Elementary Algebra - Problem Drill 24: Word Problems in Basic Algebra No. 1 of 10 1. John is an ultra-marathoner and his training runs are many hours long. He leaves for a training run at 10 a.m. and he runs at a constant rate of 6 miles per hour. His wife leaves from the same location in her car at 4 p.m. and drives at an average speed of 60 miles per hour. Both of them follow the same route. At what time will they meet? (A) 4:30 p.m. (B) 4:40 p.m. (C) 4:50 p.m. (D) 5:00 p.m. (E) 5:20 p.m. In 30, his wife will have only driven ½ hour 60 mph = 30 miles, but he will have run 6½ hours 6 mph = 39 miles. B. Correct! In 40, his wife will have driven 2/3 hour 60 mph = 40 miles, and he will have run 6 2/3 hours 6 mph = 40 miles. In 50, his wife will have driven 5/6 hour 60 mph = 50 miles, but he will only have run 6 5/6 hours 6 mph = 41 miles. In 1 hour, his wife will have driven 1 hour 60 mph = 60 miles, but he will only have run 7 hours 6 mph = 42 miles. In 80 hour, his wife will have driven 4/3 hours 60 mph = 80 miles, but he will only have run 7 1/3 hours 6 mph = 44 miles. The best strategy for this problem is to use a table to determine the time they meet. Time John s time John s distance Wife s time 4:10pm 6 1/6 hours 37 miles 10 4:20pm 6 1/3 hours 38 miles 20 4:30pm 6 1/2 hours 39 miles 30 4:40p m 6 2/3 hours 40 miles 40 4:50pm 6 5/6 hours 41 miles 50 5:00pm 7 hours 42 miles 60 Wife s distance 10 miles 20 miles 30 miles 40 miles 50 miles 60 miles From the table, it can be seen that John and his wife met at 4:40pm, at which point they both will have traveled 40 miles. The correct answer is choice B 4:40pm
No. 2 of 10 2. A class with 34 students has 6 more girls than boys. How many boys are in the class? (A) 11 (B) 14 (C) 17 (D) 20 (E) 28 If there were 11 boys, there would be 23 girls, which would mean 12 more girls than boys. B. Correct! If there were 14 boys, there would be 20 girls, which would mean 6 more girls than boys. If there were 17 boys, there would be 17 girls, which would mean the same number of girls and boys. If there were 20 boys, there would be 14 girls, which would mean 6 more boys than girls. If there were 28 boys, there would be 6 girls, which would mean 22 more boys than girls Guess, check, and revise are a good strategy to use for this problem, since it is possible to make a reasonable first guess. The number of boys must be slightly less than half the number of students in the class, so guess that there are 15 boys. There would then be 19 girls, which would mean 4 more girls than boys, and that s not a big enough difference. So, lower the guess to 14 boys. There would then be 20 girls, which would mean 6 more girls than boys, and the problem is solved. As a check, an equation could be used for this problem. If b is the number of boys, then (34 b) is the number of girls, resulting in the following equation: (34 b) b = 6 34 2b = 6 2b = 28 b = 14 This confirms the result above, so there must be 14 boys in the class. (B)14
No. 3 of 10 3. Solve the equation: x + 10 = 4 3 (A) -6 (B) 2 (C) 33 (D) 54 (E) 71 Substituting x = -6 into the equation results in a false statement. Substituting x = 2 into the equation results in a false statement. Substituting x = 33 into the equation results in a false statement. D. Correct! Substituting x = 54 into the equation results in a true statement. Substituting x = 71 into the equation results in a false statement. To solve for x, first apply the exponent. Then, solve the equation using the rules of algebra. x + 10 = 4 3 x + 10 = 64 x + 10 10 = 64 10 x = 54 The correct answer is choice D.
No. 4 of 10 4. What is the y-intercept of a line with slope 2 that passes through the point (6, 3 4)? (A) (0, 0) 2 (B) 0, 3 (C) (0, 4) (D) (0, 6) (E) (0, 10) A. Correct! Using point-slope form, the equation of the line is y 4 = 2 (x 6), which can be 3 rewritten as y = 2 x. This line passes through the origin, so the y-intercept is 0. 3 Use the point-slope form for the equation of a line to find the y-intercept. Use the point-slope form for the equation of a line to find the y-intercept. Use the point-slope form for the equation of a line to find the y-intercept. Use the point-slope form for the equation of a line to find the y-intercept. To solve this problem, you need to use the formula for point-slope form. Substituting the coordinates of the point and the slope, the equation of the line is y 4 = 2 3 (x 6), which can be rewritten as y = 2 x. This line passes through the 3 origin, so the y-intercept is 0. (A) (0, 0)
No. 5 of 10 5. Lindsay found an artist who can print uniform shirts for her team. The cost is $100 plus $8 per shirt. Write a rational function for the average cost per shirt, where s is the number of shirts. (A) 100 + 8s (B) 108s (C) 8 + 100s s (D) (E) 8 100 + s 100+ 8s s Use the fact that the total cost is 100 + 8s to find the average cost per shirt. Use the fact that the total cost is 100 + 8s to find the average cost per shirt. Use the fact that the total cost is 100 + 8s to find the average cost per shirt. Use the fact that the total cost is 100 + 8s to find the average cost per shirt. E. Correct! The total cost is 100 + 8s. To find the average cost per shirt, you must divide by the number of shirts, s. The total cost is 100 + 8s. To find the average cost per shirt, divide by the number + of shirts s. The rational function for average cost is. 100 8s s (E) 100+ 8s s
