Let's determine the correct equation that models the value of Bobby's collection given he has [tex]$d$[/tex] dimes and [tex]$q$[/tex] quarters, and the total value is [tex]$63.75.
1. First, understand the value of each type of coin:
- Each dime is worth $[/tex]0.10.
- Each quarter is worth [tex]$0.25.
2. Next, the total value of Bobby's dimes and quarters can be expressed as:
- The value of d dimes: \( 0.10 \times d \).
- The value of q quarters: \( 0.25 \times q \).
3. Since the total value of Bobby's collection is $[/tex]63.75, we can write the equation that combines these values:
[tex]\[ 0.10d + 0.25q = 63.75 \][/tex]
Let's now evaluate each given option to identify the correct one:
- Option A: [tex]\( 63.75 = d + q \)[/tex]
This implies that the total number of coins equals 63.75, which is incorrect because it does not account for the different values of dimes and quarters.
- Option B: [tex]\( 63.75 = 25d + 10q \)[/tex]
This equation implies that a dime is worth [tex]$25 and a quarter is worth $[/tex]10, which is not correct.
- Option C: [tex]\( 63.75 = 10d + 25q \)[/tex]
This suggests that a dime is worth [tex]$10 and a quarter is worth $[/tex]25, which again is incorrect.
- Option D: [tex]\( 63.75 = 0.10d + 0.25q \)[/tex]
This correctly models the total value of the coins, as it accounts for the values of both dimes and quarters.
Therefore, the correct equation that represents the total value of Bobby's collection is:
Option D: [tex]\( 63.75 = 0.10d + 0.25q \)[/tex]
Hence, the equation corresponding to Bobby's collection value is given by Option D.
