Lesson Find the number using a single linear equation
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<H2>Find the number using a single linear equation</H2> 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. <H3>Problem 1</H3>Twice a number is added to the number, giving 90. Find the number. <B>Solution</B> <pre> 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 = {{{90/3}}} = 30. <U>Answer</U>. The unknown number is 30. </pre> <H3>Problem 2</H3>When 6 is subtracted from a number, the result is seven times the number. Find the the number. <B>Solution</B> <pre> 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 = {{{(-6)/6}}} = -1. <U>Answer</U>. The unknown number is -1. <U>Check</U>. When 6 is subtracted from -1, the result is -1 -6 = -7, which is seven times (-1). The solution is correct ! </pre> <H3>Problem 3</H3>Four times a number is equal to twelve more than the number. Find the number. <B>Solution</B> <pre> 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 = {{{12/4}}} = 3. <U>Answer</U>. The unknown number is 3. </pre> <H3>Problem 4</H3>Twice a number is twenty less than the product of seven and the number. Find the number. <B>Solution</B> <pre> 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 = {{{(-20)/(-5)}}} = 4. <U>Answer</U>. The unknown number is 4. <U>Check</U>. 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 ! </pre> <H3>Problem 5</H3>One number is 17 less than another and their sum is 125. Find the numbers. <B>Solution</B> <pre> 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 = {{{108/2}}} = 54. <U>Answer</U>. The numbers are 54 and 54+17 = 71. <U>Check</U>. 54 + 71 = 125. ! Correct ! </pre> <H3>Problem 6</H3>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. <B>Solution</B> <pre> 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 = {{{216/27}}} = 8. <U>ANSWER</U>. The ones digit is 8. The tens digit is 11-8 = 3. The number itself is 38. </pre> <H3>Problem 7</H3>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. <B>Solution</B> <pre> 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. <U>ANSWER</U>. The smaller number is 170; the larger one is 170 + 406 = 576. </pre> <H3>Problem 8</H3>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. <B>Solution</B> <pre> 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. <U>ANSWER</U> </pre> My other lessons in this site for word problems on finding numbers are - <A HREF=https://www.algebra.com/algebra/homework/word/numbers/Simple-and-simplest-word-problems-on-finding-numbers-solved-by-different-methods.lesson>Simple and simplest word problems on finding numbers solved by different methods</A> - <A HREF=https://www.algebra.com/algebra/homework/word/numbers/Find-the-number-using-quadratic-equations.lesson>Find the number using quadratic equation</A> - <A HREF=https://www.algebra.com/algebra/homework/word/numbers/ind-the-numbers-using-systems-of-equations.lesson>Find the numbers using system of equations</A> - <A HREF=https://www.algebra.com/algebra/homework/word/numbers/Digit-problems.lesson>Digit problems - Find the number using system of equations</A> - <A HREF=https://www.algebra.com/algebra/homework/word/numbers/Entertainment-problems-on-finding-numbers.lesson>Entertainment problems on finding numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/word/numbers/OVERVIEW-of-lessons-for-word-problems-on-finding-numbers.lesson>OVERVIEW of lessons for word problems on finding numbers</A> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-I.
