Lesson Find the number using a single linear equation

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Find the number using a single linear equation

The problems of the type considered at this lesson are among the most simple word problems.

Why ? - Because their conditions directly instruct students on how to write equations.

Problem 1

Twice a number is added to the number, giving 90. Find the number.

Solution

Let x be the unknown number.

When doubled, it becomes 2x, and then the condition says

x + 2x = 90.


Simplify and solve for x:


3x = 90  ====>  x =  = 30.


Answer.  The unknown number is 30.

Problem 2

When 6 is subtracted from a number, the result is seven times the number. Find the the number.

Solution

Let x be the unknown number.


When 6 is subtracted from a number, it becomes (x-6), and then the condition says


x -6 = 7x.


Simplify and solve for x:


-6 = 7x - x,

-6 = 6x   ====>  x =  = -1.


Answer.  The unknown number is -1.

Check.   When 6 is subtracted from -1, the result is -1 -6 = -7,  which is seven times (-1).  

         The solution is correct !

Problem 3

Four times a number is equal to twelve more than the number. Find the number.

Solution

Let x be the unknown number.


Four times the number is 4x, while twelve more than the number is x+12. 

Thus the condition gives you an equation


4x = x + 12.


Simplify and solve for x:


4x - x = 12  ====>  x =  = 3.


Answer.  The unknown number is 3.

Problem 4

Twice a number is twenty less than the product of seven and the number. Find the number.

Solution

Let x be the unknown number.


Twice the number is 2x, while "twenty less than the product of seven and the number" is 7x-20. 

Thus the condition gives you an equation


2x = 7x - 20.


Simplify and solve for x:


2x - 7x = -20  

-5x = -20  ====>  x =  = 4.


Answer.  The unknown number is 4.


Check.   Twice the number is 2*4 = 8.

         Twenty less than the product of 7 and the number is  7*4-20 = 28-20 = 8.  

         The solution is correct !

Problem 5

One number is 17 less than another and their sum is 125. Find the numbers.

Solution

Let x be the lesser number.


Then the greater numbr is (x+17), and the sum of these numbers is x + (x+17). 

Thus the condition gives you an equation


x + (x+17) = 125.


Simplify and solve for x:


2x + 17 = 125,  

2x = 125 - 17  

2x = 108  ====>  x =  = 54.


Answer.  The numbers are 54 and  54+17 = 71.


Check.   54 + 71 = 125.    ! Correct !

Problem 6

The sum of the digits of a two digit number is 11. The new number obtained when the digits are reversed
is 7 more than twice the original number. Find the original number.

Solution

Let the ones digit of the number is A; then the tens digit is (11-A).

Hence, the digit itself is  10*(11-A) + A.


Then the digits are reversed in the number, the new number is  10A +(11-A).


The new number and the original number are related each to other by this equation


10A + (11-A) = 2*(10*(11-A) + A) + 7.


We just completed the setup.
We have now one equation for single unknown A.


Simplify and solve for A:


10A + 11 - A = 2*(110 - 10A + A) + 7

9A + 11 = 220 - 18A + 7

9A + 18A = 227 - 11

27A = 216

A =  = 8.


ANSWER.  The ones digit is  8.  The tens digit is  11-8 = 3.  The number itself is  38.

Problem 7

One number is bigger than another by 406. If the bigger number is divided by the smaller one, you’ll get the quotient 3 and remainder 66.
Find the numbers.

Solution

The condition means


    x + 406 = 3x + 66.


Solve for x, which stands for the smaller number


    406 - 66 = 3x - 2x

    340 = 2x   ====================>  x = 340/2 = 170.


ANSWER.  The smaller number is 170;  the larger one is  170 + 406 = 576.

Problem 8

The unit digit of a two digit number is 1 less than the tens digit.
If the number is increased by 8 and then divided by the sum of the digits, the result is 8.
Find the number.

Solution

Let the "tens" digit be t.

Then the "units" digit is (t-1), according to the condition.


Hence, the number itself is  N = 10t + (t-1).


Then the number N+8 is  10t + (t-1) + 8 = 10t + t + 7 = 11t + 7.


From the last statement of the problem, we have this equation


    N+8 = 8*(t+u),

or

    11t + 7 = 8*(t+(t-1)).


Simplify and find t


    11t + 7 = 8*(2t-1)

    11t + 7 = 16t - 8

      7 + 8 = 16t - 11t

       15   =    5t

        t   =    15/5 = 3.


Thus the tens digit is 3;  the units digit is 3-1 = 2;  and the number itself is  32.    ANSWER

My other lessons in this site for word problems on finding numbers are
    - Simple and simplest word problems on finding numbers solved by different methods
    - Find the number using quadratic equation
    - Find the numbers using system of equations
    - Digit problems - Find the number using system of equations
    - Entertainment problems on finding numbers
    - OVERVIEW of lessons for word problems on finding numbers

Use this file/link ALGEBRA-I - YOUR ONLINE TEXTBOOK to navigate over all topics and lessons of the online textbook ALGEBRA-I.

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