Section 1.4
Word Problems
Examples:
a. Perimeter: Determine the lengths of the sides of the triangle so that the perimeter is 105 inches. Solution:\begin{align} (2x+7)+(2x-1)+(x-1)&=105\\ 5x+5&=105\\ 5x&=100\\ x&=20 \end{align} The sides of the triangle are 47 inches, 39, inches, and 19 inches. |
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b. Equal Lengths: In the figure to the right, the two segments have the same length. How long is each one? Solution: \begin{align} 6+(3x-2)&=(2x+1)+7\\ 3x+4&=2x+8\\ x&=4 \end{align} Each of the long segments is 16 units long. |
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c. Fractions of the Whole: Mr. Lang brings a lot of Munchkins to his class for a treat. His students draw slips out of a hat to let them know how many Munchkins each will get. Janet draws a slip given her one-fourth of all the Munchkins. Jack draws a slip giving him one-sixth of all the Munchkins. Each of the remaining eight students draws a slip worth seven Munchkins. How many Munchkins did Mr. Lang bring to class? Solution: Let $x$ represent the total number of Munchkins brought to class. Then Janet's share is $\frac{1}{4}x$. Jack's share is $\frac{1}{6}x$. If we add up all the Munchkins awarded to all ten students, we have the total number of Munchkins. That is: \begin{align} \frac{1}{4}x+\frac{1}{6}x+8\cdot 7&=x\\ \frac{5}{12}x+56&=x\\ 56&=\frac{7}{12}x\\ 96&=x \end{align} Mr. Lang brings 96 Munchkins to class. |
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d. The Barnyard: A standard problem will be what we call "the barnyard problem." A common diagram will be like that appearing to the right, though more complicated figures will occur occasionally. There is a rectangular barn. Adjacent is a fenced barnyard (the fencing is only along the lines that are dark in the figure, since the walls of the barn help contain the livestock). All the angles are right angles. The first problem is to obtain a simplified expression for th enumber of feet necessary to enclose the barnyard. This requires careful use of parentheses along with minus signs. That is the whole point of the problem. In the case pictured here, we have: | |
\begin{align} Top:& &&=2x-3\\ Left:& &&=3x+5\\ Bottom:&(2x-3)-(x+1)&&=x-4\\ Right:&(3x+5)-(x-2)&&=2x+7\\ Total:& &&=8x+5 \end{align} Of course, a slight varaiation is to determine the value of $x$ if the total number of feet of fencing is some specified number---say 205 feet. That leads to a simple linear equation: \begin{align} 8x+5&=205\\ 8x&=200\\ x&=25\text{ feet} \end{align} |
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e. People's Age: Dave is 25 years older than Jeff. Twelve years ago, Dave was twice as old as Jeff. How old are Jeff and Dave now? Solution: Let $x$ represent Jeff's age now. Then Dave's age now is $x+25$. Twelve years ago, Jeff's age was $x-12$ and Dave's age was $(x+25)-12$. The problem tells us the connection between these two ages twelve years ago. $$(x+25)-12=2(x-12)$$ Simplify and solve for $x$. \begin{align} (x+25)-12&=2(x-12)\\ x+13&=2x-24\\ 37&=x \end{align} Jeff is 37 years old; Dave is 62 years old. |
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f. Value of Coins: I have a collection of nickels and dimes. There are 13 more dimes than nickels. The total value of the coins is $\$1.90$. How many of each kind of coin do I have? Solution: Let $x$ represent the number of nickels I have. Then the number of dimes is $x+13$ Our equation is about the value of the coins (not the number of coins). \begin{align} .05x+.10(x+13)&=1.90\\ 5x+10(x+13)&=190\\ 5x+10x+130&=190\\ 15x&=60\\ x&=4 \end{align} I have 4 nickels and 17 dimes. |
Exercises:
1. Determine the lengths of the sides of the triangle shown below if its perimeter is 38 inches.
2. The isosceles triangle shown has a perimeter of 71. How long are the equal sides?
3. The rectangle pictured below has a perimeter of 194 inches. What are the dimensions of the rectangle?
4. The triangle pictured below has a perimeter of 57 centimeters. What are the lengths of the three sides?
5. In the rectangle shown, the perimeter is 84 feet. What are the dimensions of the rectangle?
6. A triangle has two sides that are the same length. The third side is 5 inches longer than each of the equal sides. The perimeter of the triangle is 44 inches. How long are the three sides?
