To solve this problem, we need to find the number of purple marbles, [tex]$p$[/tex], given the total number of blue and purple marbles is 24 and the number of purple marbles is one less than four times the number of blue marbles. Here's how we can systematically solve this step-by-step:
1. Start with the Given Information:
- Total marbles: [tex]$b + p = 24$[/tex]
- Relationship: [tex]$p = 4b - 1$[/tex]
2. Set up the Equations:
- From the relationship, substitute [tex]$p = 4b - 1$[/tex] into [tex]$b + p = 24$[/tex]:
[tex]\[
b + (4b - 1) = 24
\][/tex]
3. Simplify and Solve for [tex]$b$[/tex]:
- Combine like terms:
[tex]\[
b + 4b - 1 = 24
\][/tex]
- Simplify further:
[tex]\[
5b - 1 = 24
\][/tex]
- Add 1 to both sides:
[tex]\[
5b = 25
\][/tex]
- Divide by 5:
[tex]\[
b = 5
\][/tex]
4. Find the Number of Purple Marbles [tex]$p$[/tex]:
- Now, plug [tex]$b = 5$[/tex] back into the relationship [tex]$p = 4b - 1$[/tex]:
[tex]\[
p = 4(5) - 1
\][/tex]
- Simplify it:
[tex]\[
p = 20 - 1
\][/tex]
[tex]\[
p = 19
\][/tex]
So, the number of purple marbles is 19. To confirm, we can substitute these values back into our original equations:
- [tex]$b + p = 24$[/tex]:
[tex]\[
5 + 19 = 24
\][/tex]
This confirms that our answer is correct.
Lastly, let's match the given table with our computed values:
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline \multicolumn{2}{|c|}{ Number of Marbles } \\
\hline[tex]$b$[/tex] & [tex]$p$[/tex] & [tex]$b + p - 24$[/tex] & [tex]$p - 4b - 1$[/tex] \\
\hline 5 & 19 & 0 & 0 \\
\hline
\end{tabular}
\][/tex]
The number of purple marbles, [tex]$p$[/tex], is indeed 19.
