Sure, let's solve this problem step-by-step.
First, let's define our variables:
- [tex]\( p \)[/tex]: the number of pennies.
- [tex]\( d \)[/tex]: the number of dimes.
We are given two pieces of information:
1. The total value of the pennies and dimes is [tex]$\$[/tex]1.50$.
2. There are five times as many pennies as dimes.
We can set up the following equations based on this information:
1. The total value equation:
[tex]\[
0.01p + 0.10d = 1.50
\][/tex]
2. The relationship between the number of pennies and dimes:
[tex]\[
p = 5d
\][/tex]
Now let's solve these equations step-by-step.
### Step 1: Substitute [tex]\( p = 5d \)[/tex] into the total value equation
Given:
[tex]\[
p = 5d
\][/tex]
Substitute [tex]\( p \)[/tex] in the first equation:
[tex]\[
0.01(5d) + 0.10d = 1.50
\][/tex]
### Step 2: Simplify the equation
Now, carry out the multiplication:
[tex]\[
0.05d + 0.10d = 1.50
\][/tex]
Combine like terms:
[tex]\[
0.15d = 1.50
\][/tex]
### Step 3: Solve for [tex]\( d \)[/tex]
Divide both sides of the equation by [tex]\( 0.15 \)[/tex]:
[tex]\[
d = \frac{1.50}{0.15}
\][/tex]
[tex]\[
d = 10
\][/tex]
So, we have found that there are [tex]\( 10 \)[/tex] dimes.
### Step 4: Find the number of pennies
Since [tex]\( p = 5d \)[/tex]:
[tex]\[
p = 5 \times 10
\][/tex]
[tex]\[
p = 50
\][/tex]
So, there are [tex]\( 50 \)[/tex] pennies.
### Conclusion
Amber has:
- [tex]\( 50 \)[/tex] pennies
- [tex]\( 10 \)[/tex] dimes
