To solve this problem, we need to translate the given word problem into a mathematical equation and solve it step by step.
1. Identify and define variables:
Let [tex]\( x \)[/tex] be the number of shells Mary has.
2. Express the number of shells Gracie has:
- According to the problem, Gracie has 5 more than [tex]\(1 \frac{1}{4} \)[/tex] (or 1.25) times the number of shells Mary has.
- Mathematically, this is: [tex]\[ \text{Gracie's shells} = 1.25x + 5 \][/tex]
3. Express the number of shells Nancy has:
- According to the problem, Nancy has 1 more than [tex]\(1 \frac{1}{2} \)[/tex] (or 1.5) times the number of shells Mary has.
- Mathematically, this is: [tex]\[ \text{Nancy's shells} = 1.5x + 1 \][/tex]
4. Set up the equation using the information that Gracie and Nancy have the same number of shells:
- We equate the expressions for Gracie's and Nancy's shells:
[tex]\[ 1.25x + 5 = 1.5x + 1 \][/tex]
5. Solve for [tex]\( x \)[/tex]:
- Rearrange the equation to isolate [tex]\( x \)[/tex]:
[tex]\[ 1.25x + 5 = 1.5x + 1 \][/tex]
- Subtract [tex]\( 1.25x \)[/tex] from both sides:
[tex]\[ 5 = 0.25x + 1 \][/tex]
- Subtract 1 from both sides:
[tex]\[ 4 = 0.25x \][/tex]
- Divide both sides by 0.25:
[tex]\[ x = 16 \][/tex]
6. Verify Gracie's and Nancy's number of shells using [tex]\( x = 16 \)[/tex]:
- Gracie's shells: [tex]\( 1.25 \times 16 + 5 = 20 + 5 = 25 \)[/tex]
- Nancy's shells: [tex]\( 1.5 \times 16 + 1 = 24 + 1 = 25 \)[/tex]
Since both Gracie's and Nancy's shells equal 25 when [tex]\( x = 16 \)[/tex], they have the same number of shells.
Summary:
- The correct equation representing the situation is: [tex]\[ 1.25x + 5 = 1.5x + 1 \][/tex]
- The solution for the number of shells Mary has is: [tex]\[ x = 16 \][/tex]
So, the correct equations and solution from the given options are:
- [tex]\(\frac{3}{2} x+1=\frac{5}{4} x+5\)[/tex]
- [tex]\( x = 16 \)[/tex]