7. A rectangle is 7 feet longer than it is wide. The perimeter of the rectangle is 66 feet. What are the dimensions of the rectangle?
8. The sides of a triangle are $x$, $x – 3$, and $2x – 15$. The perimeter of the triangle is 102. What is the length of the shortest side of the triangle?
9. A pentagon (a figure with five sides) has sides of length $2x – 2$, $3x + 4$, $x + 5$, 12, and $3x – 6$. The perimeter of the pentagon is 76 inches. How many of the sides of the pentagon have the same length?
10. Determine the value of $x$ in the figure below.
11. How long are the two segments shown below?
12. How long are the two segments shown below?
13. John ate one-third of all the strawberries in a basket while Joan ate 25 strawberries from the same basket. At this point, three-quarters of the berries were gone. How many strawberries were in the basket to start with?
14. A farmer owns chickens, sheep, and cows. Two-fifths of his animals are sheep; one-quarter of his animals are cows; and he has 126 chickens. How many animals does he own in all?
15. Mr. Lang is treating his class to Munchkins again. Joe draws a slip giving him one-tenth of all the Munchkins. JoAnn draws a slip giving her one-eighth of all the Munchkins. Jackie draws a slip giving her one-sixth of all the Munchkins. Each of the remaining seven students draws a slip worth ten Munchkins. That leaves three Munchkins for Mr. Lang. How many Munchkins did he bring to class?
16. Jack started in Andover, biking toward Salem. At the same time, Jill started in Salem, headed toward Andover. After 1 hour, Jack was one-quarter of the way from Andover to Salem, while Jill was one-sixth of the way from Salem to Andover. At this point, Jack and Jill were 14 miles apart. What is the distance between Andover and Salem?
17. Judy took an airplane trip. She read for the first half of the flight. She slept for the next hour. Then she worked on crossword puzzles for the last one-third of the flight. How long a flight was it?
18. In the barnyard problem pictured below, it takes 405 feet of fencing to enclose the barnyard. What is the value of $x$?
19. This time it takes 141 feet of fencing to enclose the barnyard. What is the value of $x$?
20. The green line shows where the fencing is around the big new weird barnyard. As usual, all corners are right angles. Find $x$ if the total length of fencing is 200 feet.
21. In the barnyard problem below, it takes 124 feet of fencing to enclose the barnyard. What is $x$?
22. Domenic is ten years older than Chris. In seven years the sum of their ages will be 116. How old is each one now?
23. Rachel is 28 years older than Becky. Twenty years ago she was three times as old as Becky. How old is Becky now?
24. Nat taught at P.A. eight years longer than Bob and twenty-one years longer than C-Y. When the three of them retired, they had a combined 91 years of teaching in the Math Department. How long was each of them at P.A.?
25. Marten is three years older than Lukas and two years younger than Stefan. In five years, the sum of all three of their ages will be 125. How old is each one now?
26. Walt is 18 years older than Mickey. In eight years, he will be 1.6 times as old as Mickey will be. How old is each one now?
27. There is a pile of dimes and quarters, with seven more quarters than dimes. The total value of the coins is $\$29.05$. How many coins of each sort are there?
28. Hector puts nickels and quarters (only) into his piggy bank. When he emptied his bank last week, he found that the number of quarters was 7 more than three times the number of nickels. The value of the coins was $\$68.15$. How many nickels and how many quarters were in the piggy bank?
29. Corrine has a big pile of coins — nickels and dimes. The number of dimes is 53 more than the number of nickels, and the total value of all the coins is $\$30.35$. How many nickels and how many dimes does Corrine have?
30. Zack has a bunch of coins — three more dimes than nickels and 8 more quarters than nickels. The value of all the coins is $\$9.10$. How many coins of each sort does Zack have?
31. Hailee has nickels and quarters — thirty-seven coins in all. The value of her coins is $\$5.05$. How many nickels does Hailee have?
32. I have 500 coins — nickels and dimes. The value of the coins is $\$42.00$. How many nickels and how many dimes are there?