Distributive Property and Geometric Mean Formula

Distributive Property

The distributive property is one of the most frequently used properties in basic Mathematics. In general, it refers to the distributive property of multiplication over addition or subtraction. It is also known as the distributive law of multiplication. When we distribute something, we are dividing it into parts. In math, the distributive property helps to simplify difficult problems. This is because it breaks down expressions into the sum or difference of two numbers. In this topic, we will learn about the distributive property and its examples. Let us begin it!

Distributive Property

What is the distributive property?

The distributive property will allow multiplying a sum value by multiplying each addend separately. And, then add the products. Multiplying the number immediately outside the parentheses with those given inside values. And, then adding the products together.

When an algebraic expression is having parentheses with variables. Here variable is a quantity that may change within the given context of a mathematical problem. It is usually represented by a single letter.

Two main laws under this property:

1. Distributive property of multiplication over addition:

It is a fact that whether we use the distributive property or follow the order of given operations, we will arrive at the same answer. This is the main concept. So, we can see it as given below:

Using the distributive law, we will Multiply, or distribute, the outer term to the inner terms. Then combine like terms, and solve the equation.

This property states that:

m × (n + o) = (m × n) + (m ×o)

Here variables m, n and o are the real numbers.

Let’s use a real-life example to help make this clearer. Imagine one student and her two friends each have seven mangos and four bananas. How many pieces of fruit do all three students will have in total?

In their lunch bags i.e. as the parentheses, each student will have seven mangos and four bananas. To know the number of pieces of fruit in total, students will need to multiply the whole thing by 3.

When we break it down, we are multiplying 7seven mangos and four bananas by 3 students. So, we end up with 21 mangos and 12 bananas for a total of 33 pieces of fruit.

2. Distributive property of multiplication over subtraction:

Similar to the above operation with addition, performing the distributive property with subtraction follows the same rules. But here exception is that we are finding the difference instead of the sum. Although, it doesn’t matter if the operation is addition or subtraction, keep whichever one is in the parentheses.

This property states that:

m×(n−o)=(m×n)–(m×o)

Where m, n and o are the real numbers.

 Solved examples for You

Example-1: Prove the distributive property for the following expression:

3×(4+8)=(3×4)+(3×8)

Solution:

LHS: 3 × (4+8)

=3 × (12)

=36

RHS: (3×4) +(3×8)

=(12)+(24)

=36

Since  LHS = RHS

Hence Proved.

Example-2: Prove the distributive property for the following expression:

5×(7−3)=(5×7)–(5×3)

Solution:

LHS: 5 × (7-3)

=20

RHS: (5×7)−(5×3)

=(35)-(15)

=20

Since  LHS = RHS

Hence Proved.

Geometric Mean Formula

We have used the arithmetic mean in many data-related problems. Here we will see another such term frequently used with data analysis. A geometric mean formula is used to calculate the geometric mean of a set of numbers. It is a type of mean that indicates the central tendency of a set of numbers by using the product of their values. It is also defined as the nth root of the product of n numbers. The geometric mean is properly defined only for a positive set of real numbers. In this article, we will discuss the geometric mean formula with examples. Let us begin learning!

Geometric Mean Formula

What is the Geometric Mean?

The geometric mean is the mean value of a set of products. Its calculation is commonly used to determine the performance results of an investment or portfolio. It can be stated as “the nth root value of the product of n numbers.”

The geometric mean should be used when working with percentages, which are derived from values. The geometric mean is a very useful tool for calculating portfolio performance. It is because it takes into account the effects of compounding.

The formula for Geometric Mean

The geometric mean is used as a proportion in geometry and therefore it is sometimes called the “mean proportional”. The mean proportional of two positive numbers a and b, will e the positive number x, so that:

ax=xb

i.e. after doing cross multiplication we get

x=a×b−−−−√

In general for n multiple numbers as a_1,a_2, a_3,…..,a_n then geometric mean GM will be the nth root of the product of the numbers. In terms of formula it is:

GM = a1×a2×a3×…….×an−−−−−−−−−−−−−−−−−−−−−√n

Some real-life uses of geometric mean:

1. Aspect Ratios:

The geometric mean has been used in film and video also to find the appropriate aspect ratios i.e. the proportion of the width to the height of a screen or image. It is used to find an appropriate balancing between the two aspect ratios as well as for distorting or cropping both ratios equally.

2. Computer Science:

Computers use mind-boggling amounts of large data which generally requires the summarization for many applications using various statistical measurements.

3. Medicine:

The Geometric Mean has many applications in the medical industry also. It is known as the “gold standard” for some measurements, including for the calculation of gastric emptying time.

4. Proportional Growth:

It is very useful in finding the growth rate. The geometric mean is used for calculating the proportional growth as well as demand growth.

Solved Example

1: Find the geometric mean of 4 and 3?

Solution: Using the formula for G.M.,

a=4  and b=3

Geometric Mean will be:

x= √(4×3)

= 2√3

So, GM will be 3.46

Example-2: Find the geometric mean of 5 numbers as 4, 8, 3, 9 and 17?

Solution:

Here multiple numbers are taken.

n = 5

Find geometric mean using the formula:

GM = a1×a2×a3×…….×an−−−−−−−−−−−−−−−−−−−−−√n

Putting values of numbers,

We get

GM = 4×8×3×9×17−−−−−−−−−−−−−−−√5

i.e. GM = 4×8×3×9×17−−−−−−−−−−−−−−−√5

i.e. GM= 14688−−−−−√5

So, geometric mean = 6.81