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
4b = 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 Letx1, 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 Sn = n2[a1 + an] where n is the number of terms, a1 is the first term and an 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
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: Alinear 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
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 acronymPEMDAS (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
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:
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
Percentages are used to express how large one quantity is, relative to another quantity. The first quantity usually represents a part of, or a change in, the second quantity, which should be greater than zero. For example, an increase of $ 0.15 on a price of $ 2.50 is an increase by a fraction of 0.15 / 2.50 = 0.06. Expressed as a percentage, this is therefore a 6% increase.
Let's see a word problem on it.
Question:-
A waitress earn an average tip of 12% of the cost of the food she serves.If she serves $ 375 worth of food in one evening ,how much money in tips will she earn on average.
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable.
Linear equations can have one or more variables. Linear equations occur with great regularity in applied mathematics. While they arise quite naturally when modeling many phenomena, they are particularly useful since many non-linear equations may be reduced to linear equations by assuming that quantities of interest vary to only a small extent from some "background" state.
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