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Sample Question 14:
Step 1: Start by using as many smaller value stamps as possible since Nicolas wants to use the maximum number of stamps to make $7.10.
Step 2: Use 20 of the 5-cent stamps (nickels) and 20 of the 10-cent stamps (dimes). This totals up to $1.50 and $2.00 respectively, combining to make $3.50.
Step 3: Subtract this $3.50 from the total required postage of $7.10, remaining $3.60 that still needs to be paid for.
Step 4: Now, when attempting to use the 25-cent stamps (quarters), we observe that $3.60 is not a multiple of 25 cents, so we need to manually adjust our previous decision of using all dimes and nickels. We need to give back 15 cents ($7.10 - $3.25 = $3.85$ needs to be paid by dimes and nickels, but $4.00$ is, so subtracting the two gives $0.15$) to make our remaining amount exactly a multiple of 25 cents.
Step 5: To maximize the number of stamps used, remove one dime and one nickel rather than removing three nickels. This leaves us with $3.85$ paid by dimes and nickels, for a total of 38 stamps.
Step 6: We can now use $\frac{425}{25} = 17$ quarters to make up the remaining amount.
Step 7: Adding this to the previous total, we have a grand total of 38 + 17 = 55 stamps.
So, the answer is \(\boxed{\textbf{(E)}}\ 55\).
Nicolas is planning to send a package to his friend Anton, who is a stamp collector. To pay for the postage, Nicolas would like to cover the package with a large number of stamps. Suppose he has a collection of \(5\)-cent, \(10\)-cent, and \(25\)-cent stamps, with exactly \(20\) of each type. What is the greatest number of stamps Nicolas can use to make exactly \(\$7.10\) in postage? (Note: The amount \(\$7.10\) corresponds to \(7\) dollars and \(10\) cents. One dollar is worth \(100\) cents.)
\(\textbf{(A)}\ 45 \qquad \textbf{(B)}\ 46 \qquad \textbf{(C)}\ 51 \qquad \textbf{(D)}\ 54\qquad \textbf{(E)}\ 55\)
Answer Keys
Question 14: ESolutions
Question 14Step 1: Start by using as many smaller value stamps as possible since Nicolas wants to use the maximum number of stamps to make $7.10.
Step 2: Use 20 of the 5-cent stamps (nickels) and 20 of the 10-cent stamps (dimes). This totals up to $1.50 and $2.00 respectively, combining to make $3.50.
Step 3: Subtract this $3.50 from the total required postage of $7.10, remaining $3.60 that still needs to be paid for.
Step 4: Now, when attempting to use the 25-cent stamps (quarters), we observe that $3.60 is not a multiple of 25 cents, so we need to manually adjust our previous decision of using all dimes and nickels. We need to give back 15 cents ($7.10 - $3.25 = $3.85$ needs to be paid by dimes and nickels, but $4.00$ is, so subtracting the two gives $0.15$) to make our remaining amount exactly a multiple of 25 cents.
Step 5: To maximize the number of stamps used, remove one dime and one nickel rather than removing three nickels. This leaves us with $3.85$ paid by dimes and nickels, for a total of 38 stamps.
Step 6: We can now use $\frac{425}{25} = 17$ quarters to make up the remaining amount.
Step 7: Adding this to the previous total, we have a grand total of 38 + 17 = 55 stamps.
So, the answer is \(\boxed{\textbf{(E)}}\ 55\).