Algebra Word Problems


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1 WORKPLACE LINK: Nancy works at a clothing store. A customer wants to know the original price of a pair of slacks that are now on sale for 40% off. The sale price is $6.50. Nancy knows that 40% of the original price subtracted from the original price will equal the sale price. Using for the original price she writes: Then she solves for. Algebra Word Problems Many algebra problems are about number relationships. In most word problems, one number is defined by describing its relationship to another number. One other fact, such as the sum or product of the numbers, is also given. To solve the problem, you need to find a way to epress both numbers using the same variable. 8See how to write an equation about two amounts using one variable. Eample: Together, Victor and Tami Vargas earn $33,280 per year. Tami earns $4,60 more per year than Victor earns. How much do Victor and Tami each earn per year? MATHEMATICS You are asked to find two unknown amounts. Victor s earnings: Represent the amounts using algebra. Tami s earnings: 4,60 Write an equation showing that the sum of the two amounts is $33,280. Solve the equation. 4,60 33,280 Combine like terms. 2 4,60 33,280 Subtract 4,60 from both sides of the equation. 2 4,60 4,60 33,280 4, ,20 Divide both sides by , ,560 Now go back to the beginning, when you first wrote the amounts in algebraic language. Since represents Victor s earnings, you know that Victor earns $4,560 per year. Tami s earnings are represented by 4,60. Add: 4,560 4,60 8,720. Tami earns $8,720 per year. Answer: Victor earns $4,560, and Tami earns $8,720. Check: Return to the original word problem and see whether these amounts satisfy the conditions of the problem. The sum of the amounts is $33,280, and $8,720 is $4,60 more than $4,560. The answer is correct. 8Learn how to apply algebraic thinking to problems about age. Eample: Erica is four times as old as Blair. Nicole is three years older than Erica. The sum of their ages is 2. How old is Erica? 262 The problem concerns three ages. Let equal Blair s age. Blair s age: Represent the amounts using the same variable. Erica s age: 4 Nicole s age: 4 3 Write an equation showing the sum equal to
2 Solve the equation The variable is equal to 2, but that doesn t answer the question posed in the 9 8 problem. The problem asks you to find Erica s age, which is equal to 4, or 4(2). 2 Answer: Erica is 8 years old. Check: Blair is 2, Erica is 8, and Nicole is. The ages total 2. This worked eample does not show every step. As you gain eperience, you will do many of the steps mentally. 8Learn how to write equations for consecutive number problems. Eample: The sum of three consecutive numbers is 75. Name the numbers. Consecutive numbers are numbers in counting order. To solve problems of this type, let equal the first number. The second and third numbers can be epressed as and 2. Write an equation Solve Answer: The numbers are 24, 25, and 26. Check: The numbers are consecutive, and their sum is 75. S K I L L P R A C T I C E For each problem, write an equation and solve. Check your answer.. Name two numbers if one number is 3 more than twice another, and their sum is Erin is 8 years less than twice Paula s age. The sum of their ages is 40. How old is Erin? 3. Lyle and Roy do landscaping. They recently earned $840 for a project. If Lyle earned $4 for every $ earned by Roy, how much of the money went to Lyle? 4. The sum of four consecutive numbers is 626. Find the four numbers. 5. A movie theater sold 5 times as many children s tickets as adult tickets to an afternoon show. If 32 tickets were sold in all, how many were children s tickets? 6. Together, Grace and Carlo spent $5 on a gift. If Grace contributed twice as much money as Carlo, how much did Carlo spend? 3 7. Fahi s age is 4 of Mia s age. The sum of their ages is 9. How much older is Mia than Fahi? 8. The sum of two consecutive odd numbers is 64. Name the numbers. (Hint: Let represent the first number and 2 the second number.) 9. One number is 8 more than 2 of another number. The sum of the numbers is 23. What are the numbers? 0. Adena, Julius, and Tia volunteered to read to children at the public library. Julius worked two hours less than Tia. Adena worked twice as many hours as Julius. Altogether they worked 58 hours. How many hours did Adena work? () 4 (4) 42 (2) 6 (5) 46 (3) 28 Answers and eplanations start on page 329. PROGRAM 38: Introduction to Algebra 263
