landon has some nickels and some dimes. he has at most 25 coins worth a minimum of $2 combined. if landon…

landon has some nickels and some dimes. he has at most 25 coins worth a minimum of $2 combined. if landon has 17 nickels, determine the minimum number of dimes that he could have.

landon has some nickels and some dimes. he has at most 25 coins worth a minimum of $2 combined. if landon has 17 nickels, determine the minimum number of dimes that he could have.

Answer

Explanation:

Step1: Set up variables

Let (n) be the number of nickels ((n = 17)), (d) be the number of dimes. The total number of coins (C=n + d), and (C\leq25). The total value (V = 5n+10d), and (V\geq200) (since (2) dollars (= 200) cents).

Step2: Use the coin - number inequality

From (C=n + d\leq25), substituting (n = 17), we get (17 + d\leq25), so (d\leq8).

Step3: Use the value inequality

Substitute (n = 17) into (V = 5n+10d\geq200). We have (5\times17+10d\geq200), which simplifies to (85 + 10d\geq200). Then (10d\geq200 - 85=115), and (d\geq\frac{115}{10}=11.5).

Since (d) is an integer, and we want to satisfy both the value and coin - number conditions (we made a wrong start with the coin - number inequality, we should focus on the value inequality).

Let's start from the value formula (V=5n + 10d\geq200), with (n = 17). [ \begin{align*} 5\times17+10d&\geq200\ 85+10d&\geq200\ 10d&\geq200 - 85\ 10d&\geq115\ d&\geq11.5 \end{align*} ] Since (d) is a non - negative integer, (d\geq12)

Answer:

(12)