If an event can happen in ecactly m ways, and if following it, a second event can happen in exactly n ways, then the two events in succession can happen in exactly mn ways.
Illustration.
Suppose there are five routs from A to B and three routs from B to C. In how many ways a person can go from A to C?
Since there are five different routs from A to b, the person can go the first part of his journey in 5 different ways. Having completed in any one of the 5 different ways , he has 3 different ways to complete the second part of the journey fro B to C. Thus each way of going from A to B give rise to 3 different ways of going from B to C.
There fore the total number of ways of completing the whole journey = number of ways for the first part x number of ways for the second part.
= 5 x 3=15.
Generalisation
If an event can occur in m different ways, a second event in n different ways, a third event in exactly p different ways and so on, then the total number of ways in which all events can occur in succession is mnp....
Labels: Arithmetic, Definitions
Today we have a questions on percentage.
Ana is working in a restaurant as a bearer.
As a penalty Ana’s wages were decreased by 50%.
After one month the reduced wages were increased by 50%.
Find her loss.
Hint: The salary increased is the 50% of the decreased salary.
Answer:
New salary is 50+25=75
Therefore loss=25%
Labels: Arithmetic, Problems and Solutions
Fundamental Theorem of Arithmetic :
Every composite number can be expressed ( factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.
The Fundamental Theorem of Arithmetic says that every composite number can be factorised as a product of primes. Actually it says more. It says that given any composite number it can be factorised as a product of prime numbers in a‘unique’ way, except for the order in which the primes occur. That is, given any composite number there is one and only one way to write it as a product of primes,as long as we are not particular about the order in which the primes occur.
So, for example, we regard 2 × 3 × 5 × 7 as the same as 3 × 5 × 7 × 2, or any other possible order in which these primes are written. This fact is also stated in the following form:
The prime factorisation of a natural number is unique, except for the order of its factors.
In general, given a composite number x, we factorise it as x = p1p2 ... pn, where p1, p2,..., pn are primes and written in ascending order
If we combine the same primes, we will get powers of primes.
Once we have decided that the order will be ascending, then the way the number is factorised, is unique.
The Fundamental Theorem of Arithmetic has many applications, both within mathematics and in other fields.
Labels: Algebra, Arithmetic, Theorems
If a clock takes 6 seconds to strike 6. How long the same clock take to strike 12.
Can you find the answer?
Labels: Arithmetic, Puzzles
Euclid's Division Lemma
Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0≤r
This result was perhaps known for a long time, but was first recorded in Book VII of Euclid’s Elements. Euclid’s division algorithm is based on this lemma.
Euclid’s division algorithm is a technique to compute the Highest Common Factor (HCF) of two given positive integers. Recall that the HCF of two positive integers a and b is the largest positive integer d that divides both a and b.
Let us see how the algorithm works, through an example first. Suppose we need to find the HCF of the integers 455 and 42.
We start with the larger integer, that is, 455.
Then we use Euclid’s lemma to get 455 = 42 × 10 + 35.
Now consider the divisor 42 and the remainder 35, and apply the division lemma to get 42 = 35 × 1 + 7.
Now consider the divisor 35 and the remainder 7, and apply the division lemma to get 35 = 7 × 5 + 0.
Notice that the remainder has become zero, and we cannot proceed any further. We claim that the HCF of 455 and 42 is the divisor at this stage, i.e., 7. You can easily verify this by listing all the factors of 455 and 42.Why does this method work? It works because of the following result. So, let us state Euclid’s division algorithm clearly.To obtain the HCF of two positive integers, say c and d, with c> d, follow the steps below:
Step 1 : Apply Euclid’s division lemma, to c and d. So, we find whole numbers, q and r such that c = dq + r, 0 ≤r
Step 3 : Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.
This algorithm works because HCF (c, d) = HCF (d, r) where the symbol HCF (c, d) denotes the HCF of c and d, etc.Euclid’s division algorithm is not only useful for calculating the HCF of very large numbers, but also because it is one of the earliest examples of an algorithm that a computer had been programmed to carry out.
Remarks :
1. Euclid’s division lemma and algorithm are so closely interlinked that people often call former as the division algorithm also.
