Answered

Westonci.ca is the premier destination for reliable answers to your questions, provided by a community of experts. Get immediate and reliable solutions to your questions from a community of experienced experts on our Q&A platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.

5. Tierra, Nico, and Alex had [tex]$\$[/tex]865[tex]$ altogether. Tierra spent $[/tex]\frac{2}{5}[tex]$ of her money. Nico spent $[/tex]\[tex]$40$[/tex], and Alex spent twice as much as Tierra. If the three friends had the same amount of money left, how much money did Alex have in the beginning?

[tex]\[
\begin{array}{l}
T=\frac{3}{5}x \\
N=x-40
\end{array}
\][/tex]

Sagot :

GinnyAnswer
To determine how much money Alex had initially, let’s go through the problem step by step.

### Step 1: Define the Variables
Let [tex]\( T \)[/tex], [tex]\( N \)[/tex], and [tex]\( A \)[/tex] be the initial amounts of money Tierra, Nico, and Alex had, respectively. Let [tex]\( x \)[/tex] be the amount of money they each had left after spending.

### Step 2: Set Up the Equations
We know the total amount of money they had initially:
[tex]\[ T + N + A = 865 \][/tex]

We are given information about how much each person spent and how much they had left:
- Tierra spent [tex]\(\frac{2}{5}\)[/tex] of her money:
[tex]\[ T_{left} = T - \frac{2}{5}T = \frac{3}{5}T \][/tex]
- Nico spent [tex]\( \$40 \)[/tex]:
[tex]\[ N_{left} = N - 40 \][/tex]
- Alex spent twice as much as Tierra:
[tex]\[ A_{spent} = 2 \times \frac{2}{5}T = \frac{4}{5}T \][/tex]
So, Alex’s remaining money is:
[tex]\[ A_{left} = A - \frac{4}{5}T \][/tex]

Since after spending, they all had the same amount of money left:
[tex]\[ \frac{3}{5}T = N - 40 = A - \frac{4}{5}T \][/tex]

### Step 3: Solve the System of Equations
We now have three key equations:
[tex]\[ T + N + A = 865 \quad \text{(Equation 1)} \][/tex]
[tex]\[ \frac{3}{5}T = N - 40 \quad \text{(Equation 2)} \][/tex]
[tex]\[ \frac{3}{5}T = A - \frac{4}{5}T \quad \text{(Equation 3)} \][/tex]

#### Simplify Equation 3:
[tex]\[ \frac{3}{5}T + \frac{4}{5}T = A \][/tex]
[tex]\[ \frac{7}{5}T = A \][/tex]
[tex]\[ A = \frac{7}{5}T \quad \text{(Simplified form of Equation 3)} \][/tex]

#### Use Equation 2:
[tex]\[ N = \frac{3}{5}T + 40 \quad \text{(Rearranged form of Equation 2)} \][/tex]

#### Substitute [tex]\(N\)[/tex] and [tex]\(A\)[/tex] back into Equation 1:
[tex]\[ T + \left(\frac{3}{5}T + 40\right) + \frac{7}{5}T = 865 \][/tex]
[tex]\[ T + \frac{3}{5}T + 40 + \frac{7}{5}T = 865 \][/tex]
Combine like terms:
[tex]\[ T + \frac{3}{5}T + \frac{7}{5}T + 40 = 865 \][/tex]
[tex]\[ T + 2T + 40 = 865 \][/tex]
[tex]\[ 3T + 40 = 865 \][/tex]
Solve for [tex]\(T\)[/tex]:
[tex]\[ 3T = 825 \][/tex]
[tex]\[ T = 275 \][/tex]

Now use the value of [tex]\(T\)[/tex] to find [tex]\(N\)[/tex] and [tex]\(A\)[/tex]:
[tex]\[ N = \frac{3}{5} \times 275 + 40 \][/tex]
[tex]\[ N = 165 + 40 \][/tex]
[tex]\[ N = 205 \][/tex]

[tex]\[ A = \frac{7}{5} \times 275 \][/tex]
[tex]\[ A = 385 \][/tex]

### Conclusion
Therefore, Alex initially had:
[tex]\[ \boxed{385} \][/tex]

So, Alex had [tex]\( \$385 \)[/tex] in the beginning.
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.