Question 1172991: 2. Mr. Agoncillo has at least Php150,000 deposited in two banks. One bank gives an annual interest of 4% while the other bank gives 6%. In a year, Mr. Agoncillo receives at most Php12,000.
a. What mathematical statements represent the given situation?
b. Graph the system.
c. Give two possible amounts Mr. Agoncillo could have deposited in each bank. Justify your answers.
d. If you were Mr. Agoncillo, in what bank account would you place greater amount of money? Why?
thanks so much in advance!
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem step-by-step.
**a. Mathematical Statements (Inequalities)**
* **Variables:**
* Let 'x' be the amount deposited in the 4% interest bank.
* Let 'y' be the amount deposited in the 6% interest bank.
* **Inequalities:**
1. **Total Deposit:** The total deposit is at least Php 150,000.
* x + y ≥ 150,000
2. **Total Interest:** The total interest is at most Php 12,000.
* 0.04x + 0.06y ≤ 12,000
3. **Non-negativity:** Deposits cannot be negative.
* x ≥ 0
* y ≥ 0
**b. Graph the System**
1. **x + y ≥ 150,000**
* Graph the line x + y = 150,000.
* Intercepts: (150,000, 0) and (0, 150,000).
* Shade the region above the line.
2. **0.04x + 0.06y ≤ 12,000**
* Simplify: Multiply by 100: 4x + 6y ≤ 1,200,000.
* Simplify further: Divide by 2: 2x + 3y ≤ 600,000.
* Graph the line 2x + 3y = 600,000.
* Intercepts: (300,000, 0) and (0, 200,000).
* Shade the region below the line.
3. **x ≥ 0 and y ≥ 0**
* This restricts the solution to the first quadrant.
4. **The Solution Region:** The solution is the area where all shaded regions overlap.
**c. Two Possible Amounts**
1. **Point 1: (150,000, 0)**
* x = 150,000 (4% bank)
* y = 0 (6% bank)
* Total deposit: 150,000
* Total interest: 0.04(150,000) + 0.06(0) = 6,000
* This point satisfies both conditions.
2. **Point 2: (100,000, 200,000)**
* x=100,000 (4% bank)
* y=200,000 (6% bank)
* Total deposit: 300,000
* Total interest: 0.04(100,000) + 0.06(200,000) = 4,000 + 12,000 = 16,000. This point does not satisfy the total interest constraint.
* We need a point that is within the solution set.
* Let's test the point (200,000, 100,000)
* x=200,000 (4% bank)
* y=100,000 (6% bank)
* Total deposit: 300,000
* Total interest: 0.04(200,000) + 0.06(100,000) = 8,000 + 6,000 = 14,000. This point does not satisfy the total interest constraint.
* Let's test the point (250,000, 50,000)
* x=250,000 (4% bank)
* y=50,000 (6% bank)
* Total deposit: 300,000
* Total interest: 0.04(250,000) + 0.06(50,000) = 10,000 + 3,000 = 13,000. This point does not satisfy the total interest constraint.
* Let's test the point (180,000, 20,000)
* x=180,000 (4% bank)
* y=20,000 (6% bank)
* Total deposit: 200,000
* Total interest: 0.04(180,000) + 0.06(20,000) = 7,200 + 1,200 = 8,400. This point satisfies the constraints.
**d. Bank Account Preference**
* If I were Mr. Agoncillo, I would place a greater amount of money in the 6% interest bank.
* **Reasoning:**
* The 6% interest bank provides a higher return on investment.
* Even though there is a constraint on the total interest, maximizing the amount in the 6% bank within the constraints will yield a higher overall return.
* However, if Mr. Agoncillo has other considerations such as risk tolerance or liquidity needs, he might opt for a different allocation.
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