Lesson Typical word problems to solve using a single linear equation
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<H2>Typical word problems to solve using a single linear equation</H2> <H3>Problem 1</H3>One weekend members of a cycling club rode on a trail uphill. The uphill climb reduces the cyclists usual average speed by 6 km/h and they took 4 hours to get the lake. On the return trip the downhill ride increased the cyclists usual average speed by 4 km/h. The return trip took 2 hours. What was the usual speed? What was the one way distance they traveled? <B>Solution</B> <pre> Let x = the usual speed, in km/h. Then the condition says (x-6)*4 = (x+4)*2 <<<---=== stating that each way distances are the same ! It is your basic equation. Now you simplify and solve it for x. 4x - 24 = 2x + 8, 4x - 2x = 8 + 24. 2x = 32 ====> x = {{{32/2}}} = 16. Thus we found usual speed = 16 km/h. The distance is (x-6)*4 = (16-6)*4 = 10*4 = 40 kilometers. </pre> <H3>Problem 2</H3>A concert venue sold 1700 tickets. One evening tickets cost $25 for a covered pavilion seats and $15 for a lawn seats, a total of $33000. How many of each type of ticket were sold ? <B>Solution</B> <pre> Let x be the number of the $25 tickets. Then the number of the $15 tickets is (1700-x). The "money" equation is 25x + 15*(1700-x) = 33000 dollars. Simplify and solve for x: 25x + 25500 - 15x = 33000 ====> 10x = 33000 - 25500 = 7500 ====> x = {{{7500/10}}} = 750. <U>Answer</U>. 750 tickets by $25 and the rest 1700-750 = 950 tickets by $15. <U>Check</U>. 750*25 + 950*15 = 33000. ! Correct ! </pre> <H3>Problem 3</H3>Isko wants to mix raisins worth 14 pesos per pound and nuts worth 22 pesos per pound to make 25 pounds of a mixture worth 16 pesos per pound. How many pounds of raisins and how many pounds of nuts should he use? <B>Solution</B> <pre> Let x be the amount of raisins (in pounds) to be mix. Then the amount of nuts is (25-x) pounds. x pounds of raisin cost 14x pesos. (25-x) pounds of nuts cost 22*(25-x) pesos. The total cost for ingredients is 14x + 22*(25-x). Isko wants 25 pounds of the mixture cost 16 pesos per pound. In other words, {{{(14x + 22*(25-x))/25}}} = 16 pesos per pound. It is your basic "money" (or "price") equation. To solve it, multiply both sides by 25. You will get 14x + 22*(25-x) = 16*25, or 14x + 550 - 22x = 400, or -8x = 400 - 550 = 150 ====> x = {{{(-150)/(-8)}}} = 18.75. <U>Answer</U>. Isko should mix 18.75 pounds of raisins with 25-18.75 = 6.25 pounds of nuts. <U>Check</U>. (18.75*14 + 6.25*22)/25 = 16 pesos per pound. ! Correct ! </pre> <H3>Problem 4</H3>A bank contains 8 more pennies than nickels and 3 more dimes than nickels. If the total amount of money in the bank is $3.10, find the number of dimes in the bank. <B>Solution</B> <pre> Let "n" be the number of nickels. Then the number of pennies is (n+8) and the number of dimes is (n+3). Each penny contributes 1 cent to the total. In all, (n+8) pennies contribute (n+8) cents to the total. Each nickel contributes 5 cents to the total. "n" nickels contribute 5n cents to the total. Each dime contributes 10 cents to the total. (n+3) dimes contribute 10(n+3) cents to the total. Therefore, your "value" equation is (n+8) + 5n + 10(n+3) = 310 cents. n + 8 + 5n + 10n + 30 = 310, 16n + 38 = 310 ---> 16n = 310-38 ---> 16n = 272 ---> n = {{{272/16}}} = 17. Thus the number of nickels is 17. Then the number of pennies is (17+8) = 25 and the number of dimes is (17+3) = 20. <U>Check</U>. 25 + 17*5 + 10*20 = <U>Answer</U>. The number of pennies is 25, the number of nickels is 17 and the number of dimes is 20. </pre> <H3>Problem 5</H3>Charles took a Math exam with 20 questions. For every correct question, Charles received 10 marks. For every incorrect or unanswered question, he lost 5 marks. If his final score was 140 marks, how many questions were answered correctly ? <B>Solution</B> <pre> Let x = how many questions were answered correctly. Then the number of questions answered incorrectly is (20-x). Charles earned 10x marks and lost 5*(20-x) marks. His balance is 10x - 5*(20-x) = 140. Simplify and solve it 10x - 100 + 5x = 140 15x = 140 + 100 15x = 240 ==========> x = {{{240/15}}} = 16. <U>Answer</U>. 