3 COMMUNITY LINK: For a grant application, Jodi is sketching the proposed landscaping plan for a new community center. The plan calls for a rectangular garden. Jodi needs the dimensions of the rectangle to draw the plan. She knows that the length is two times the width and that the perimeter is 26 meters. Jodi writes an equation to find the dimensions. More Algebra Word Problems Many algebra problems are about the figures that you encounter in geometry. To solve these problems, you will need to combine your understanding of geometry and its formulas with your ability to write and solve equations. 8Learn how to solve for the dimensions of a rectangle. Eample: In the situation above, Jodi is given the perimeter of a rectangle. She also knows that the length is twice as many meters as the width. Using the formula for finding the perimeter of a rectangle, find the dimensions of the rectangle. MATHEMATICS The formula for finding the perimeter of a rectangle is P 2l 2w, where l length and w width. You will be given a page of formulas when you take the GED Math Test. A copy of that page is printed for your study on page 340 of this book. When an item on the GED Math Test describes a figure in words alone, your first step should be to make a quick sketch of the figure. Read the problem carefully, and label your sketch with the information you have been given. In this case, let represent the width and 2 the length. 2 P 26 m Now substitute the information you have into the formula and solve. P 2l 2w 26 2(2) 2() If the width () is 2 meters, then 26 6 the length (2) must be 42 meters. 2 Answer: The rectangular garden is 2 meters wide and 42 meters long. Check: Make sure the dimensions meet the condition of the problem. Substitute the dimensions into the perimeter formula: P 2(2) 2(42) meters. The answer is correct. Another common type of algebra problem involves the denominations of coins and bills. In this type of problem, you are told how many coins or bills there are in all. You are also given the denominations of the coins or bills used and the total amount of money. Your task is to find how many there are of each denomination. 264
4 8Learn how to solve denomination problems by studying this eample. Eample: Marisa is taking a cash deposit to the bank. She has $0 bills and $5 bills in a deposit pouch. Altogether, she has 30 bills with a total value of $890. How many bills of each kind are in the pouch? Let represent the number of $0 bills in the bag. Net, use the same variable to represent the number of $5 bills. If there are 30 bills in all, then 30 represents the number of $5 bills. Now you need to establish a relationship between the number of bills and their value. Using the epression above, the total value of the $0 bills is 0, and the total value of the $5 bills is 5(30 ). Write an equation and solve. 0 5(30 ) 890 Multiply both terms in parentheses by Combine like terms Subtract 650 from both sides Divide both sides by 5 to get the 48 number of $0 bills Subtract to find the number of $5 bills. Answer: There are 48 tendollar bills and 82 fivedollar bills. Check: (48 $0) (82 $5) $480 $40 $890. The answer is correct. S K I L L P R A C T I C E Solve each problem.. The length of a rectangle is 4 inches more than twice its width. If the perimeter of the rectangle is 38 inches, what is its width? 2. The perimeter of a right triangle is 60 cm. Side A is 2 cm less than half of Side B. Side C is 2 cm longer than Side B. Find the lengths of the three sides. 3. A bag contains a total of 200 quarters and dimes. If the total value of the bag s contents is $4.00, how many quarters and how many dimes are in the bag? 4. A school sold 300 tickets to a basketball game. Tickets were $9 for adults and $5 for children. If the total revenue was $2340, how many of each ticket type were sold? PROGRAM 38: Introduction to Algebra H I S T O R Y C o n n e c t i o n Seeing a pattern can help you solve algebra problems. Suppose you were asked to find the sum of the whole numbers from to 00. What would you do? In 787, tenyearold Carl Gauss was given this problem. As the other students began adding on their slates, Carl solved the problem mentally. Carl observed that by adding the highest and lowest numbers in the sequence and working toward the middle he could find pairs that equaled 0: 00 0, , , and so on. Carl knew there must be 50 pairings in all. He multiplied 0 by 50 and got 5050, the correct sum. Carl Gauss went on to become a famous mathematician. Apply Carl s method here. Find the sum of the even whole numbers from 2 to 40. Answers and eplanations start on page
5 Skill Practice, page ( 4) 8. 7 Subtracting 2 is equivalent to 4 adding 2, or Subtract inside the parentheses first: 6( 5) Evaluate the eponent, then add inside the parentheses: (8 2 20) (64 20) Problem Solver Connection, page or 6 ( 4) 2 ( 3) Skill Practice, page a 3b m 2 2m a 2 3b 5b y Do the division and multiplication first: 4y 2 2y and 2( 2y) 2 4y. Then simplify by combining all like terms First find the numerator: 2 2 y 2(9) Then find the denominator: 5 ( 5) 20 5( 3) 5 0. Divide: ,000 Instead of computing fractions of p, you can start by combining terms: 4p.25p 0.75p = 6p. Substitute and solve: 00 6p = = 2,000. Skill Practice, page b m n 8 7. (Note: ) y a In standard form, and In standard form, ,000 and = 94, sq mi 6. 3,64,000 sq mi 7. 25,000,000,000,000 miles This number is read 25 trillion times Since our place value system is based on tens, a difference of in the eponent changes the number by a factor of Skill Practice, page 26. y 4 2y 7 2y 8 y n 5n 4 9 5n 5 n 4. b 3 4b 8 2b 8 6b 3 b 5. a 30 a a 6 5 a Skill Practice, page 263. The numbers are 8 and 39. first number, second number Erin is 24. Paula s age, Erin s age ANSWER KEY 329
