Example 1: Twenty-six less than three times a number is the same as the sum of negative two and five times the number. Find the number.

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1 Section 2.4 continued : Application Problems Tips for solving application problems 1. Read the entire problem. What are you trying to find? What information is given? 2. Plan your approach to the problem. Make a drawing, list operations you will use, and/or create a table if needed. 3. Solve the problem. Identify the unknown value, call it a variable, define the variable, and then translate the problem into algebra to form an equation so you can then solve it. Question: What does it mean to define the variable? Answer: This means that you are to tell the reader what the variable represents. It s kind of like when you are going to use a dictionary to look up the definition of a word, you expect the dictionary to define, or tell you, what that word means. If you define your variable at any time before your write the equation, then please use a Let statement, e.g. Let n = the number of nickels. But if you define your variable RIGHT AFTER you write your equation, then please use a where statement, e.g. where n = the number of nickels 4. Write your answer in a complete sentence. The reader must have a clear idea of what you are talking about. 5. Check your answer. Does your answer make sense? Is it reasonable? Form an equation or inequality and solve it to answer the following problems. Write your answers in complete sentences. Don t forget to use a let statement or where statement do define the variable. Please note: The equations you form must be at minimum 1-step equations. e.g. x = this is not at least a 1-step equation, so not ok however, x 3 = 5 this is a 1-step equation, so it s perfect! Example 1: Twenty-six less than three times a number is the same as the sum of negative two and five times the number. Find the number. Math 60 Beginning Algebra Cerritos College Pg 9 Chapter 2 Lecture Notes by Maria Torres

2 Example 2: When four is subtracted from a number, the result is three more than two-thirds of the number. Find the number. Ex 3: Four times a number is three times the difference of thirty-five and the number. Find the number. Example 4: If three-eighths of a number is added to twice the number, the result is thirty-eight. Find the number. Math 60 Beginning Algebra Cerritos College Pg 10 Chapter 2 Lecture Notes by Maria Torres

3 Consecutive Numbers Consecutive Integers Consecutive Odd Integers Consecutive Even Integers Integers are consecutive if each Odd integers are consecutive if Even integers are consecutive if one is 1 more than the previous each one is 2 more than the each one is 2 more than the integer. previous integer. previous integer. Three consecutive integers can be expressed as x, where x is an integer Three consecutive odd integers can be expressed as x, where x is an odd integer Three consecutive even integers can be expressed as x, where x is an even integer e.g. 1, 2, 3, 4, 5 e.g. 1, 3, 5, 7, 9 e.g. 2, 4, 6, 8, 10 Example 5: The sum of three consecutive integers is 75. Find the integers. Example 6: The sum of four consecutive odd integers is 96. Find the integers. Math 60 Beginning Algebra Cerritos College Pg 11 Chapter 2 Lecture Notes by Maria Torres

4 Example 7: Three times the smallest of three consecutive even integers is two more than twice the largest integer. Find the integers. BP1. Bonus Problems Find three consecutive even integers such that their sum is twelve less than twice times the smallest integer. Answer: 18, 16, 14 BP2. Find three consecutive integers such that sixty-two less than four times the largest integer is the same as the sum of all three integers. Anwswer: 57, 58, 59 Math 60 Beginning Algebra Cerritos College Pg 12 Chapter 2 Lecture Notes by Maria Torres

5 Example 8: A mathematics textbook editor spent 7.5 hr making telephone calls, writing s, and attending meetings. She spent twice as much time attending meetings as making telephone calls and 0.5 hr longer writing s than making telephone calls. How many hours did she spend on each task? Math 60 Beginning Algebra Cerritos College Pg 13 Chapter 2 Lecture Notes by Maria Torres

6 Example 9: China earned a total of 88 medals at the 2012 Summer Olympics. The number of gold medals earned was 15 more than the number of bronze medals. The number of bronze medals earned was 4 fewer than the number of silver medals. How many of each kind of medal did China earn? (Source: World Almanac and Book of Facts.) Math 60 Beginning Algebra Cerritos College Pg 14 Chapter 2 Lecture Notes by Maria Torres

