Least Common Multiple

Introduction:
   In arithmetic number of theory is the least common multiple or lowest common multiple (LCM) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple of both of a and of b. Since it was a multiple, it can be divided by a and b without a reminder. If either a or b is 0, so that number is no such positive integer, then LCM(a, b) is defined to be zero.

Example:

Multiples of 4 are
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, ......................
Multiple of 7 are
7, 14, 21, ................
(add 4 to each get to next multiple)

Applications

When adding and subtracting, or comparing vulgar fractions, Useful to find the least common multiple of the denominators, is  called the lowest common denominator. For instance,
{3\over15}+{1\over4}={12\over60}+{15\over60}={9\over20},
where the denominator of 60 was used to least common multiple of 15 and 4.

Hope you like the above explanation, Please leave your comments, if you have any doubts.

How to simplify Negative Exponents

Introduction:
      Exponents are also called Powers or Indices. Exponents containing negative sign are called negative exponents. The exponent of a number  says how many times to use the number in a multiplication. Dividing is the inverse (opposite) of Multiplying. A negative exponent number says how many times to divide by the number.
     Calculate the positive exponent (aⁿ)
       Then take the Reciprocal (i.e. 1/aⁿ)
      To change the sign ( minus to plus or plus to minus) of the exponent, use the Reciprocal (i.e. 1/aⁿ).

Simplifying Negative Exponents

Simplifying Negative Exponents:

            Exponent  is a small number written near the top of another number which show how many times a number or a variable is multiplied by itself. Negative exponents  indicate the inverse of the corresponding positive exponent.

Simplifying Variables with Negative Exponents:

Consider a problem like the one shown below.
x^-4  = 1/x⁴

Simplifying Fractions with Negative Exponents:

To increase a fraction to a power, raise the numerator and denominator to that power.
            a^-3b-⁴/b^-2=a-³b^-2


Hope you like the above explanation, Please leave your comments, if you have any doubts.

Arithmetic Problems

Definition:

Arithmetic involves the study of quantity, especially as the result of combining numbers. In common usage, Arithmetic refers to the simpler properties when using the traditional operations of addition, subtraction, multiplication and division with smaller values of numbers.

1) Question: The arithmetic mean between two terms in an arithmetic sequence is 21. If one term is 13, then find the other term.

A ) 58
B ) 50
C ) 29
D ) 21

Steps to derive

1 (a+b)2
[The arithmetic mean of two numbers a and b.]

2 (13+b)2 = 21
[Substitute the values.]

3 13 + b = 42

4 b = 29

5 So, the second term is 29.



Hence the right answer is Option C

2) Question: If 6 arithmetic means are introduced between 1 and 15, then find their sum.

A ) 64
B ) 48
C ) 2
D ) 32

Steps to derive

1 Let x₁, x₂, x₃, x₄, x₅, x₆ be the six arithmetic means introduced between 1 and 15 respectively.

2 1, x₁, x₂, x₃, x₄, x₅, x₆, 15 are in arithmetic progression.

3 Sum of the 8 terms = 8 / 2[1 + 15]
[The sum of the n terms S_n = n2[a₁ + a_n] where n is the number of terms, a₁ is the first term and a_n is the last term.]

4 = 4 × 16 = 64

5 Therefore, the sum of the six arithmetic means = x₁ + x₂ + x₃ + x₄ + x₅ + x₆ = 64 - 16 = 48



Hence the right answer is Option B

Linear Equations and linear Functions

Definition:

  

Linear Equations: A linear equation is defined as an algebraic equation in which every expression is either a stable (constant) or the product of a stable and a single variable. Linear equations can have one or more than one variables.  

  Linear functions:  A linear function is defined as a function whose chart (graph) consists of sectors of one straight line during its domain.

We can establish the linear function and it acquires value f (a) at ‘a’ and f (b) at ‘b’ by the following formula:
                                 
f(x)=f(a) (x-b)/(a-b) + f(b) (x-a)/(b-a)
   Because the first term is Zero (0) when x is ‘b’ and is f (a) when x is ‘a’, while the second term is Zero (0) when x is ‘a’ and is   f(b) when x is ‘b’.
   Finally we get,     y= mx+b


1. Example of Linear Equation:
Solve 3[2m - (7 - 3m)] = m - 21.           
Solution:
First we can solve the left hand side.
3[2m-(7-3m)]
Here first solving the inside term.
3(2m-7+3m) and then,
3(5m-7)
15m -21
Now compare with right hand side.
15m- 21= m-21
15m-m= 21-21
14m= 0
Therefore m=0
Answer: m=0

2. Example of Linear Function:
Solve 3(2m - 1) = 4(m + 5).
Solution:
3(2m - 1) = 4(m + 5)
6m-3=4m+20
6m-4m= 20+3
2m= 23
  m =23/2
  m =11.5
Answer: m=11.5

