To solve the problem, we need to carefully set up equations based on the information given:
1. Defining variables:
- Let [tex]\(x\)[/tex] be the number of seashells Mary has.
- Gracie has 5 more than [tex]\(1 \frac{1}{4}\)[/tex] (or [tex]\(1.25\)[/tex]) times the number of shells Mary has.
- Nancy has 1 more than [tex]\(1 \frac{1}{2}\)[/tex] (or [tex]\(1.5\)[/tex]) times the number of shells Mary has.
2. Writing equations for Gracie and Nancy:
- Gracie's seashells:
[tex]\[
\text{Gracie's shells} = 1.25x + 5
\][/tex]
- Nancy's seashells:
[tex]\[
\text{Nancy's shells} = 1.5x + 1
\][/tex]
3. Setting equations equal since Gracie and Nancy have the same number of shells:
[tex]\[
1.5x + 1 = 1.25x + 5
\][/tex]
4. Solving the equation step-by-step:
- Subtract 1.25x from both sides:
[tex]\[
1.5x - 1.25x + 1 = 5
\][/tex]
- Simplify:
[tex]\[
0.25x + 1 = 5
\][/tex]
- Subtract 1 from both sides:
[tex]\[
0.25x = 4
\][/tex]
- Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{4}{0.25} = 16
\][/tex]
The solution to this equation is [tex]\(x = 16\)[/tex].
So, the correct equation to represent this situation is:
[tex]\[
1.5x + 1 = 1.25x + 5
\][/tex]
And the correct solution is:
[tex]\[
x = 16
\][/tex]
Now let's compare this with the provided potential answers:
- The right equation:
[tex]\[
\frac{3}{2} x + 1 = \frac{5}{4} x + 5
\][/tex]
- The solution correctly identifying [tex]\(x = 16\)[/tex].
Finally, the correct answer is the equation:
[tex]\[
\frac{3}{2} x + 1 = \frac{5}{4} x + 5
\][/tex]
with the solution:
[tex]\[
x = 16
\][/tex]
