Sure, let's solve the problem step by step.
Kandi has [tex]$2.65 in nickels and dimes and a total of 40 coins. We need to find out how many nickels she has.
1. Let's denote the number of nickels as \( n \) and the number of dimes as \( d \).
2. We know there are 40 coins in total. This gives us our first equation:
\[ n + d = 40 \]
3. We also know the total value of the coins is $[/tex]2.65. Since each nickel is worth [tex]$0.05 and each dime is worth $[/tex]0.10, we can write our second equation:
[tex]\[ 0.05n + 0.10d = 2.65 \][/tex]
4. To make calculations easier, let's multiply the second equation by 100 to eliminate the decimals:
[tex]\[ 5n + 10d = 265 \][/tex]
5. Now we have the following system of equations:
[tex]\[ n + d = 40 \][/tex]
[tex]\[ 5n + 10d = 265 \][/tex]
6. To solve this system, we'll use substitution or elimination. Let's use the substitution method. From the first equation, we can express [tex]\( d \)[/tex] in terms of [tex]\( n \)[/tex]:
[tex]\[ d = 40 - n \][/tex]
7. Substitute [tex]\( d = 40 - n \)[/tex] into the second equation:
[tex]\[ 5n + 10(40 - n) = 265 \][/tex]
8. Simplify and solve for [tex]\( n \)[/tex]:
[tex]\[ 5n + 400 - 10n = 265 \][/tex]
[tex]\[ -5n + 400 = 265 \][/tex]
[tex]\[ -5n = 265 - 400 \][/tex]
[tex]\[ -5n = -135 \][/tex]
[tex]\[ n = \frac{-135}{-5} \][/tex]
[tex]\[ n = 27 \][/tex]
Therefore, Kandi has 27 nickels.
To find the number of dimes, we substitute [tex]\( n = 27 \)[/tex] back into the first equation:
[tex]\[ n + d = 40 \][/tex]
[tex]\[ 27 + d = 40 \][/tex]
[tex]\[ d = 40 - 27 \][/tex]
[tex]\[ d = 13 \][/tex]
So, Kandi has 27 nickels and 13 dimes.
