Questions and Answers
Which of the following are steps in practical problem solving? (Select all that apply)
How many coins does Betty have if she has $3.45?
24
If x represents the smaller number, what is the larger number that is 10 times as large?
10x
Which equation can be used to solve for the smallest angle if one of the two supplementary angles is five times the other?
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If x represents the son's age two years ago, which expression represents his father's age three years from now?
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Which of the following equations cannot be used to solve for angles if one of two complementary angles is 4 degrees more than the other?
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What are the three consecutive integral numbers whose sum is 117?
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What are the numbers if one is five more than the other and their sum is three less than three times the smaller?
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If x represents the smaller number, what is the larger number that is 3 more than 2 times the smaller number whose sum is 27?
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If x represents the smaller integer, what is the expression for the larger integer if the sum of 2 consecutive odd integers is 92?
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Which equation can be used to solve for the sides if the base of an isosceles triangle is 4 inches less than one of the equal sides, and the perimeter is 32?
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Which equation can be used to solve for the smaller number if one number is 800 less than 3 times the other and their sum is 1,200?
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What are the dimensions of a rectangle if the length is 3 times the width and the perimeter is 22?
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If x represents the smaller integer, what is the expression for the larger integer if the sum of 2 consecutive numbers is 37?
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Which equation can be used to determine how many nickels and dimes a man has if he has two more nickels than dimes and a total of $1.15?
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How old are James and Joe if Joe is 10 years older than James and in 8 years, twice Joe's age will equal three times James's age?
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Study Notes
Problem Solving Steps
- Identify practical problem-solving steps: Assign a variable, write a condition, solve the equation.
- Steps include: Assigning an identifying variable to the quantity, writing a sentence about conditions, and checking the solution.
Coin Problems
- Betty has 10 more dimes than quarters; her total is $3.45, resulting in 7 quarters and 17 dimes.
- A man has 2 more nickels than dimes and a total of $1.15; this leads to the equation 5(x + 2) + 10x = 115 for the calculation.
Algebraic Representation
- For one number being 10 times another with a difference of 81, if x is the smaller number, the larger number is represented as 10X.
- In the case of supplementary angles, if one angle is 5 times the other, the equation 6X = 180 can be used to find the angles.
Age Problems
- Two years ago, a father was four times his son’s age; in three years, he will be three times the son’s age. If x is the son's age two years back, the father's age three years from now is represented as x + 3.
- Joe is 10 years older than James; in 8 years, twice Joe's age equals three times James's age, identifying ages as James = 12 years and Joe = 22 years.
Consecutive Numbers and Angles
- The sum of 3 consecutive integers is 117, yielding the numbers 38, 39, and 40.
- For complementary angles where one is 4 degrees more than the other, the unusable equation to solve the problem is 2x - 4 = 90.
Rectangles and Isosceles Triangles
- The length of a rectangle, three times its width, has a total perimeter of 22, with width and length calculated as Width = 2.75 and Length = 8.25.
- In an isosceles triangle problem, where the base is 4 inches shorter than sides, the equation to find the sides is 3x - 4 = 32.
Additional Numeric Relationships
- One number is 3 more than 2 times another, their sum being 27, results in the larger number being expressed as 2x + 3.
- When the total of two consecutive odd integers equals 92, it shows that if x is the smaller integer, the larger one is x + 2.
General Insight
- Formulating equations from verbal problem statements is essential for successfully solving mathematical problems in various contexts.
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Test your understanding of practical problem-solving steps in Algebra II with these flashcards. Focus on identifying variables and checking solutions to prepare for number problems effectively.