No. 6 of 10 6. Lindsay found a company who can print uniform shirts for her team. The cost is $200 plus $6 per shirt. How much would it cost to have 30 shirts printed? (A) $180 (B) $206 (C) $230 (D) $380 (E) $410 The cost would be $180 if the extra $200 charge did not apply. This would be the cost for just 1 shirt, not 30. This would be the cost if there were a $1 charge per shirt instead of $6 per shirt. D. Correct! The total cost is 200 + 6(30) = $380. Check your arithmetic. Algebraically, the cost is 200 + 6n, where n is the number of shirts to be printed. Lindsay wants to have 30 shirts printed, so n = 30. This gives a total cost of 200 + 6(30) = $380. (D) $380
No. 7 of 10 7. Which of the following was not a problem-solving strategy presented in the tutorial? (A) Use a formula (B) Use a calculator (C) Look for a pattern (D) Compute or simplify (E) Guess, check, and revise This was one of the problem-solving strategies presented in the tutorial. B. Correct! This was not one of the problem-solving strategies presented in the tutorial. This was one of the problem-solving strategies presented in the tutorial. This was one of the problem-solving strategies presented in the tutorial. This was one of the problem-solving strategies presented in the tutorial. The following problem-solving strategies were discussed in this tutorial: Make a Model, Diagram, or Equation Use a Formula Compute or Simplify Make a Table, Chart or List Guess, Check, and Revise Consider a Simpler Case Look for a Pattern (B) Use a calculator
No. 8 of 10 8. Temperatures in degrees Fahrenheit F are related to temperatures in degrees Celsius C by a linear function. The Fahrenheit temperature of -40 F is equivalent to the Celsius temperature of -40 C, and the temperature of 32 F is equivalent to 0 C. What linear function explains the relationship of F to C? (A) F = 9 5 C + 32 (B) C = 9 5 F + 32 (C) F = 9 5 C 32 (D) C = 9 5 F 32 (E) F = 5 9 C + 32 A. Correct! When F = -40, C = -40 and F = 32, C = 0 are substituted into this equation the results are true statements. When F = 32, C = 0 is substituted into this equation the result is a false statement. When F = 32, C = 0 is substituted into this equation the result is a false statement. When F = 40, C = 40 is substituted into this equation the result is a false statement. When F = 40, C = 40 is substituted into this equation the result is a false statement. This problem can be solved by testing the answer choices. For choice (A), the following results occur: F = -40, C = -40-40 = 9/5 (-40) + 32-40 = -72 + 32-40 = -40 F = 32, C = 0 32 = 9/5 (0) + 32 32 = 32 From these results, choice (A) seems to be correct. For verification, algebraic formulas can be used to find the exact linear function. For the points (0,32) and (-40,-40), the slope is: m = 32 (-40) / 0 (-40) = 72 / 40 = 9/5 The y-intercept can then be found as follows: y = mx + b 32 = 9/5 (0) + b 32 = b With m = 9/5 and b = 32, the linear function relating F and C is F = 9/5 C + 32.
No. 9 of 10 9. The length of a rectangle is 2 more than twice the width. The perimeter of the rectangle is 40 inches. What is the area of the rectangle? (A) 6 square inches (B) 14 square inches (C) 20 square inches (D) 80 square inches (E) 84 square inches Write expressions representing the length and width of the rectangle to find the area. Write expressions representing the length and width of the rectangle to find the area. Write expressions representing the length and width of the rectangle to find the area. Write expressions representing the length and width of the rectangle to find the area. E. Correct! The width is 6 inches, the length is 14 inches, and the area is 6 14 = 84 square inches. If the width is represented as w inches, then the length is 2w + 2 inches, as shown below. The perimeter of a rectangle is (2 length) + (2 width), which in this case is 2 (2w + 2) + 2w. Since the perimeter is known, this gives the following equation, which can be solved for w: 2 (2w + 2) + 2w = 40 4w + 4 + 2w = 40 6w = 36 w = 6 If w = 6, then l = 2w + 2 = 2(6) = 2 = 14. The area, then, is A = lw = 14 6 = 84 square inches. (E)84 square inches
No. 10 of 10 10. Alex bought 2 pencils and 1 eraser for $1.15. Eli bought 2 pencils and 2 erasers for $1.80. How much will Remy pay if he buys 1 pencil and 1 eraser? (A) $0.25 (B) $0.65 (C) $0.90 (D) $1.30 (E) $2.95 The cost of just one pencil is $0.25. The cost of just one eraser is $0.65. C. Correct! The cost of one pencil is $0.25, the cost of one eraser is $0.65, so their combined price is $0.90. This is too much. Use a system of equations to find the answer. The cost of 4 pencils and 3 erasers is $2.95. If p represents pencils and r represents erasers, then the following system of equations can be used: 2p + r = 1.15 2p + 2r = 1.80 If the first equation is subtracted from the second, the result is r = 0.65 Consequently, erasers cost $0.65 each. This value can be substituted into the first equation to find the value of a pencil: 2p + r = 1.15 2p + 0.65 = 1.15 2p = 0.50 p = 0.25 Pencils cost $0.25 each. Therefore, the cost of one pencil and one eraser is 0.25 + 0.65 = $0.90. To check this result, another result is to realize that the cost of two pencils and two erasers is given in the second equation: Dividing all terms in this equation gives: 2p + 2r = 1.80 p + r = 0.90 This equation shows that the combined price of one pencil and one eraser is $0.90. (Note, however, that this method does not allow you to find the price of either item by itself. But that s okay, since that s not what the question asked.) (C) $0.90