7 Example 10: Write an algebraic expression or inequality described by the word phrases. a. The profit p needs to be at least $100 b. The profit p needs to be more than $100 c. The cost c should be at most $100 d. The cost c should not exceed $100 Example 11: Panchito is the organizer of a jog-a-thon and has an archway of balloons constructed at the finish line of the race. A company charges a $90 setup fee and then 7 cents for every balloon. If Panchito has $225 to spend on the decorations, then up to how many balloons can he purchase? Math 60 Beginning Algebra Cerritos College Pg 15 Chapter 2 Lecture Notes by Maria Torres

8 Example 12: A taxi charges a 1-time $6 fee plus $1.85 for every mile. How far can Panchito travel with a budget of $50? Note: You cannot pay for part of a mile. Answer must be a whole number. (end of Section 2.4) Bonus Problem 3: An international phone call costs $2, plus 30 cents per minute. If Panchito has $5.60 to spend on a call, then up to how many minutes can he use the phone for? Answer: At most 12 minutes Bonus Problem 4: A taxi charges a 1-time $4 fee plus $1.45 for every mile. How far can Panchito travel with a budget of $50? Note: You cannot pay for part of a mile. Answer must be a whole number. Answer: At most 31 miles Math 60 Beginning Algebra Cerritos College Pg 16 Chapter 2 Lecture Notes by Maria Torres

9 Remember to read the textbook before attempting to do your homework. Sections 2.5: Geometry Applications A formula is an equation stating that two or more quantities are the same as one another. In other words, it is an equation used to express a specific relationship mathematically. The perimeter (P) of a figure is given by the sum of all its sides it measures the distance all around. Don t forget the units, e.g. cm, in, ft. Area (A): measure of the interior (of a flat object). Note: use square units, use units 2 (e.g. cm 2, in 2, ft 2 ) Rectangle: width length Area = length width A = L W where L = length, W = width Perimeter = 2(length) + 2(width) P = 2L + 2W Example 1: The perimeter of a rectangle is 112 in. The length is 4 in less than three times the width. Find the dimensions of the rectangle. Math 60 Beginning Algebra Cerritos College Pg 17 Chapter 2 Lecture Notes by Maria Torres

10 Example 2: The perimeter of a rectangle is 36 yd. The width is 18 yd less than twice the length. Find the dimensions of the rectangle. Example 3: A triangle has a perimeter of 31 in. The longest side is 1 in. less than twice the shortest side, and the third side is 4 in. longer than the shortest side. Find the lengths of the three sides. Math 60 Beginning Algebra Cerritos College Pg 18 Chapter 2 Lecture Notes by Maria Torres

11 Geometry Fact: For any triangle, the sum of all the three angles equals 180. Example 4: Find the measure of all three angles. x ( x + 5) x Example 5: One angle of a triangle is 20 larger than the smallest angle, and the third angle is 6 times as large as the smallest angle. Find the measures of the three angles. BP: Find the measure of all three angles. (end of Sec 2.5) x (3x 2) 90 Answer: 23, 67, 90 Math 60 Beginning Algebra Cerritos College Pg 19 Chapter 2 Lecture Notes by Maria Torres

12 Remember to read the textbook before attempting to do your homework. Sections 2.6: Ratio and Proportion In this section we will continue to solve equations. We will be solving special equations called proportions: a c = read as a to b is the same as c to d b d Question: What is a proportion? Answer: A proportion is a statement that says that two ratios or rates are the same. Question: Rumor has it that I can take a shortcut when solving equations that are proportions. How can I quickly solve a proportion? Answer: Oh that s easy! Just take their cross-products and set them equal to each other. Question: Oh really??? But why does this work? Answer: It works because for any proportion, the cross products are always equal. Moral of the story: In any proportion a b = c d a d = b c the cross products are equal. Example 1: Solve each equation. You may not use a calculator to answer. a. m + 6 m + 10 = a m + 6 m + 10 = b x 15, 000 = c. 3 12, 000 (rewrite each problem below and show all work) 6 10 = 9(2 x 4) 12( x 4) Ran out of space??? Don t worry! Just take out a sheet of paper and continue writing. Math 60 Beginning Algebra Cerritos College Pg 20 Chapter 2 Lecture Notes by Maria Torres