ---- 1 quarter, 1 nickel, 1 dime, 4 pennies
45-49 -------1 quarter, 2 nickels 1 dime, 4 pennies
50 - 54 ---- 1 quarter, 2 dimes, 1 nickel, 4 pennies
55 - 59 --- 2 quarters, 1 nickel, 4 pennies
60 - 64 ---- 2 quarters, 1 dime, 3 pennies
65 - 69 --- 2 quarters , 1 dime, 1 nickel, 4 pennies
70-74 --- 2 quarters, 2 dimes, 4 pennies
75 -79 -- 2 quarters, 2 dimes, 1 nickel, 4 pennies
80 - 84 --- 2 quarters, 2 dimes, 1 nickel, 4 pennies OR 3 quarters, 1 nickel, 4 pennies
85-89 --- 3 quarters, 1 dime, 4 pennies
90-94 --- 3 quarters, 1 dime, 1 nickel, 4 pennies
95 -99 --- 3 quarters , 2 dimes, 4 pennies
100 ------ 3 quarters, 2 dimes, 1 nickel
to do 95 - 99 with only 2 quarters would need
2 quarters, 4 dimes, 1 nickel, 4 pennies
so I could cover any amount from 1 to 100 with
a) 4 pennies, 2 nickels, 2 dimes, and 3 quarters. for a total of 11 coins
or
b) 4 pennies, 2 nickels, 4 dimes, and 2 quarters for a total of 12 coins
Therefore:
4 pennies
2 nickels
2 dimes
3 quarters or a total of 11 coins
ARRGGGHHHH!!!!
looks like I messed up in my 45-49
could be done with 1 quarter, 2 dimes, and 4 pennies.
Nowhere else do we actually need 2 nickels
so as bobpursley said
3 quarters,
2 dimes
1 nickel
4 pennies for a total of 10 coins
To find the fewest number of coins you could have had in your pocket to make exact change from one cent up to and including one dollar, you can work backward and figure out which coins you must have had.
First, let's consider the highest possible amount you could have paid without receiving change, which is one dollar. Since you cannot use a dollar coin (because you would need change for amounts less than one dollar), the only way to make exact change without receiving any additional coins is by using quarters.
Therefore, you must have had at least four quarters in your pocket, totaling up to one dollar. This means that you have already used up 4 coins.
Now, moving on to the next highest amount which requires exact change, which is 99 cents. Since you have already used four quarters, you need to make the remaining 99 cents using nickels, dimes, and pennies.
To minimize the number of coins, you can use three dimes (10 cents each) to make 30 cents. Then, you need to make up the remaining 69 cents. Since you cannot use a quarter again, you can use two nickels (5 cents each) to make 10 cents and then use four pennies (1 cent each) to make 4 cents.
So, for a total of 99 cents, you used 3 dimes, 2 nickels, and 4 pennies, which adds 9 more coins to the 4 quarters, making a total of 13 coins.
To continue the process and find the fewest number of coins for each amount, you can repeat the above steps for each remaining amount until you reach 1 cent.
Here is a breakdown of the coins you would have had in your pocket for each amount:
- 1 dollar: 4 quarters
- 99 cents: 3 dimes, 2 nickels, 4 pennies
- 98 cents: 3 dimes, 2 nickels, 3 pennies
- 97 cents: 3 dimes, 2 nickels, 2 pennies
- ...
- 9 cents: 3 dimes, 1 nickel, 4 pennies
- 8 cents: 3 dimes, 1 nickel, 3 pennies
- 7 cents: 3 dimes, 1 nickel, 2 pennies
- ...
- 1 cent: 3 dimes, 1 nickel, 1 penny
So, the fewest number of coins you could have had in your pocket to make exact change from one cent up to and including one dollar is 13 coins, consisting of 4 quarters, 3 dimes, 2 nickels, and 4 pennies.