Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Our Q&A platform offers a seamless experience for finding reliable answers from experts in various disciplines. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
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.
### 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.
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.
