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CAT Relation between AM, GM and HM - Practice Questions & MCQ

Edited By admin | Updated on Oct 04, 2023 04:20 PM | #CAT

Syllabus

Quick Facts

5 Questions around this concept.

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MEDIUM

If $a,b,c>0$ , then the minimum value of $\frac{ab}{c^{2}}+\frac{bc}{a^{2}}+\frac{ca}{b^{2}}$ is

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$\\I\! \! f\; a_{1}, a_{2}, a_{3}, a_{4}>0, then \;the \;minimum\; value\; of\\\\ \left(a_{1}+a_{2}+a_{3}+a_{4}\right)\left(\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}+\frac{1}{a_{4}}\right) \text { is }$

None of these

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Concepts Covered - 1

Relation between AM, GM and HM

Definition:
- Arithmetic Mean (AM) of two numbers a and b is given by: $\mathrm{ A M=\frac{a+b}{2}}$
- Geometric Mean (GM) of two numbers a and b is given by: $\mathrm{ \mathrm{GM}=\sqrt{a \times b}}$
- Harmonic Mean (HM) of two numbers a and b is given by: $\mathrm{\mathrm{HM}=\frac{2 a b}{a+b}}$
Key Relationship: For any two positive numbers, the AM is always greater than or equal to the GM, which in turn is always greater than or equal to the HM. Symbolically:
$\mathrm{ A M \geq G M \geq H M }$

Foundation Building Questions:

Question 1: If the arithmetic mean and geometric mean of two positive numbers are 10 and 8 respectively, find the harmonic mean.

Solution:
Given, AM = 10 and GM = 8.

Using the relationship between $\mathrm{AM}, \mathrm{GM}$, and $\mathrm{HM}$ :
$\mathrm{ H M=\frac{2 \times A M \times G M}{A M+G M}=\frac{2 \times 10 \times 8}{10+8}=\frac{160}{18}=\frac{80}{9} }$
Question 2: The harmonic mean between two numbers is 12. If one number is 15 , find the other.

Solution:
Let the other number be x.
Using the formula for $\mathrm{ \mathrm{HM}, \mathrm{HM}=\frac{2 a b}{a+b}=12 }$
Given $\mathrm{ \mathrm{a}=15, \mathrm{HM}=12. }$
Solving for b (or x ) we get $\mathrm{ x=\frac{30 \times 15}{15+12-30}=45 }$

Application of Previous Concepts:

The relationship between AM, GM, and HM can sometimes be utilised in solving complex problems that involve sequences, especially when one needs to establish a relationship between terms or find an unknown term. This concept is also often integrated with the earlier concepts on AP, GP, and HP, especially when deducing properties or solving for specific terms.

Tips and Tricks:

1. Remember the Inequality: The relationship is fundamental and comes handy in a variety of problems.

2. AM-GM-HM for Multiple Terms: When more than two numbers are involved, use their general formulas and remember that the inequality still holds.

3. Visual Representation: Plotting AM, GM, and HM on a number line can sometimes help in visualising problems and understanding the relationship better.

4. Harmonic Mean Shortcut: The harmonic mean of two numbers is just twice their product divided by their sum. This shortcut can save time in exams.

5. Application in Real-world Problems: This relationship can be used in optimization problems, such as when maximising or minimising certain quantities.

6. AM-GM Inequality Extension: The AM-GM inequality can be extended for sets of numbers. For n positive numbers: the AM of the numbers is always greater than or equal to the nth root of their product (GM).

By understanding the relationships between the Arithmetic, Geometric, and Harmonic means, students can tackle a wide variety of problems. It's crucial to remember the core inequality and how it can be applied to different scenarios, whether it's deducing relationships between terms or solving for specific quantities.