Practice Problems1. Find The Product Of Each Pair Of Positive Numbers Whose Sum Is 50.2. Identify The Pair Of Numbers That Have A Sum Of 50 And Produce The Largest Possible Product.3. Determine The Widths Of A Rectangle Whose Perimeter Is 20 Meters.
Problem 1: Finding the Product of Pairs of Positive Numbers
Introduction
In this problem, we are tasked with finding the product of each pair of positive numbers whose sum is 50. This involves understanding the relationship between the sum and product of two numbers, as well as how to approach a problem that requires multiple calculations.
Solution
To solve this problem, we can start by listing all possible pairs of positive numbers that add up to 50. We can then calculate the product of each pair and identify the products.
Let's consider the pairs of positive numbers that add up to 50:
- (1, 49)
- (2, 48)
- (3, 47)
- (4, 46)
- (5, 45)
- (6, 44)
- (7, 43)
- (8, 42)
- (9, 41)
- (10, 40)
- (11, 39)
- (12, 38)
- (13, 37)
- (14, 36)
- (15, 35)
- (16, 34)
- (17, 33)
- (18, 32)
- (19, 31)
- (20, 30)
- (21, 29)
- (22, 28)
- (23, 27)
- (24, 26)
Now, let's calculate the product of each pair:
- (1, 49) = 49
- (2, 48) = 96
- (3, 47) = 141
- (4, 46) = 184
- (5, 45) = 225
- (6, 44) = 264
- (7, 43) = 301
- (8, 42) = 336
- (9, 41) = 369
- (10, 40) = 400
- (11, 39) = 429
- (12, 38) = 456
- (13, 37) = 481
- (14, 36) = 504
- (15, 35) = 525
- (16, 34) = 544
- (17, 33) = 561
- (18, 32) = 576
- (19, 31) = 589
- (20, 30) = 600
- (21, 29) = 609
- (22, 28) = 616
- (23, 27) = 621
- (24, 26) = 624
Conclusion
In this problem, we found the product of each pair of positive numbers whose sum is 50. The products range from 49 to 624.
Problem 2: Identifying the Pair of Numbers with the Largest Product
Introduction
In this problem, we are tasked with identifying the pair of numbers that have a sum of 50 and produce the largest possible product. This involves understanding how to maximize the product of two numbers given a constraint on their sum.
Solution
To solve this problem, we can use the concept of maximizing the product of two numbers given a constraint on their sum. We can start by listing all possible pairs of positive numbers that add up to 50 and then calculate their products.
Let's consider the pairs of positive numbers that add up to 50:
- (1, 49)
- (2, 48)
- (3, 47)
- (4, 46)
- (5, 45)
- (6, 44)
- (7, 43)
- (8, 42)
- (9, 41)
- (10, 40)
- (11, 39)
- (12, 38)
- (13, 37)
- (14, 36)
- (15, 35)
- (16, 34)
- (17, 33)
- (18, 32)
- (19, 31)
- (20, 30)
- (21, 29)
- (22, 28)
- (23, 27)
- (24, 26)
Now, let's calculate the product of each pair:
- (1, 49) = 49
- (2, 48) = 96
- (3, 47) = 141
- (4, 46) = 184
- (5, 45) = 225
- (6, 44) = 264
- (7, 43) = 301
- (8, 42) = 336
- (9, 41) = 369
- (10, 40) = 400
- (11, 39) = 429
- (12, 38) = 456
- (13, 37) = 481
- (14, 36) = 504
- (15, 35) = 525
- (16, 34) = 544
- (17, 33) = 561
- (18, 32) = 576
- (19, 31) = 589
- (20, 30) = 600
- (21, 29) = 609
- (22, 28) = 616
- (23, 27) = 621
- (24, 26) = 624
Conclusion
In this problem, we identified the pair of numbers that have a sum of 50 and produce the largest possible product. The pair (24, 26) produces the largest product of 624.
Problem 3: Determining the Widths of a Rectangle
Introduction
In this problem, we are tasked with determining the widths of a rectangle whose perimeter is 20 meters. This involves understanding the relationship between the perimeter and the dimensions of a rectangle.
Solution
To solve this problem, we can use the formula for the perimeter of a rectangle:
Perimeter = 2(length + width)
We are given that the perimeter is 20 meters, so we can set up the equation:
20 = 2(length + width)
Now, we can simplify the equation:
10 = length + width
We can solve for the width by subtracting the length from both sides:
width = 10 - length
Conclusion
In this problem, we determined the widths of a rectangle whose perimeter is 20 meters. The width is equal to 10 - length.
Conclusion
Q: What is the product of each pair of positive numbers whose sum is 50?
A: The products of each pair of positive numbers whose sum is 50 are:
- (1, 49) = 49
- (2, 48) = 96
- (3, 47) = 141
- (4, 46) = 184
- (5, 45) = 225
- (6, 44) = 264
- (7, 43) = 301
- (8, 42) = 336
- (9, 41) = 369
- (10, 40) = 400
- (11, 39) = 429
- (12, 38) = 456
- (13, 37) = 481
- (14, 36) = 504
- (15, 35) = 525
- (16, 34) = 544
- (17, 33) = 561
- (18, 32) = 576
- (19, 31) = 589
- (20, 30) = 600
- (21, 29) = 609
- (22, 28) = 616
- (23, 27) = 621
- (24, 26) = 624
Q: Which pair of numbers has a sum of 50 and produces the largest possible product?
A: The pair of numbers that has a sum of 50 and produces the largest possible product is (24, 26), which produces a product of 624.
Q: How do you determine the widths of a rectangle whose perimeter is 20 meters?
A: To determine the widths of a rectangle whose perimeter is 20 meters, you can use the formula for the perimeter of a rectangle:
Perimeter = 2(length + width)
You can set up the equation:
20 = 2(length + width)
Simplifying the equation, you get:
10 = length + width
You can solve for the width by subtracting the length from both sides:
width = 10 - length
Q: What is the relationship between the perimeter and the dimensions of a rectangle?
A: The perimeter of a rectangle is equal to the sum of the lengths of all its sides. The formula for the perimeter of a rectangle is:
Perimeter = 2(length + width)
This means that the perimeter is directly proportional to the sum of the length and width of the rectangle.
Q: How do you maximize the product of two numbers given a constraint on their sum?
A: To maximize the product of two numbers given a constraint on their sum, you can use the concept of maximizing the product of two numbers given a constraint on their sum. This involves finding the pair of numbers that has the largest product while still satisfying the constraint on their sum.
Q: What are some common techniques for solving problems in mathematics?
A: Some common techniques for solving problems in mathematics include:
- Using formulas and equations to represent relationships between variables
- Solving systems of equations
- Using algebraic manipulations to simplify expressions
- Using geometric shapes and visualizations to represent problems
- Using logical reasoning and problem-solving strategies to approach problems
Q: How do you develop problem-solving skills in mathematics?
A: Developing problem-solving skills in mathematics involves:
- Practicing solving problems and exercises
- Developing a deep understanding of mathematical concepts and techniques
- Learning to approach problems in a logical and systematic way
- Developing the ability to analyze and interpret mathematical information
- Learning to communicate mathematical ideas and solutions effectively
Conclusion
In this article, we answered some common questions about practice problems in mathematics. We discussed the product of each pair of positive numbers whose sum is 50, the pair of numbers that has a sum of 50 and produces the largest possible product, and the widths of a rectangle whose perimeter is 20 meters. We also discussed some common techniques for solving problems in mathematics and how to develop problem-solving skills in mathematics.