What is PEMDAS rule? in Order of Operations

In Number Theory Tutorial ,an expression or string of symbols is intended to represent a numerical value; a properly-formed expression may be evaluated in an unambiguous way. But in practice, an expression with multiple terms and operators may be written without parentheses, in which case the intended value of the expression is determined by convention. When we evaluate the expression is both preceded and followed by an operator such as minus or times, a convention is needed to clarify which operator should be applied first; this rule is known as a precedence rule, or more informally order of operation. From the earliest use of mathematical notation[citation needed], multiplication took precedence over addition, whichever side of a number it appeared on. Thus 3 + 4 × 5 = 5 × 4 + 3 = 23. When exponents were first introduced, in the 16th and 17th centuries, exponents took precedence over both addition and multiplication, and could be placed only as a superscript to the right of their base. Thus 3 + 5 2 = 28 and 3 × 5 2 = 75. To change the order of operations, a vinculum (an overline or underline) was originally used. Today one uses parentheses (). Thus, if one wants to force addition to precede multiplication, one writes (3 + 4) × 5 = 35.We can use order of operations calculator as well.
Mnemonics are often used to help students remember the rules. One common strategy is to build an acronym from the names of the operations. For instance, in the United States, the acronym PEMDAS (for Parentheses, Exponentiation, Multiplication/Division, Addition/Subtraction) is used, sometimes expressed as the sentence "Please Excuse My Dear Aunt Sally" or one of many other variations. In other English speaking countries, Parentheses may be called Brackets, and Exponentiation may be called Indices, Powers or Orders. Also, as Multiplication and Division are of equal precedence, M and D may be interchanged in the mnemonic. (This does not mean that multiplication and division can be performed in any arbitrary order, just that it is wrong to say "M always comes before D", or "D comes before M". See below.) The same comments apply to Addition and Subtraction.

Question:-

Solve 2 - [ (7-6) + (9-19)]

Answer:-

2 - [ (7-6) + (9-19)]

Here we have many signs.

When we have more than one sign ,we have to apply PEMDAS rule

= 2 - [ (1) + (-10)]

We have to solve inside the parenthesis first

Here (+) (-) = -

= 2 - [ 1 - 10]

= 2 - [-9]

Here (-) (-) = +

= 2+9

= 11 is the Answer

A problem on simple interest

Interest is a fee paid on borrowed assets. It is the price paid for the use of borrowed money, or, money earned by deposited funds.Assets that are sometimes lent with interest include money, shares, consumer goods through hire purchase, major assets such as aircraft, and even entire factories in finance lease arrangements. The interest is calculated upon the value of the assets in the same manner as upon money. Interest can be thought of as "rent of money". For example, if you want to borrow money from the bank, there is a certain rate you have to pay according to how much you want loaned to you.

Simple interest is calculated only on the principal amount, or on that portion of the principal amount which remains unpaid.

The amount of simple interest is calculated according to the following simple interest formula:

Let's see a problem on this from 8th grade math worksheets

Question:-

What is the interest earned on # 97500 when rate of interest is 8% and for 5 months?

Answer:-

Here is the solution from answers to math problems



This problem based on simple interest.

The formula for simple interest is

    p x T x R
I= ------------
       100

p= principle=$97500

R=rate of interest =8%

T=Time of year=5 months
=5/12 in years

I=interest amount

We have to put these values in formula

  97500 x 8 x 5
I= --------------
     100 x 12

     3900000
I = ----------
      1200

 I=3250

So the total interest earned on $97500 is $3250.

This is a part of elementary algebra practice .

Problem on mean median mode range

This math help is for mean median mode range worksheets

Mean, median, and mode are three kinds of "averages".

The "mean" is the "average" you're used to, where you add up all the numbers and then divide by the number of numbers.

The "median" is the "middle" value in the list of numbers. To find the median, your numbers have to be listed in numerical order, so you may have to rewrite your list first.


The "mode" is the value that occurs most often. If no number is repeated, then there is no mode for the list.

The "range" is just the difference between the largest and smallest values.

EXAMPLE:-

Find the mean, median, mode, and range for the following list of values:
8, 9, 10, 10, 10, 11, 11, 11, 12, 13

The mean is the usual average:
(8 + 9 + 10 + 10 + 10 + 11 + 11 + 11 + 12 + 13) ÷ 10 = 105 ÷ 10 = 10.5

The median is the middle value. In a list of ten values, that will be the (10 + 1) ÷ 2 = 5.5th value; that is, average of the fifth and sixth numbers to find the median:
(10 + 11) ÷ 2 = 21 ÷ 2 = 10.5

The mode is the number repeated most often. This list has two values that are repeated three times.10 and 11.

The largest value is 13 and the smallest is 8,
so the range is 13 – 8 = 5.

So
mean: 10.5
median: 10.5
modes: 10 and 11
range: 5


For more help on this ,you can reply me.