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Math Homework Help

A Sales Chart

published on February 17th, 2008 . by Vanaja

Sarah went to a grocery shop. There was a sales chart.

Items Price
Cabbage $1.8 for 2
Carrot $0.6 for 1.k.g
Onion $1.75 for 1 carton
Potato $2.05 for 1 carton
Tomato $1 for 1 k.


Sarah bought 4 carton potatoes and 1 cabbage. If she gave $10, how much money she got back?



Hint: 10-{(4×2.05)+.9}

The Fundamental Principle of Counting

published on January 13th, 2008 . by Vanaja

If an event can happen in exactly 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….

Ana-The bearer

published on May 30th, 2007 . by Vanaja

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%

Fundamental Theorem of Arithmetic

published on April 13th, 2007 . by Vanaja

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.

The Gong

published on April 11th, 2007 . by Vanaja

If a clock takes 6 seconds to strike 6. How long the same clock take to strike 12.


Can you find the answer?

Euclid’s Division Lemma

published on April 10th, 2007 . by Vanaja

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 ?rStep 2 : If r = 0, d is the HCF of c and d. If r ? 0, apply the division lemma to d and 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.

Algorithm & Lemma

published on April 3rd, 2007 . by Vanaja

An algorithm is a series of well defined steps
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.

A lemma is a proven statement used for
proving another statement.

Limits

published on January 9th, 2007 . by Vanaja

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

Some Properties Of Ratios

published on December 9th, 2006 . by Vanaja

1) Let a:b>c:d and c:d be two ratios. Then,

i) a:b > c:d, if ad>bc,

ii) a:b< ?xml:namespace prefix = c /> <>

iii) a:b = c:d, if ad=bc

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.

6) If the same positive quantity is added to both the terms of a ratio of greater inequality, then the ratio is increased.

Ratio

published on December 5th, 2006 . by Vanaja

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 to $50 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 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.

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