Arithmetic Mean

Step 1: Recall the formulas for arithmetic mean (AM) and geometric mean (GM).

The arithmetic mean of two numbers and is given by:

The geometric mean of two numbers and is given by:

Step 2: Use the given values of AM and GM to set up equations.

From the problem, the arithmetic mean is 75:

From the problem, the geometric mean is 21:

Step 3: Solve the system of equations.

We now have the following system of equations:

Let and be the roots of a quadratic equation. The quadratic equation with roots and is:

Substitute and :

Step 4: Solve the quadratic equation.

Use the quadratic formula:

where , , and . Substitute these values:

Simplify:

Compute the two roots:

Step 5: Identify the numbers.

The two numbers are and .

Final Answer: The numbers are , which corresponds to option .

Correct answer: a

Explanation:
The arithmetic mean of a set of numbers is given by the sum of the numbers divided by the number of terms. Here, the terms are , , , and . The arithmetic mean is: Simplifying the numerator: So, the arithmetic mean is:

Hence, the arithmetic mean is a.

Correct answer: 13, 7

Explanation:
Let the two numbers be and .
Given:
Arithmetic mean = 10: Product = 91: Now, try pairs that add to 20 and multiply to 91.
and

Hence, the two numbers are 13 and 7.

Correct answer: 2:1

Explanation:
In a triangle, the centroid is the point where the three medians intersect.
The centroid divides each median in the ratio: That is, the centroid is located two-thirds of the way from a vertex along the median.

Hence, the correct ratio is 2:1.