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  1. Columbus State Community College Chapter 3 Section 4 Solving Application Problems with Two Unknown Quantities

  2. Solving Application Problems: Two Unknown Quantities • Solve application problems with two unknown quantities.

  3. Solving an Application Problem Solving an Application Problem Step 1 Readthe problem once to see what it is about. Read it carefully a second time. As you read, make a sketch or write word phrases that identify the known and the unknown parts of the problem. Step 2 (a)If there is one unknown quantity, assign a variable to represent it. Write down what your variable represents. Step 2 (b)If there is more than one unknown quantity, assign a variable to represent “the thing you know the least about.” Then write variable expression(s), using the same variable, to show the relationship of the other unknown quantities to the first one. continued…

  4. Solving an Application Problem Solving an Application Problem (continued) Step 3 Write an equation, using your sketch or word phrases as the guide. Step 4 Solve the equation. Step 5 State the answer to the question in the problem and label your answer. Step 6 Check whether your answer fits all the facts given in the original statement of the problem. If it does, you are done. If it doesn’t, start again at Step 1.

  5. Solving an Application Problem: Two Unknown Quantities EXAMPLE 1 Application Problems: Two Unknown Quantities Last month, Holly worked 30 hours more than Traci. Together they worked a total of 360 hours. Find the number of hours each person worked last month. Step 1 Readthe problem. It is about the number of hours worked by Holly and Traci. Unknowns: hours worked by Holly; hours worked by Traci Known: Holly worked 30 hours more than Traci; 360 hours total for Holly and Traci

  6. Solving an Application Problem: Two Unknown Quantities EXAMPLE 1 Application Problems: Two Unknown Quantities Last month, Holly worked 30 hours more than Traci. Together they worked a total of 360 hours. Find the number of hours each person worked last month. Step 2(b)There are two unknowns so assign a variable to represent “the thing you know the least about.” You know least about the hours worked by Traci, so let trepresent Traci’s hours. Holly worked 30 hours more than Traci, so her hours are t + 30, that is, Traci’s hours, ( t ) plus 30 more.

  7. Solving an Application Problem: Two Unknown Quantities EXAMPLE 1 Application Problems: Two Unknown Quantities Last month, Holly worked 30 hours more than Traci. Together they worked a total of 360 hours. Find the number of hours each person worked last month. Step 3 Write an equation. Hours worked by Holly Hours worked by Traci Total hours worked t + t + 30 = 360

  8. Solving an Application Problem: Two Unknown Quantities EXAMPLE 1 Application Problems: Two Unknown Quantities Last month, Holly worked 30 hours more than Traci. Together they worked a total of 360 hours. Find the number of hours each person worked last month. Step 4 Solve the equation. t + t + 30 = 360 2t + 30 = 360 – 30 – 30 2t = 330 2 2 t = 165

  9. Solving an Application Problem: Two Unknown Quantities EXAMPLE 1 Application Problems: Two Unknown Quantities Last month, Holly worked 30 hours more than Traci. Together they worked a total of 360 hours. Find the number of hours each person worked last month. Step 5 State the answer. Because t represents Traci’s hours, and the solution of the equation is t = 165, Traci worked 165 hours. t + 30 represents Holly’s hours. Replace t with 165. 165 + 30 = 195, so Holly worked 195 hours.

  10. Solving an Application Problem: Two Unknown Quantities EXAMPLE 1 Application Problems: Two Unknown Quantities Last month, Holly worked 30 hours more than Traci. Together they worked a total of 360 hours. Find the number of hours each person worked last month. Step 6 Check the solution using the original problem. “Holly worked 30 hours more than Traci.” Holly’s 195 hours are 30 more than Traci’s 165 hours, so the solution checks. “Together they worked total of 360 hours.” Holly’s 195 hours + Traci’s 165 hours = 360 hours so the solution checks.

  11. Solving an Application Problem: Two Unknown Quantities EXAMPLE 1 Application Problems: Two Unknown Quantities Last month, Holly worked 30 hours more than Traci. Together they worked a total of 360 hours. Find the number of hours each person worked last month. Step 6 Check the solution using the original problem. We’ve answered the question correctly because 165 hours and 195 hours fit all the facts given in the problem.

  12. Solving an Application Problem: Two Unknown Quantities EXAMPLE 2 Geometric Problems: Two Unknown Quantities The length of a rectangle is 9 cm more than twice its width. The perimeter is 108 cm. Find the length and width of this rectangle. Step 1 Readthe problem. It’s about a rectangle. Make a sketch of a rectangle. Unknowns: length of a rectangle; width of a rectangle Known: Length is 9 cm more than twice its width; perimeter is 108 cm Perimeter is 108 cm width length

  13. Solving an Application Problem: Two Unknown Quantities EXAMPLE 2 Geometric Problems: Two Unknown Quantities The length of a rectangle is 9 cm more than twice its width. The perimeter is 108 cm. Find the length and width of this rectangle. Step 2(b)There are two unknowns so assign a variable to represent “the thing you know least about.” You know the least about the width, so let w represent the width. The length is 9 cm more than twice the width, so the length is 2w + 9. Perimeter is 108 cm w width 2w + 9 length

  14. Solving an Application Problem: Two Unknown Quantities EXAMPLE 2 Geometric Problems: Two Unknown Quantities The length of a rectangle is 9 cm more than twice its width. The perimeter is 108 cm. Find the length and width of this rectangle. Step 3 Write an equation. Use the formula for perimeter of a rectangle, P = 2l + 2w, to help you write the equation. Perimeter is 108 cm w 2w + 9 P = 2 l+ 2 w 108 = 2(2w + 9) + 2 • w

  15. Solving an Application Problem: Two Unknown Quantities EXAMPLE 2 Geometric Problems: Two Unknown Quantities The length of a rectangle is 9 cm more than twice its width. The perimeter is 108 cm. Find the length and width of this rectangle. Step 4 Solve. 108 = 2 ( 2w + 9 ) + 2 • w 108 = 4w+ 18 + 2w 108 = 6w + 18 – 18 – 18 90 = 6w 6 6 15= w

  16. Solving an Application Problem: Two Unknown Quantities EXAMPLE 2 Geometric Problems: Two Unknown Quantities The length of a rectangle is 9 cm more than twice its width. The perimeter is 108 cm. Find the length and width of this rectangle. Step 5 State the answer. w represents the width, and w = 15 so the width is 15 cm. 2w + 9 represents the length. Replace w with 15. 2 ( 15 ) + 9 = 39, so the length is 39 cm. The label for both answers is cm. The final answer is: The width is 15 cm and the length is 39 cm.

  17. Solving an Application Problem: Two Unknown Quantities EXAMPLE 2 Geometric Problems: Two Unknown Quantities The length of a rectangle is 9 cm more than twice its width. The perimeter is 108 cm. Find the length and width of this rectangle. Step 6 Check the solution using your sketch and the original problem. “The length of a rectangle is 9 cm more than twice it’s width.” 39 cm is 9 cm more than twice 15 cm. “The perimeter is 108 cm.” P = 2 • 39 cm + 2 • 15 cm P = 78 cm + 30 cm P = 108 cm 15 cm 39 cm The answers satisfy the facts in the original problem, so the solution checks.

  18. Solving Application Problems: Two Unknown Quantities Chapter 3 Section 4 – Completed Written by John T. Wallace