Question

The arithmetic mean between two positive numbers a and b where (a > b) is twice their geometric mean. Then \(\frac{\text { a }}{\mathrm{b}}\) (a) \(2+\sqrt{3}\) (b) \(7+4 \sqrt{3}\) (c) \(2-\sqrt{3}\) (d) \(7-4 \sqrt{3}\)

Short answer

Answer: (b) \(7+4 \sqrt{3}\)

Step-by-step solution

Write the relationship between arithmetic mean and geometric mean

The arithmetic mean of two positive numbers a and b is given by: $$ \frac{a+b}{2} $$ The geometric mean of the positive numbers a and b is given by: $$ \sqrt{ab} $$ We are given that the arithmetic mean is twice the geometric mean, so we have: $$ \frac{a+b}{2} = 2\sqrt{ab} $$

Simplify the equation and solve for a/b

To simplify the equation, we can multiply both sides by 2 to get rid of the denominators: $$ a+b = 4\sqrt{ab} $$ Now, let \(k=\frac{a}{b}\), so that \(a = kb\). Substitute this into the equation: $$ kb+b = 4\sqrt{kb \cdot b} $$ Divide by b to remove it from the equation: $$ k+1 = 4\sqrt{k} $$ Square both sides of the equation to get rid of the square root: $$ (k+1)^2 = 16k $$

Solve the quadratic equation for k

Expand the left side of the equation: $$ k^2 + 2k + 1 = 16k $$ Rearrange the equation to have all terms on one side and solve for k: $$ k^2 - 14k + 1 = 0 $$ We have a quadratic equation of the form \(k^2 - 14k + 1 = 0\). Use the quadratic formula to find the roots of the equation: $$ k = \frac{-(-14) \pm \sqrt{(-14)^2 - 4(1)(1)}}{2(1)} $$ Solving, we get the two possible values for k: $$ k = 7 \pm 4\sqrt{3} $$

Identify the appropriate solution

The two possible values of k (\(k = 7 + 4\sqrt{3}\) or \(k = 7 - 4\sqrt{3}\)) correspond to the values \(\frac{a}{b}\) for their respective pairs of a and b. Since a > b, we need the larger of the two given values. Our solution is therefore: $$ \frac{a}{b} = 7 + 4\sqrt{3} $$ The correct answer is (b) \(7+4 \sqrt{3}\).

Key concepts

Arithmetic Mean

The arithmetic mean, often referred to as the average, is a concept used to describe the central value of a set of numbers. It is calculated by adding all the numbers together and then dividing by the count of the numbers. For two numbers, this is expressed as \( \frac{a+b}{2} \).

In the context of the given problem, the arithmetic mean represents a midway point between the two numbers, and it is particularly significant because it relates directly to the geometric mean through a given ratio. Understanding this relationship is essential in solving problems that involve comparing these means.

Geometric Mean

The geometric mean, on the other hand, is a measure of central tendency, which is calculated by multiplying all the numbers together and then taking the nth root of the total—where n is the total number of values. For two positive numbers, the geometric mean is \( \sqrt{ab} \).

The geometric mean is used especially when dealing with proportions and growth rates, like in finance or biology. In our exercise, it serves as a comparative benchmark to the arithmetic mean, forming the basis of the algebraic equation that we must solve. Despite the arithmetic mean being a more commonly known concept, the geometric mean is equally important for understanding the nature of different kinds of averages.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, generally written in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \eq 0 \). They are solved using methods like factoring, completing the square, or applying the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

For the problem at hand, solving the quadratic equation after substituting \(k = \frac{a}{b}\) allows us to find the ratio that we are looking for. Quadratic equations are foundational in algebra and appear in many areas of mathematics and science, making them a crucial concept to understand.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They are used to describe relationships and to solve for unknowns. Simplifying algebraic expressions by performing operations such as addition, subtraction, multiplication, division, and root extraction is key to solving algebraic problems.

In our exercise, an understanding of how to manipulate algebraic expressions is necessary to isolate the variable \( k \) and ultimately find the value that answers the original question. Mastery in handling algebraic expressions empowers students to tackle a wide range of mathematical problems beyond just this exercise.