Solving the Problem of Positive Integers with Given Product and Ratio
In mathematics, the solution of problems involving the product and ratio of positive integers is a fundamental concept. This article addresses a specific problem where the product of two positive integers is 1575 and their ratio is 9:7. We will explore different methods to find the value of the smaller integer and demonstrate the steps required to solve the problem accurately.
Understanding the Problem
Given the product of two positive integers is 1575 and their ratio is 9:7, our goal is to find the value of the smaller integer. This problem can be broken down using the following steps:
Solution Method 1
**Step 1:** Let the first number be (9x) and the second number be (7x).
**Step 2:** The product of these two numbers is given by:
[9x times 7x 1575]
**Step 3:** Simplifying the equation, we get:
[63x^2 1575]
**Step 4:** Solving for (x^2):
[x^2 frac{1575}{63} 25]
**Step 5:** Taking the square root of both sides:
[x 5]
**Step 6:** Therefore, the numbers can be determined as:
[7x 7 times 5 35]
[9x 9 times 5 45]
So, the smaller integer is 35.
Solution Method 2
**Step 1:** Let the two numbers be (9x) and (7x).
**Step 2:** The product of these numbers is given by:
[9x times 7x 1575]
**Step 3:** Simplifying the equation, we get:
[63x^2 1575]
**Step 4:** Solving for (x^2):
[x^2 frac{1575}{63} 25]
**Step 5:** Taking the square root of both sides:
[x 5] or [x -5]
**Step 6:** Since the integers are positive, we consider (x 5).
**Step 7:** Therefore, the numbers are:
[7x 7 times 5 35]
[9x 9 times 5 45]
So, the smaller integer is 35.
Solution Method 3
**Step 1:** Let the integers (x) and (y) be represented as (x 9a) and (y 7a) where (a) is the greatest common divisor of (x) and (y).
**Step 2:** Given the ratio (x:y 9:7).
**Step 3:** The product of (x) and (y) is given by:
[x times y 1575]
**Step 4:** Substituting (x 9a) and (y 7a) in the equation:
[9a times 7a 1575]
**Step 5:** Simplifying the equation, we get:
[63a^2 1575]
**Step 6:** Solving for (a^2):
[a^2 frac{1575}{63} 25]
**Step 7:** Taking the square root of both sides:
[a 5]
**Step 8:** Therefore, the numbers are:
[x 9a 9 times 5 45]
[y 7a 7 times 5 35]
So, the smaller integer is 35.
Solution Method 4
**Step 1:** Let the numbers be (a) and (b).
**Step 2:** From the given ratio, let (a 9x) and (b 7x).
**Step 3:** The product (a times b 1575), thus:
[9x times 7x 1575]
**Step 4:** Simplifying the equation, we get:
[63x^2 1575]
**Step 5:** Solving for (x^2):
[x^2 frac{1575}{63} 25]
**Step 6:** Taking the square root of both sides:
[x 5] or [x -5]
**Step 7:** Since the integers are positive, we consider (x 5).
**Step 8:** Therefore, the numbers are:
[7x 7 times 5 35]
[9x 9 times 5 45]
So, the smaller integer is 35.
Solution Method 5
**Step 1:** Let the two numbers be (9k) and (7k).
**Step 2:** The product of these two numbers is given by:
[9k times 7k 1575]
**Step 3:** Simplifying the equation, we get:
[63k^2 1575]
**Step 4:** Solving for (k^2):
[k^2 frac{1575}{63} 25]
**Step 5:** Taking the square root of both sides:
[k 5]
**Step 6:** Therefore, the numbers are:
[7k 7 times 5 35]
[9k 9 times 5 45]
So, the smaller integer is 35.
Conclusion
Through various methods, we have determined that the smaller integer is 35 when the product of two positive integers is 1575 and their ratio is 9:7. This problem reinforces the importance of understanding the relationships between products, ratios, and basic algebraic operations.