We have been solving linear equation by several methods like substitution, elimination, comparing etc. The problems related to the applications of Linear Equations are discussed in various possible ways. Linear equation problems can be general word problems also where equation has to be framed using given data. So, it is required to read the equation properly in order to get the desired info and then converting that info to numeric expressions. For instance, let us consider the following word problem based on ages of people and making the linear equation:
Example: The ages of two friends is in the Ratio 5 is to 6, after how many years will the ages be in the ratio 7 is to 8?
Solution: According to given problem ratio of ages of two friends is 5 to 6 initially. We have to calculate the number of years between the change of their age ratio from 5 : 6 to 7 : 8. Let their present ages be 5a and 6a years respectively. After 'b' years their ages will be in ratio of 7 to 8. So, by using these constraints we can define following equation:
(5 a + b) / (6 a + b) = 7 /8,
Applying cross multiplication on the above equation we get:
8 (5 a + b) =7 (6 a + b),
Or 40 a+ 8 b = 42 a + 7 b,
Or b = 2 a,
This shows that age of second friend is double the age of first friend. So, after 2 times the greatest common factor of their present ages, their ratio would be 7 : 8 or we can say in 2 years their ratio will be 7 to 8.
Scientific Notations are generally used to represent the Numbers in mathematics that seem to be very insignificant or very big. This type of number will possess just 2 factors out which first one appears in between numbers 1 and 10. Second factor is usually written in form of an exponent of 10. For example, 1012 means exponent of 10 is 12. This is what we call as scientific representation of any number. We are calling two numbers as factors because numbers that are multiplied to each other to get a product are called as factors only.
Let us consider an example to understand the concept of converting any number into its scientific form: We have to represent 0 Point 0035 times 10 to the power of negative 2 in scientific notation. Remember first factor has to lie between 1 and 10. This makes necessary for us to keep on shifting the decimal point until we get a digit lying between 1 and 10. Here, in our example the number is 0.0035. We can see that digit that lies between 1 and 10 is lying to the right of decimal at thousands place. So, at least 3 times the decimal point has to be shifted to get desired factor. That is we have to multiply 0.0035 by 103 to get 3 in the units place to the left side of decimal. We get:
0.0035 * 103 / 103 = 3.5 * 10^{-3},
Also number has to be multiplied by 10-2 and then final scientific notation will be decided. On multiplying 10-2 by 3.5 * 10-3, we get following result:
3.5 * 10^{- (3 + 2)} = 3.5 * 10^{-5},
So, second factor is 10^{-5} and scientific notation for desired number is given as:
3.5 * 10^{-5}.
In mathematical word problems involvement of operations like addition, subtraction, multiplication and division seem to be much in use. Word problems are basically expressed in the form of statements. These statements are essentially not mathematical expressions or Numbers. We ourselves have to understand complete word problem properly and then extracting the necessary information that can be used for evaluation purpose. Next part of word problems is the solution. In a word problem all dimensions for various quantities have to be handled in a proper way. If conversions are required, do them purposefully. Let us consider an example of such problems to understand their working in Math properly.
Suppose we have following example word problem:
Q. The sum of two numbers is 15 if one of the numbers is X, the other would be best represented by which of the following? Options are:
A. X + 15
B. X – 15
C. 15 – X
D. X
Solution: Word problem says that numbers when added give result 15 and one number out of two is X. Let us assume that second number is given as Y. According to the word problem, only information we are available with is their sum, i.e.
X + Y = 15,
Taking 'X' to another side as it is also a known quantity; we get value of 'Y' as:
Y = 15 – X,
Thus correct representation of value of 'Y' is given by option C i.e. 15 – X. Thus we see, to find the value of one missing variable in the problem we need just relevant information. Similarly, for word problems involving more than 1 variables, we need that information.
The descending arrangement of the name of shop marked as Numbers is shown below:
Chocolate shop, jewelry shop, mobile shop, pizza shop, toy shop. i.e. 10, 9, 6, 2, 1.
So, here also we can see that there is a miscellaneous arrangement in the list of score of candidates.
To make it ordered in a specific pattern, let us start from largest score of candidates like 54.7 then the score smaller than 54.7 is 12.5 then we have 9.8 another smaller than all of them is 6 then we have 5.9 and the smallest is 4.3.