6 MATHEMATICS Lyle earned $672. Roy s earnings, Lyle s earnings , 56, 57, and adult tickets, children s tickets $7 Carlo s contribution, Grace s contribution Mia is 3 years older than Fahi. Mia s age, Fahi s age Mia is 52 and Fahi is and and 3 first number, second number (3) 28 Tia s hours, Julius s hours 2, Adena s hours 2( 2) 2 2( 2) Substitute 6 for in the epression for Adena s hours. Skill Practice, page inches width, length 2 4 P 2l 2w 38 2(2 4) The sides of the triangle measure 0, 24, and 26 cm. side B, side A 2 2, side C 2 P a b c Side B is 24 cm. Substitute 24 cm for in the epressions for sides A and C quarters and 60 dimes number of quarters number of dimes (200 ) adult tickets and 90 children s tickets number of tickets sold to adults = number of tickets sold to children = (300 ) History Connection, page There are 20 pairings of even numbers from 2 to 40 that have the same sum , , , and so on. Multiply: GED Practice, pages (2) B Since is greater than 2, it must lie to the right of 2. Since is less than 0, it must lie to the left of 0. Only point B is both greater than 2 and less than (2) 78 (a 7b) The quantity of a plus 7b will be a sum, which is to be subtracted from (5) 96 Remembering that two negative numbers multiplied together result in a positive product, your answer is (2) B First convert 7 to 3 7, which you can estimate is a little more than 3 2. Being a 4 negative number, 3 7 will be just to the left of (3) $450 $20 $340 $845 Each deposit represents a positive; the transfer represents a negative. 6. (4) 48 Substitute the known values. (d 7 is etraneous information.) a 4(2)( 5) a 8(6) (5) 2(2y) 2(3) The formula for finding the perimeter of a rectangle is P 2l 2w. Substitute 2y for l and 3 for w. 8. (4) 288 The formula for finding the area of a rectangle is A lw. The length of the rectangle is 2y or 2(2), which equals 24. The width of the rectangle is 3w or 3(4), which equals 2. Thus, the area of the rectangle is A = 24(2) = 288.
7 FAMILY LINK: Robert is helping his son do his math homework. I know the answer is $20. I added $5 and $55 and multiplied by 3. But I have to show my work, and I don t know how to write what I did, his son said. Robert helped his son write each step. He put the sum in parentheses and then multiplied. The problem setup was written 3($5 + $55). Solving Setup Problems Some items on the GED Math Test ask you to select a correct method for solving a problem instead of finding the solution. These problems do not require calculations. Instead, you need to find an epression that shows the correct numbers, operations, and order of steps. 8Think about the method you would use to solve this problem. MATHEMATICS Eample: Marie earns $6 per hour for overtime hours. She worked 8 hours overtime last week and 6 hours overtime this week. Which epression could be used to find Marie s overtime pay for the twoweek period? () $6 8 6 (4) 6($6 8) (2) $6(8 6) (5) 8(6) 8($6) (3) 8(6) $6 You re right if you chose (2) $6(8 6). In this epression, $6 (the overtime hourly wage) is multiplied by the sum of 8 and 6 (the total overtime hours worked). There is another way to work the problem. You could find the overtime pay for each week and add: $6(8) $6(6). Both choice (2) and this epression yield the same result. Both are correct. $6(8 6) $6(4) $224 $6(8) $6(6) $28 $96 $224 To solve GED setup problems, think about how you would calculate an answer to the problem. Put your ideas into words. Then substitute numbers and operations symbols for the words. Properties of Operations To recognize the correct setup, you should be familiar with these important properties. The commutative property applies only to addition OR and multiplication. It means that you can add or multiply OR numbers in any order without affecting the result. The associative property works only for addition and (5 8) multiplication. It means that when you add or multiply OR more than two numbers, you can group the numbers 5 (8 0) any way that you like without affecting the result. (6 2) OR 6 (2 3)