16 questions were answered correctly. </pre> <H3>Problem 6</H3>Mark had 4 times as many sweets as John. When Mark gave John 15 sweets, both of them had the same number of sweets. How many sweets did they have altogether? <B>Solution</B> <pre> John had x sweets; Mark had 4x sweets. After sharing sweets 4x - 15 = x + 15. Simplify and find x 4x - x = 15 + 15 3x = 30 x = 30/3 = 10. <U>ANSWER</U>. Altogether, they have x + 4x = 5x = 5*10 = 50 sweets. </pre> <H3>Problem 7</H3>Shannon grossed $725 one week by working 52 hours. Her employer pays time-and-a half for all hours worked in excess of 40 hours. Find the Shannon's regular hourly wage. <B>Solution</B> <pre> Let x be the regular hourly wage; then overtime hourly rate is 1.5x. The equation is 40x + (52-40)*1.5x = 725. Simplify 40x + 12*1.5x = 725 40x + 18x = 725 58x = 725 x = {{{725/58}}} = 12.50 dollars per hour. <U>ANSWER</U> </pre> On solving single linear equations and relevant word problems see the lessons - <A HREF=https://www.algebra.com/algebra/homework/equations/How-to-solve-a-linear-equation.lesson>HOW TO solve a linear equation</A> - <A HREF=https://www.algebra.com/algebra/homework/equations/Using-linear-equations-to-solve-word-problems.lesson>Simple word problems to solve using a single linear equation</A> - <A HREF=https://www.algebra.com/algebra/homework/equations/More-complicated-word-problems-to-solve-using-single-linear-equation.lesson>More complicated word problems to solve using a single linear equation</A> - <A HREF=https://www.algebra.com/algebra/homework/equations/Typical-problems-on-selling-and-buying-items.lesson>Typical problems on buying and selling items</A> - <A HREF=https://www.algebra.com/algebra/homework/equations/Typical-investment-problems.lesson>Typical investment problems</A> - <A HREF=https://www.algebra.com/algebra/homework/equations/Advanced-word-problems-to-solve-by-reduction-to-single-linear-equation.lesson>Advanced word problems to solve using a single linear equation</A> - <A HREF=https://www.algebra.com/algebra/homework/equations/HOW-TO-algebreze-and-solve-these-problems-using-one-eqn-in-one-unknown.lesson>HOW TO algebraize and solve these problems using one equation in one unknown</A> - <A HREF=https://www.algebra.com/algebra/homework/equations/Challenging-word-problems-to-solve-using-a-single-linear-equation.lesson>Challenging word problems to solve using a single linear equation</A> - <A HREF=https://www.algebra.com/algebra/homework/equations/Selected-word-problems-to-solve-by-reducing-to-single-linear-equation.lesson>Selected word problems to solve by reducing to single linear equation</A> - <A HREF=https://www.algebra.com/algebra/homework/equations/Solving-some-business-related-problems.lesson>Solving some business-related problems</A> - <A HREF=https://www.algebra.com/algebra/homework/equations/HOW-TO-solve-these-simple-word-problems-MENTALLY-without-using-equations.lesson>HOW TO solve these simple word problems MENTALLY without using equations</A> - <A HREF=https://www.algebra.com/algebra/homework/equations/Time-equation-to-solve-some-Travel-and-Distance-problems.lesson>Using time equation to solve some Travel and Distance problems</A> - <A HREF=https://www.algebra.com/algebra/homework/equations/Using-price-equation-to-solve-some-business-related-problems.lesson>Using price equation to solve some business related problems</A> - <A HREF=https://www.algebra.com/algebra/homework/equations/Solving-simple-and-simplest-problems-by-the-backward-method.lesson>Solving problems by the backward method</A> - <A HREF=https://www.algebra.com/algebra/homework/equations/Solving-problems-on-the-remaining-amount.lesson>Solving more complicated problems by the backward method</A> - <A HREF=https://www.algebra.com/algebra/homework/equations/Solving-entertainment-problems-on-shortage-money.lesson>Solving entertainment problems on shortage of money</A> - <A HREF=https://www.algebra.com/algebra/homework/equations/OVERVIEW-of-lessons-on-solving-linear-equations-and-word-problems-in-one-unknown.lesson>OVERVIEW of lessons on solving linear equations and word problems in one unknown</A> in this site.
