Monday, October 30th, 2017
Algebra is a very powerful branch of Mathematics. It provides solution to realworld problems. It helps in transforming word problems into mathematical expressions in form of equations using variables to denote unknown quantities or parameters and thus, providing numerous of techniques to solve these mathematical equations and hence, determining the answer to the problem. Representation of problem it the key step of finding the solution because, until we have carefully described the actual problem into mathematical statements and have correctly form the relationship between the variables and constants using various techniques to solve these equations will be a futile activity. As, if we made mistake in forming up the relationship in mathematical terms then the solution that will be obtained would be based on the equation you have used and thus, will be worthless and not true. So, this step is a very important and crucial step and in this blog, we will learn how to form the correct relationship using Linear Equations. A Linear equation is the one which contains either constant or a combination of constant and variable. A single variable linear equation is generally denoted as “ax + b = 0”, where a and b are constants and a ≠ 0. A linear equation can contain more than one variable, two variable linear equation is usually written as ‘ax + by + c = 0’ and in general nvariable linear equation is “a_{1}x_{1} + a_{2}x_{2 }+ …+ a_{n}x_{n } = 0”, where a_{1, }a_{2}, …, a_{n} are constants. Here, linear doesn’t mean linear in variable but linear in the power of variable i.e. the power of x, y, … should be 1 and not more than that. Therefore, an equation such as x^{2 }+ 3 = 0 cannot be called as Linear Equation, it is Quadratic equation.
We all have come across Linear equation in our high school and most of us find them terrifying, not solving them but forming these equations. The language of the word problem is at times tricky and we end up making mistakes and forming incorrect equations thus, leading to faulty solutions. And Trust me! In CAT the problem you will be getting very tricky problems and there would be multiple instance where you have to make use of linear equation to find the solution of the problem. So, to successfully apply linear equations, we first need to learn to aptly translate the word problem into them. The following steps with the help of an example will guide you about how to form Linear Equations.
Let’s now move on to an example and I will illustrate you how to use these steps to formulate linear equations. Consider this,
The sum of digits of twodigit number is 7. When the digits are reversed, the number is increased by 27. Find the numbers.
Now what?
Let’s read the question several times and note down all the key points and translate them into mathematical equations. I know it’s an easy one but still you should read it at least twice.
Here, the unknown no. is a twodigit no. whose sum is 7. So, we use x for tens place and y for units place and as given
x + y = 7
Also, it’s been provided in the question that when we swap the digits of original the no. is increased by 27 i.e. if x is on tens place and y on units place then the original no. would be
10x + 1y
And if we swap the digits then y will be now at tens place and x at units but the new no. is bigger from the original by 27. This means,
10x + 1y = 10y + 1x – 27
We are left we 2 equations and 2 unknown we can solve it using elimination method. Rearranging and simplifying the second equation in the form “y – x = 3” and solving both the equations we get,
x = 2 and y = 5
Now let’s check once are we getting the right answer if x = 2 and y = 5 then the original no. would be 25 and if reversed the new no. would be 52 and yes, their difference is 27. So, our answer is correct.
Since, Linear equations has wide range of application such as problem on ages, numbers, time, speed and distance problems, Timework problems, Functions, Arithmetic Progression etc. In fact, it can be used anywhere where a relationship is defined for some unknown value and our aim is evaluate those parameters we can make use of it. But, today in this blog, we will be focused on only one type of problems i.e. the word problems based on age. These problems are very confusing and the language is a bit complex and we end up usually making up errors in the formulation of the equation. So, I’ll discuss and try various different type of questions on this concept that will give you a thorough understanding of how to form Linear equations and solve them.
Now, Problem on ages can be categorized into three types, i.e. Questions based on calculating the present age, Questions to determine the age of person after k years and questions that calculate age of a person before k years. These three types may cover cases of various types with different combination of ratios, fractions etc.
Before moving forth with the above given types of problems keep in mind these quick tips:
Age after k years will be x + k
Age before k years will be x – k
The age k times would be kx
Given below are tables that contain various cases of the 3different type of problems of ages
Problems based on present age  Formulation of Linear Equations using the steps mentioned earlier  Final Equations and their solution 
Case 1: Whole number form
Numerical: The age of mother is thrice that of her daughter. After 12 years, the age of the mother will be twice that of her daughter. What is the present age of the daughter and mother? 
Solution: Let the age of daughter be x and that of mother will be y. And the first line clearly states that at present the age of mother is thrice of daughter. Therefore, y = 3x. Now, after 12 years i.e. x + 12, mother’s age after twelve years would be (y + 12) and also it will be twice of her daughter, y + 12 = 2(x + 12).  The final equations are:
y = 3x y – 2x = 12 Substituting first equation into second and solving we get, x = 12 years and y = 36 
Case 2: Fractional Form
Numerical: Sneh’s age is 1/6^{th} of her father age. Sneh’s father age will be twice of Vimal age after 10 years. If Vimal’s 8^{th} birthday was celebrated 2 years ago. Then what is the present age of Sneh?

