You have a nickel, a dime, a quarter and a half
dollar. A clerk shows you several items, each a
different price, any one of which you could buy with
your coins without receiving change. What is the
largest number of articles he could have shown you?
a. 8
b. 10
c. 13
d. 15
e. 21
please answer and explain
21
To find the largest number of articles the clerk could have shown you, we need to consider the possible combinations of coins that can be used to purchase an item without receiving change.
Let's start by examining the highest value coin, which is the half dollar. Since the half dollar is the highest value coin, you can only buy items priced at exactly half a dollar.
Now, let's move on to the quarter. The quarter is worth 25 cents, which means you can buy items priced at multiples of 25 cents (25 cents, 50 cents, 75 cents, etc.).
Next, let's consider the dime. The dime is worth 10 cents, so we can buy items priced at multiples of 10 cents (10 cents, 20 cents, 30 cents, etc.).
Finally, let's look at the nickel. The nickel is worth 5 cents, so we can buy items priced at multiples of 5 cents (5 cents, 10 cents, 15 cents, etc.).
By looking at these denominations, we can see that the largest number of articles the clerk could have shown you is when the prices are multiples of the highest value coin, which in this case is the half dollar. In this situation, the other coins will not be used, as the price is exactly equal to the half dollar.
Since there are 21 multiples of 50 cents (ranging from 50 cents to 10.50 dollars, with increments of 50 cents), the largest number of articles the clerk could have shown you is 21.
Therefore, the correct answer is e. 21.
To find the largest number of articles the clerk could have shown you while buying without receiving change, we need to consider the possible combinations of coins and determine which combination gives the largest total price.
We can start by analyzing the values of the coins.
- A nickel is worth $0.05
- A dime is worth $0.10
- A quarter is worth $0.25
- A half dollar is worth $0.50
To maximize the number of articles, we want to use the highest value coins first. This means we should try to use the least number of coins possible with the highest denominations.
Since we need to purchase items without receiving change, the total value of the coins must match the price of each item.
Let's consider the possible combinations:
Using only the half dollar coin ($0.50):
The maximum price we can buy is $0.50. So, we can only buy one item with a price of $0.50.
Using the half dollar and the quarter coins ($0.50 + $0.25):
The maximum price we can buy is $0.75. We can buy an item costing $0.75.
Using the half dollar, the quarter, and the dime coins ($0.50 + $0.25 + $0.10):
The maximum price we can buy is $0.85. We cannot buy an item costing $0.85, as it exceeds the total value of the available coins.
Using the half dollar, the quarter, the dime, and the nickel coins ($0.50 + $0.25 + $0.10 + $0.05):
The maximum price we can buy is $0.90. We cannot buy an item costing $0.90, as it exceeds the total value of the available coins.
Using all the coins ($0.50 + $0.25 + $0.10 + $0.05):
The maximum price we can buy is $0.90. We cannot buy an item costing $0.90, as it exceeds the total value of the available coins.
From the analysis above, we can conclude that the largest number of articles the clerk could have shown us is one, since we can only purchase one item with a price of $0.50 using the half dollar coin.
Therefore, the correct answer is option: a. 1.
Note: The options provided do not include the number "1" as a choice. Therefore, there may be an error in the answer choices or the options provided are not consistent with the question.