8 The distributive property says that a number outside 8(8 6) 8(8) 8(6) parentheses can be multiplied by each number inside the 8(4) parentheses. Then the products are added or subtracted 2 2 according to the operation symbol. On the GED Math Test, your setup may not seem to be among 5(2 3) 5(2) 5(3) the answer choices. Try applying these properties to the 5(9) 60 5 choices to see whether the setup is written a different way S K I L L P R A C T I C E Solve each problem.. In January, 58 parents attended the PTA meeting. In February, 72 parents attended, and 3 in March. Which epression represents the total attendance for the 3 months? () (2) 3( ) (3) (4) 58(72 3) (5) 3(3 58) Which of the following epressions is the same as (5 5) (5 8)? () 5 (5 5) 8 (2) 5(5 8) (3) 5(5 8) (4) (5 8) 5 (5) 5(5 8) 3. Lamar earned $20,800 last year, and his wife, Neva, earned $22,880. Which epression could be used to find the amount they earned per month last year? () 2($22,880 $20,800) (2) ($22,800 $20,800) 2 (3) 2($22,880 $20,800) (4) ($22,880 $20,800) 2 (5) ($20,800 $22,800) 2 4. Alea sold 2 sweatshirts at $23 each and 28 Tshirts at $4 each. Which epression represents total sales? () 2($23) 28($4) (2) 2($23 $4) (3) 28($4 $23) (4) 2 $23 28 $4 (5) 28(2) $23($4) PROGRAM 29: Problem Solving C A L C U L A T O R C o n n e c t i o n Suppose you are solving a setup problem. You know how to do the math, but you don t recognize any of the setups offered as answer choices. You may have used a correct, but different, approach. Your calculator can help. Eample: Phyllis buys two software programs, each costing $50. There is a $25 rebate on each program. Which epression shows the combined cost? () 2($50)($25) (4) $50 2($25) (2) 2($50) + 2($25) (5) 2($50) $25 (3) 2($50) 2($25) Step. Think about how you would solve this problem. Perhaps you would add and subtract: $50 + $50 $25 $25. Your approach is not among the choices, however. Step 2. Do the math using your approach: $50 + $50 $25 $25 = $250. Now use your calculator to evaluate each setup until you find the one that yields your answer. Which epression is correct? Answers and eplanations start on page
9 MATHEMATICS 7. (3) 262,000 sq mi The digit is in the thousands place of the number 26,94. The digit to the right is 9. Add to the thousands place and replace the digits to the right with zeros. The number 26,94 rounds to 262, (4) 8 hundred thousand The place value for 8 in this number means 8 00,000, or 800, (4) $5,000 $4,000 Use frontend estimation by looking only at the value of the first digit. Subtract the smaller number from the larger. 20. (2) less than Karen Hall. Comparing the values of the numbers given, you will see that $38,054 (Ed s commissions) is less than $38,450 (Karen s commissions). 2. (3) $,000 You can eyeball or estimate the difference by looking at the two salaries or rounding them to the nearest thousand: $43,000 $42,000 $,000. PROGRAM 29: PROBLEM SOLVING Basic Operations Review Skill Practice, page 47., $93 3., (4) $0,608 Multiply: , , (4) $6,63 Add: $3,940 $,45 $,528 Solving Word Problems Skill Practice, page 53. (3) $20,707 Add: $3,682 $7, (5) Not enough information is given. You would need the amount of the annual goal to find how much the association has left to raise. 3. (2) $6,764 Subtract: $3,682 $6,98 Skill Practice, page 55. $ $9 $ $79 $432 8 $ bottles $432 $ miles miles 5. () 575 miles per hour Use the formula: d rt. Substitute the values you know: 725 r 3. Divide to find the rate: miles per hour. Science Connection, page 55 9 psf Divide,620 pounds by 80 square feet. Solving MultiStep Problems Skill Practice, page miles (5 2) miles (5 2 4) Skill Practice, page r r chairs Divide: 2, Ignore the remainder, since Laurie can t buy part of a chair. 6. $67 Subtract: $,000 $ $,338 Multiply: $ $,420 Multiply to find the cost of the book trucks: $89 5 $945. Then add the cost of the cart: $945 $475 $, (2) 690 Divide: $2,500 $8 694 r8, which rounds to 690. Skill Practice, page 5 You may use any letter for the variable.. $25 $36 $6 or $36 $6 2. $66 $624 = $ hours h $9 $342 or $9 h $ books b 6 42 Problem Solver Connection, page 5 To count up from $85 to $500, think: $5 gets me to $90, $0 more gets me to $200. Then I need $300 to get to $500. Add: $5 $0 $300 $35. Skill Practice, page 59. (3) Add to find the total. 2. (5) 5(5 8) This problem applies the distributive property. 3. (4) ($22,880 $20,800) 2 Add their annual wages and divide by 2, the number of months in a year. 4. () 2($23) 28($4) Apply the cost formula for each item and add the results. Calculator Connection, page 59 (3) 2($50) 2($25) This is the only epression that equals $250. GED Practice, pages (5) $2 Since the shirts each cost the same amount, divide $84 by 4 shirts. The information about the jeans is unnecessary, or etraneous, for this calculation. 2. () 84 6 Divide the number of employees per team into the total number of employees. 3. () Subtract the morning and the afternoon work periods (2 2 and 4 hours) from the total number of seminar hours (8 hours). 4. (2) $7.00 Subtract: $35 $28 = $7.
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