2. Although Euclid’s Division Algorithm is stated for only positive integers, it can be extended for all integers except zero, i.e., b ≠ 0.
Labels: Algebra, Arithmetic
which gives a procedure for solving a type of
problem।
The word algorithm comes from the name
of the 9th century Persian mathematician
al-Khwarizmi. In fact, even the word ‘algebra’
is derived from a book, he wrote, called Hisab
al-jabr w’al-muqabala.
proving another statement.
Labels: Arithmetic, Definitions
The notion of limit is one of the most basic and powerful concepts in all of mathematics. Differentiation and Integration, which comprise the core of study in calculus, are both products of the limit. The concept of limit is the foundation stone of calculus and as such is the basis of all that follows it. It is extremely important that you get a good understanding of the notion of limit of a function if you have a desire to fully understand calculus at the entry level.
You can find some basic results on limits here
Labels: Arithmetic, Calculus, Functions
1) Let a:b>c:d and c:d be two ratios. Then,
i) a:b> c:d, if ad>bc,
ii) a:b 2) A ratio a:b is called a ratio of i) greater inequality ifa>b, ii) less inequality if a < b iii) equality if a=b 3) If the same positive quantity is added to both the terms of a ratio of greater inequality, then the ratio is decreased. 4) If the same positive quantity is added to both the terms of a ratio of less inequality, then the ratio is decreased. 5) If the same positive quantity is subtracted to both the terms of a ratio of greater inequality, then the ratio is increased.
Labels: Algebra, Arithmetic
The ratio of two quantities of the same kind and in the same units is a comparison by division of the measure of two quantities.
In other words ,the ratio of two quantities of the same kind is the relation between their measures and determines how many times one quantity is greater than or less than the other quantity.
The ratio of a to b is the fraction a/b, and is generally written as a:b.
- Example 1: The ratio of 25ドル to 50ドル is 25:50 or25/50 or 1:2
- Example 2: The ratio of 2m to 80 cm is 200:80 or 200/80 or 5:2
- Example 3: There is no ratio between 10ドル and 5 meter.
Since the ratio of two quantities of the same kind determines how many times one quantity contains other, is an abstract quantity. In other words, ratio has no unit or it is independent of the units used in the quantities compared.
For the ratio a:b, a and b are called terms of the ratio. The former a is called the first term or antecedent and the later b is known as the second term or consequent.
Labels: Algebra, Arithmetic, Definitions
Addition and Subtraction When we divide Fractions, we actually multiply the numerator by the reciprocal of the denominator. A reciprocal is a fraction turned upside down. For example, 3/4 divided by 5/6 = 3/4*6/5 = 18/20 = 9/10
When adding or subtracting fractions, if the denominators are same, simply add or subtract the numerators and write the denominator given.
If the denominators are different, convert each fraction so that its denominator is equal to the Lowest Common Multiple of the denominators of the given fractions. This means multiplying both the numerator and denominator by the same term. For example, say the LCM is 24 and one fraction in the problem is 5/6, then to convert that fraction to denominator of 24 you must multiply numerator and denominator by4, so the fraction becomes 20/24. You follow this same procedure for each term, add or subtract as indicated by the sign of each term and divide the total by the LCM
For example:
Multiplication of Fractions is straightforward. You just multiply the numerators and multiply the denominators and then reduce the fraction to its lowest term, if possible
Labels: Algebra, Arithmetic
It appears that many students, even some students of grade 9 and grade 10 are struggling with fractions. So I have prepared this lesson.Not in a thorough but simplified manner.
A term which is of the form a/b is called a fraction. The top Numbers a’ is called the numerator and the bottom number ‘b’ is called the denominator.
Proper fraction: If the numerator is less than the denominator, it is called a proper fraction
Example:1/2,4/7
Improper Fraction: If the numerator is greater than the denominator, it is called an improper fraction.
Example:5/2,9/7
Mixed Fraction: - A term consisting of an integer (5) followed by a proper fraction (2/3), written as 5 2/3 is called a mixed fraction.
see you tomorrow
Labels: Algebra, Arithmetic
It is possible with a little trickery to prove that 2=3 by using algebra.
Let x=y=5
Labels: Arithmetic, Puzzles