So we have list as 54.7, 12.5, 9.8, 6, 5.9, 4.3.
This list is arranged in a format which we will term as descending order because here we have list which is basically in decreasing order format of arrangement.
We have to find the score of player, who has scored the highest in the match, then list of scores should be arranged in specific order, so that it will become easy for us to find any of the scores in the list.
So, for arranging the list of scores in descending order we should start with largest score made by the players of the team. So the largest score is 120 so it should be placed at first Position in the list, now the score which is smaller than 120 and larger than any other number in the list is 100. Similarly we will compare every score with the score in the list which is the largest in unordered list and smallest than the scores in the ordered list. So after matching all the scores we will get the list of scores arranged in descending order as shown below:
120, 100, 54, 34, 32, 10, 5, 4, 1, 0.
So if we have to find any score either highest or second highest. It will become a simple task for us.
Given Numbers are 8, 10, 12, 20, 60, 33, 64, 55, 210, 130, 530, 999, 888 and 450.
The order of numbers in which they are written from smallest to largest is known as ascending order. So in ascending order we can write it as:
Ascending order - 8, 10, 12, 20, 33, 55, 60, 64, 130, 210, 450, 530, 888 and 999.
Given Numbers are 4, 5, 8, 1, 10, 12, 2, 9, 20, 60, 33, 64, 55 and 250.
The order of numbers in which they are written from smallest to largest is known as ascending order. So in ascending order we can write it as:
Ascending order - 1, 2, 4, 5, 8, 9, 10, 12, 20, 33, 55, 60, 64, 250.
Given Numbers are 50, 80, 10, 100, 120, 20, 90, 200, 600, 330, 640, 550, 2100, 1110 and 290.
The order of numbers in which they are written from smallest to largest is known as ascending order. So in ascending order we can write it as:
Ascending order - 10, 20, 50, 80, 90, 100, 120, 200, 290, 330, 550, 600, 640, 1110, 2100.
Given Numbers are 15, 18, 10, 100, 102, 12, 91, 201, 160, 133, 604, 155, 21, 116 and 129.
So in ascending order we can write it as:
Ascending order – 10, 12, 15, 18, 21, 91, 100, 102, 116, 129, 133, 155, 160, 201, 604.
Given Numbers are 5, 8, 1, 10, 12, 2, 9, 20, 60, 33, 64, 55, 210, 111 and 29.
The order of numbers in which they are written from smallest to largest is known as ascending order. So in ascending order we can write it as:
Ascending order- 1, 2, 5, 8, 9, 10, 12, 20, 29, 33, 55, 60, 64, 111, 210.
The formula for finding the total surface area of Right Circular Cone is given by:
TSA of a right circular cone = âŠĽr (r + √ r^{2} + h^{2});
Here ‘r’ is the radius of right circular cone and ‘h’ is the height of right circular cone.
Given, radius = 9 inch;
Height = 15 inch.
And we know that the value of âŠĽ is 3.14;
TSA =?
On putting these values in the given formula we get the TSA of right circular cone.
TSA of a right circular cone = âŠĽr (r + √ r^{2} + h^{2});
TSA = 3.14 * 9 (9 + √ (9)^{2} + (15)^{2};
TSA = 3.14 * 9 (9 + √ 81 + 225;
TSA = 3.14 * 9 (9 + √ 306;
TSA = 3.14 * 5(5 + 17.49);
TSA = 3.14 * 112.46;
TSA = 353.13;
So the total surface area of right circular cone is 353.13 inch^{2}.
We first arrange the product in form of column i.e hundreds, tens and ones and then proceed:
Hundreds tens ones
4 6 3
X 1 2
____________________________
We first multiply 463 by 2, we get
Hundreds tens ones
4 6 3
X 1 2
____________________________
8 12 6. Now we find that there is 12 on tens, so we retain 2 on tens place and carry 1 to hundreds place.
Hundreds tens ones
4 6 3
X 1 2
____________________________
9 2 6 now we take 463 by 1 which is at tens place, by putting X at ones place
46 3 X
___________________________
5 5 5 6 We add both the product and get the result