Solution: If you read the question. You’ll find three people here and you must be worried how to solve equations of 3 variables. But do not panic. Just make the equations. First, let age of Sneh be x and of his father be y. And it’s Cleary stated that the age of Sneh is 1/6^{th} of his father. Therefore, x = 1/6y. Now assume Vimal’s age be z, so after 10 years Vimal age would be z+10 and Sneh’s father age will be twice of Vimal. Hence, y+10 = 2(z+10). Also, Vimal was of 8 years 2 years ago. Thus, present age of Vimal = 8 + 2 = 10 = z.  The final equations are
x = 1/6y y + 10 = 2 (z+ 10) z = 10 Substituting z = 10 in 2 equation we can easily get the age of Sneh’s father to be 30 and then we can calculate present age of Sneh. x = 5 
Case 3: Combination of Ratio and Fraction form
The ratio between Neelam and Shiny is 5:6 respectively. If the ratio between 1/3^{rd} age of Neelam and half of Shiny’s age is 5: 9. Then what will be Shiny’s present age? 
Since, the ratio between Neelam and Shiny is 5:6. Therefore, their present age would be 5x and 6x respectively. Also, the ratio between 1/3^{rd} age of Neelam i.e. 1/3 * 5x and half age of shiny i.e. ½ * 6x is 5:9.  Therefore, the final equation using the given info would be
5x/3/3x = 5/9 Solving the above equation, we get, 1 = 1 Thus, the present ages cannot be determined with the given information. 
Problems based on age before k years  Formulation of Linear Equations using the steps mentioned earlier  Final Equations and their solution 
Case 1: Fractional Form
Numerical: Farah got married 8 years ago. Today her age is 1 2/7^{th} times her age at the time of her marriage. At present her daughter’s age is 1/6^{th} of her age. What was her daughter’s age 3 years ago? 
Solution: Let present age of Farha be x. At present, his age is 9/7^{th} of the age when she got married and it’s been 8 years since she got married. Therefore, her age would be x – 8 when she got married and her present age is 9/7(x – 8). Also, currently her daughter’s age is 1/6^{th} of her. Let the present age of her daughter be y.  The final equations are
x = 9/7* (x – 8) y = x/6 Solving the first equation, we get age of Farah = 36 years. Hence her daughter’s present age is 6 years but we need her age 3 years back. So, she would have been 3 years old. 
Case 2:
Numerical: The present age of Amit and his father are in the ratio 2:5. Four year hence the ratio of their ages becomes 5:11 respectively. What is the father’s age five years ago?

Solution: It’s given in question that the current age of Amit and his father are in ratio 2:5. Their present age would be 2x and 5x resp. Four years from now i.e. 2x + 4 and 5x + 4 the ratio between their ages become 5:11  The final equation is
(2x + 4)/ 5 = (5x + 4) / 11 22x + 44 = 25x + 20 X = 8 Age of Amit and his father is 16 and 40 resp. Therefore, five years ago Amit’s father age = 40 – 5 = 35 years. 
Problems based on age after k years  Formulation of Linear Equations using the steps mentioned earlier  Final Equations and their solution 
Case 1: Whole number form
Numerical: The sum of present ages of father and son is 8 years more than the present age of the mother. The mother is 22 years older than the son. What will be the age of father after 4 years? 
Solution: Again, there are 3 people in this question. Thus, three variables. Let present age of father, son and mother be x, y and z resp. Since, the sum of the present ages of father and son is 8 years more than the mother i.e. x + y = 8 + z.
Also, mother is 22 years older than son, z = 22 + x. We need to find age of father after 4 years i.e. y + 4 
The final equations are
x + y = 8 + z z = 22 + x Substituting the value of second equation in first we get, x + y = 8 + 22 + x y = 30 Therefore, y + 4 = 34 years. 
Case 2: Ratio Form
Numerical: The ages of A and B are in the ratio 6:5 and the sum of their ages is 44 years. What will be the ratio of their ages after 8 years?

Solution: This one is a very simple question. The ratio between ages of A and B is 6:5. Thus, their present age is 5x and 6x resp. And also, the sum of their ages is 44 i.e. 5x + 6x = 44.  Solving the equation,
6x + 5x = 44 X = 4 But, we need to find the ratio of their ages 8 years from now. Thus, their present ages are 24 and 20 respectively. After 8 years, their ages will be 32 and 28 and hence, the ratio would be 32:28 i.e. 8:7

Case 3: Combination of age after k years and before k years.
Numerical: The ratio between the present ages of A and B is 5:3. The ratio between A’s age 4 years ago and B’s age 4 years hence is 1:1. What is the between A’s age 4 years hence and B’s age 4 years ago?

Solution: You may find this question a bit confusing coz of its language. No worries, it’s not that difficult. Just read the question and form equations using the information given like this. It’s been given the ratio between the present age of A and B is 5:3. Thus, their present age would be 5x and 3x respectively. Now, 4 years ago the ratio between A’s age and B’s age 4 years hence is 1:1. The age of A 4 years ago would be 5x – 4 and age of B 4 years from now will be 3x + 4. And the ratio of this is 1:1 i.e. (5x – 4)/ (3x + 4) = 1/1. But we need to calculate the ratio between A and B such that (5x + 4) :(3x – 4)  The equation is as follows,
(5x – 4)/ (3x + 4) = 1/1 Solving it we get, x = 4 Hence, A’s current age is 20 and B’s present age is 12 Now putting this value in (5x + 4) :(3x – 4) We get, 24:8 = 3:1 
I hope, your concept would have been cleared and you now evolve better understanding about it and will be able to solve questions on your own. The above given combinations are just few types, there could many like that, you will get better hold of it if you practice them.
You can also see: Basic Probability Concepts for CAT Preparation
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