# Problems on Application of Linear Equations

Problems which are expressed in words are known as word problems or applied problems.  If we practice word problems or applied problems then we understand the simple techniques of translating them into equations.

A word (or applied) problem involving unknown number (or quantity) can be translated into a linear equation consisting of one unknown number (or quantity). The equation is formed by using the conditions of the problem. By solving the resulting equation, the unknown quantity can be found.

Solving a word problem by using linear equation in one variable

Steps to solve a word problem:

(i) Read the statement of the word problems carefully and repeatedly to determine the unknown quantity which is to be found.

(ii) Represent the unknown quantity by a variable.

(iii) Use the conditions given in the problem to frame an equation in the unknown variable.

(iv) Solve the equation thus obtained.

(v) Verify if the value of the unknown variable satisfies the conditions of the problem.

Problems on Application of Linear Equations in one variable:

1. The sum of two numbers is 80. The greater number exceeds the smaller number by twice the smaller number. Find the numbers.

Solution:

Let the smaller number be x

Therefore the greater number = 80 – x

According to the problem,

(80 - x) - x = 2x

80 - x - x = 2x

80 - 2x = 2x

80 - 2x + 2x = 2x + 2x

4x = 80

4x/4 = 80/4

x = 20

Now substitute the value of x = 20 in 80 - x

80 - 20 = 60

Therefore, the smaller number is 20 and the greater number is 60.

2. Find the number whose one-fifth is less than the one-fourth by 3.

Solution:

Let the unknown number be x

According to the problem, one-fifth of x is less than the one-fourth of x by 3

Therefore, x/4 – x/5 = 3

Multiplying both sides by 20 (LCM of denominators 4 and 5 is 20)

5x – 4x = 3  20

x = 60

Therefore, the unknown number is 60.

3. A boat covers a certain distance downstream in 2 hours and it covers the same distance upstream in 3 hours. If the speed of the stream is 2 km/hr, find the speed of the boat.

Solution:

Let the speed of the boat be x km/hr

The speed of the stream = 2 km/hr

Speed of the boat downstream = (x + 2) km/hr

Speed of the boat upstream = (x - 2) km/hr

Distance covered in both the cases is same.

2(x + 2) = 3(x - 2)

2x + 4 = 3x – 6

2x – 2x + 4 = 3x – 2x – 6

4 = x – 6

4 + 6 = x – 6 + 6

x = 10

Therefore, the speed of the boat is 10 km/hr.