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Question 1172925: Mrs. Diez has savings account in two banks. The combined amount of these savings is at least Php 150,000. One bank gives an interest of 4% while the other bank gives 6%. In a year, Mrs. Diez receives at most Php12,000.
a. What mathematical statements represent the given situation?
b. Draw the graphs of the mathematical statements.
c. How will you determine the amount of savings in each bank account?
d. Give one possible amount of savings in both accounts.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Absolutely! Let's break down this problem step-by-step.
**a. Mathematical Statements**
Let:
* `x` = amount of savings in the bank with 4% interest
* `y` = amount of savings in the bank with 6% interest
1. **Combined savings:**
* The combined amount of savings is at least Php 150,000.
* Mathematical statement: `x + y ≥ 150000`
2. **Total interest:**
* The total interest received is at most Php 12,000.
* Interest from the 4% bank: 0.04x
* Interest from the 6% bank: 0.06y
* Mathematical statement: `0.04x + 0.06y ≤ 12000`
3. **Non-negative savings:**
* Savings cannot be negative.
* Mathematical statements: `x ≥ 0` and `y ≥ 0`
**b. Graphs of the Mathematical Statements**
1. **x + y ≥ 150000:**
* To graph this, we first graph the line `x + y = 150000`.
* Find the intercepts:
* If x = 0, then y = 150000. (0, 150000)
* If y = 0, then x = 150000. (150000, 0)
* Draw the line connecting these points.
* Since it's `x + y ≥ 150000`, we shade the region above the line.
2. **0.04x + 0.06y ≤ 12000:**
* To make it easier, multiply by 100 to remove decimals: `4x + 6y ≤ 1200000`
* Divide by 2 to simplify: `2x + 3y ≤ 600000`
* Graph the line `2x + 3y = 600000`.
* Find the intercepts:
* If x = 0, then 3y = 600000, so y = 200000. (0, 200000)
* If y = 0, then 2x = 600000, so x = 300000. (300000, 0)
* Draw the line connecting these points.
* Since it's `2x + 3y ≤ 600000`, we shade the region below the line.
3. **x ≥ 0 and y ≥ 0:**
* This means we are only considering the first quadrant of the coordinate plane.
To fully graph this, plot the points and shade the appropriate regions. The solution region is where all shaded areas overlap within the first quadrant.
**c. Determining the Amount of Savings**
1. **Graphing and Intersection:**
* The solution to the system of inequalities is the region where the graphs of all inequalities overlap.
* The corner points of this region are important. They represent possible solutions.
* The exact amounts of savings can be determined by finding the coordinates of the intersection point of the lines `x + y = 150000` and `2x + 3y = 600000`.
2. **Solving the System of Equations:**
* Solve the system of equations:
* `x + y = 150000`
* `2x + 3y = 600000`
* From the first equation, `x = 150000 - y`.
* Substitute this into the second equation: `2(150000 - y) + 3y = 600000`
* `300000 - 2y + 3y = 600000`
* `y = 300000`
* Substitute `y = 300000` back into `x = 150000 - y`: `x = 150000 - 300000 = -150000`
* Because x is negative, this corner point is outside of the feasible region. Therefore the intersection of the two lines is outside of the solution set. We must find another method.
* Test corner points that are created by the intersection of the individual lines, with the x or y axis.
* Test the point (150000,0).
* 150000+0 = 150000. 0.04(150000) + 0.06(0) = 6000. This point is valid.
* Test the point (0,150000).
* 0+150000 = 150000. 0.04(0) + 0.06(150000) = 9000. This point is valid.
* Test the point (300000,0).
* 300000+0 = 300000. 0.04(300000) + 0.06(0) = 12000. This point is valid.
* Test the point (0,200000).
* 0+200000 = 200000. 0.04(0)+0.06(200000) = 12000. This point is valid.
**d. Possible Amounts of Savings**
Here's one possible solution:
* Mrs. Diez could have Php 150,000 in the 4% bank (x = 150000) and Php 0 in the 6% bank (y = 0).
* Total savings: 150000 + 0 = 150000
* Total interest: (0.04 * 150000) + (0.06 * 0) = 6000
Another possible solution.
* Mrs. Diez could have Php 0 in the 4% bank (x = 0) and Php 150,000 in the 6% bank (y = 150000).
* Total savings: 0 + 150000 = 150000
* Total interest: (0.04 * 0) + (0.06 * 150000) = 9000.
Another possible solution.
* Mrs. Diez could have Php 300,000 in the 4% bank (x=300000) and Php 0 in the 6% bank (y=0).
* Total savings: 300000+0=300000
* Total interest: (0.04 * 300000) + (0.06 * 0) = 12000.
Another possible solution.
* Mrs. Diez could have Php 0 in the 4% bank (x=0) and Php 200000 in the 6% bank (y=200000).
* Total savings: 0+200000=200000
* Total interest: (0.04 * 0) + (0.06 * 200000) = 12000.
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