Alright, let's break this problem down step by step.
1. Initial Amount of Shampoo:
Nico starts with a bottle of shampoo that contains [tex]\(30 \frac{1}{2}\)[/tex] ounces. We can convert this mixed number into an improper fraction for easier manipulation, or simply a decimal. Converting [tex]\(30 \frac{1}{2}\)[/tex] to a decimal, we get:
[tex]\[
30 \frac{1}{2} = 30.5 \text{ ounces}
\][/tex]
2. Spill Incident:
Nico accidentally spills 3 ounces of shampoo. Thus, the effective amount of shampoo left in the bottle is:
[tex]\[
30.5 - 3 = 27.5 \text{ ounces}
\][/tex]
3. Shampoo Per Bath:
Each time Nico shampoos his dog, he uses [tex]\(1 \frac{1}{4}\)[/tex] ounces of shampoo. Again, converting this to a decimal:
[tex]\[
1 \frac{1}{4} = 1.25 \text{ ounces}
\][/tex]
4. Equation Setup:
Let [tex]\( x \)[/tex] be the number of times Nico can shampoo his dog. We set up the equation based on the remaining shampoo and the amount used per bath.
5. Formulating the Equation:
The effective remaining shampoo is [tex]\( 27.5 \)[/tex] ounces, and this needs to be equal to the [tex]\(1.25x\)[/tex] ounces used per bath:
[tex]\[
1.25x = 27.5
\][/tex]
6. Solving for x:
[tex]\[
x = \frac{27.5}{1.25}
\][/tex]
Performing the division:
[tex]\[
x = 22
\][/tex]
Thus, Nico can bathe his dog 22 times with the remaining shampoo.
7. Verify the Correct Equation:
Based on the given choices, we identify the correct equation and validate our value of [tex]\( x \)[/tex]:
- [tex]\( 1 \frac{1}{4} x - 3 = 30 \frac{1}{2} \)[/tex] when [tex]\( x = 22 \)[/tex]:
Substituting [tex]\( x = 22 \)[/tex]:
[tex]\[
1.25 \times 22 - 3 = 30.5
\][/tex]
[tex]\[
27.5 - 3 = 30.5
\][/tex]
[tex]\[
1.25 \times 22 - 3 = 30.5 \quad \text{(True)}
\][/tex]
Given that the correct equation is:
[tex]\[
1.25x - 3 = 30.5
\][/tex]
And the value of [tex]\( x \)[/tex] is:
[tex]\[
x = 22
\][/tex]
So the correct equation and value for [tex]\( x \)[/tex] are:
[tex]\[
1 \frac{1}{4} x-3=30 \frac{1}{2} ; x=22 \text{ times}
\][/tex]
