The product of two positive integers Is 1575 and their ratio is `9:7`. The smaller integer is

Options

A. a) 25
B. b) 35
C. c) 45
D. d) 70

Answer

To solve the problem step by step, we will follow the given information about the product of two integers and their ratio.

### Step 1: Understand the given information
We are given:
- The product of two positive integers is 1575.
- The ratio of the two integers is 9:7.

### Step 2: Express the integers in terms of a variable
Let the two integers be represented as:
- \( x = 9a \) (the larger integer)
- \( y = 7a \) (the smaller integer)

### Step 3: Set up the equation based on the product
According to the problem, the product of the two integers is:
\[ x \cdot y = 1575 \]
Substituting the expressions for \( x \) and \( y \):
\[ (9a) \cdot (7a) = 1575 \]
This simplifies to:
\[ 63a^2 = 1575 \]

### Step 4: Solve for \( a^2 \)
To find \( a^2 \), divide both sides of the equation by 63:
\[ a^2 = \frac{1575}{63} \]

### Step 5: Calculate \( \frac{1575}{63} \)
Now, we perform the division:
\[ a^2 = 25 \]

### Step 6: Find the value of \( a \)
Taking the square root of both sides gives:
\[ a = 5 \]

### Step 7: Calculate the integers
Now we can find the values of \( x \) and \( y \):
- \( x = 9a = 9 \cdot 5 = 45 \)
- \( y = 7a = 7 \cdot 5 = 35 \)

### Step 8: Identify the smaller integer
The smaller integer is:
\[ y = 35 \]

### Final Answer
The smaller integer is **35**.

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