Properties of the Transformation

Oct 13, 2025 by miodeya.com 95 views

Compound Interest

Calculating Compound Interest

In this section, we will calculate the interest earned on a principal amount of $5200 at a rate of 2.5% per annum after 5 years in a compound interest account.

Step 1: Convert the interest rate to a decimal

The interest rate is given as 2.5%. To convert it to a decimal, we divide by 100:

2.5 ÷ 100 = 0.025

Step 2: Calculate the interest earned per year

The formula for compound interest is:

A = P(1 + r)^n

Where:

A = the future value of the investment/loan, including interest
P = principal investment amount (the initial deposit or loan amount)
r = annual interest rate (in decimal)
n = number of years the money is invested or borrowed for

However, since we are only interested in the interest earned, we can use the formula:

Interest = P * r * n

Plugging in the values, we get:

Interest = 5200 * 0.025 * 5

Step 3: Calculate the total interest earned

Interest = 5200 * 0.025 * 5 Interest = 650

So, the total interest earned on $5200 at 2.5% per annum after 5 years in a compound interest account is $650.

Example Use Case

Suppose you deposit $5200 into a savings account that earns a compound interest rate of 2.5% per annum. After 5 years, you want to know how much interest you have earned. Using the formula above, you can calculate the interest earned and determine the total amount in your account.

Transformations

Reflection and Rotation

In this section, we will explore the transformation of a triangle using the rule (x, y) → (-y, x).

Step 1: Understand the transformation

The transformation rule (x, y) → (-y, x) means that for every point (x, y) on the original triangle, the corresponding point on the transformed triangle is (-y, x).

Step 2: Apply the transformation to the triangle

Let's consider a triangle ABC with vertices A(1, 2), B(3, 4), and C(5, 6). To apply the transformation, we replace x with -y and y with x for each vertex:

A(1, 2) → A'(-2, 1) B(3, 4) → B'(-4, 3) C(5, 6) → C'(-6, 5)

Step 3: Plot the transformed triangle

Plotting the transformed vertices A', B', and C', we get a new triangle A'B'C'.

Properties of the Transformation

The transformation (x, y) → (-y, x) is a reflection across the line y = -x followed by a rotation of 90 degrees clockwise.

Example Use Case

Suppose you have a triangle with vertices A(2, 3), B(4, 5), and C(6, 7). You want to apply the transformation (x, y) → (-y, x) to the triangle. Using the formula above, you can calculate the new vertices of the transformed triangle and plot the new triangle.

Conclusion

In this weekly practice, we calculated the interest earned on a principal amount of $5200 at a rate of 2.5% per annum after 5 years in a compound interest account. We also explored the transformation of a triangle using the rule (x, y) → (-y, x) and applied the transformation to a sample triangle.

Key Takeaways

The formula for compound interest is A = P(1 + r)^n, but we can use the formula Interest = P * r * n to calculate the interest earned.
The transformation (x, y) → (-y, x) is a reflection across the line y = -x followed by a rotation of 90 degrees clockwise.
To apply the transformation to a triangle, we replace x with -y and y with x for each vertex.

Practice Problems

Calculate the interest earned on a principal amount of $8000 at a rate of 3.5% per annum after 10 years in a compound interest account.
Apply the transformation (x, y) → (-y, x) to the triangle with vertices A(1, 2), B(3, 4), and C(5, 6).
Plot the transformed triangle and determine its properties.

Answer Key

Interest = 8000 * 0.035 * 10 = 2800
A'(-2, 1), B'(-4, 3), C'(-6, 5)
The transformed triangle is a reflection of the original triangle across the line y = -x followed by a rotation of 90 degrees clockwise.
Weekly Practice 3.10: Compound Interest and Transformations - Q&A ===========================================================

Q&A: Compound Interest

Q: What is compound interest?

A: Compound interest is the interest earned on both the principal amount and any accrued interest over time. It is calculated using the formula A = P(1 + r)^n, where A is the future value of the investment/loan, P is the principal investment amount, r is the annual interest rate, and n is the number of years.

Q: How is compound interest different from simple interest?

A: Simple interest is calculated only on the principal amount, whereas compound interest is calculated on both the principal amount and any accrued interest. This means that compound interest grows faster over time than simple interest.

Q: What is the formula for calculating compound interest?

A: The formula for calculating compound interest is A = P(1 + r)^n, where A is the future value of the investment/loan, P is the principal investment amount, r is the annual interest rate, and n is the number of years.

Q: How do I calculate the interest earned on a principal amount using compound interest?

A: To calculate the interest earned on a principal amount using compound interest, you can use the formula Interest = P * r * n, where P is the principal investment amount, r is the annual interest rate, and n is the number of years.

Q: What is the effect of compounding frequency on compound interest?

A: The compounding frequency refers to how often the interest is compounded per year. For example, monthly compounding means that the interest is compounded 12 times per year. The more frequently the interest is compounded, the faster the compound interest grows.

Q: Can I use compound interest to calculate the future value of an investment?

A: Yes, you can use compound interest to calculate the future value of an investment. Simply plug in the values for the principal amount, interest rate, and number of years into the formula A = P(1 + r)^n.

Q&A: Transformations

Q: What is a transformation in mathematics?

A: A transformation is a way of changing the position, size, or shape of a geometric figure. It can be a translation, rotation, reflection, or dilation.

Q: What is the rule (x, y) → (-y, x) in mathematics?

A: The rule (x, y) → (-y, x) is a transformation that reflects a point across the line y = -x and then rotates it 90 degrees clockwise.

Q: How do I apply the transformation (x, y) → (-y, x) to a triangle?

A: To apply the transformation (x, y) → (-y, x) to a triangle, you need to replace x with -y and y with x for each vertex of the triangle.

Q: What is the effect of the transformation (x, y) → (-y, x) on the coordinates of a point?

A: The transformation (x, y) → (-y, x) changes the coordinates of a point from (x, y) to (-y, x).

Q: Can I use the transformation (x, y) → (-y, x) to solve problems involving reflections and rotations?

A: Yes, you can use the transformation (x, y) → (-y, x) to solve problems involving reflections and rotations.

Practice Problems

Calculate the interest earned on a principal amount of $10000 at a rate of 4% per annum after 15 years in a compound interest account.
Apply the transformation (x, y) → (-y, x) to the triangle with vertices A(2, 3), B(4, 5), and C(6, 7).
Plot the transformed triangle and determine its properties.

Answer Key

Interest = 10000 * 0.04 * 15 = 6000
A'(-3, 2), B'(-5, 4), C'(-7, 6)
The transformed triangle is a reflection of the original triangle across the line y = -x followed by a rotation of 90 degrees clockwise.