Answered

Select the correct equations.

Gracie, Mary, and Nancy each have a small collection of seashells. Gracie has 5 more than [tex]$1 \frac{1}{4}$[/tex] times the number of shells Mary has. Nancy has 1 more than [tex]$1 \frac{1}{2}$[/tex] times the number of shells Mary has. Gracie and Nancy have the same number of shells. If [tex][tex]$x$[/tex][/tex] is the number of shells Mary has, identify the equation that represents this situation and identify its solution.

A. [tex]\frac{3}{2} x + 1 = \frac{5}{4} x + 5[/tex]
B. [tex]x = 24[/tex]
C. [tex]\frac{3}{2} x - 1 = \frac{5}{4} x + 5[/tex]
D. [tex]\frac{3}{2} x + 1 = \frac{5}{4} x - 5[/tex]
E. [tex]x = 18[/tex]
F. [tex]x = 16[/tex]



Answer :

GinnyAnswer
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]