Integers Ratio Problem Setup

Let the two integers be $x$ and $y$. We are given that the difference, the sum, and the product of these two integers are in the proportion $1 : 3 : 10$. Without loss of generality, let's assume $x \ge y$ so the difference is non-negative.

According to the problem statement, we have the following relationships:

• Difference: $x - y$
• Sum: $x + y$
• Product: $xy$

The ratio is given as $(x-y) : (x+y) : (xy) = 1 : 3 : 10$. This implies that there exists a constant $k$ such that:

$\frac{x-y}{1} = \frac{x+y}{3} = \frac{xy}{10} = k$

Formulating Equations from Ratio

From the ratio relationship, we can write a system of three equations:

1. $x - y = k \quad \cdots (1)$
2. $x + y = 3k \quad \cdots (2)$
3. $xy = 10k \quad \cdots (3)$

We need to find the values of $x$ and $y$ that satisfy these equations.

Solving for the Integers

Let's solve the first two equations for $x$ and $y$ in terms of $k$.

Add equation (1) and equation (2):

$(x - y) + (x + y) = k + 3k$

$2x = 4k$

$x = 2k$

Subtract equation (1) from equation (2):

$(x + y) - (x - y) = 3k - k$

$2y = 2k$

$y = k$

Now we have $x = 2k$ and $y = k$. Since we assumed $x \ge y$, this is consistent as long as $k \ge 0$.

Determining the Constant k

Substitute the expressions for $x$ and $y$ into equation (3):

$xy = 10k$

$(2k)(k) = 10k$

$2k^2 = 10k$

Rearrange the equation to solve for $k$:

$2k^2 - 10k = 0$

$2k(k - 5) = 0$

This equation gives two possible values for $k$:

• $2k = 0 \implies k = 0$
• $k - 5 = 0 \implies k = 5$

Finding the Specific Integers

Let's consider the two cases for $k$:

Case 1: $k = 0$

If $k = 0$, then $x = 2(0) = 0$ and $y = 0$.

Difference: $0 - 0 = 0$

Sum: $0 + 0 = 0$

Product: $0 \times 0 = 0$

The ratio is $0 : 0 : 0$. This is not in the proportion $1 : 3 : 10$, so $k=0$ is not a valid solution for this problem.

Case 2: $k = 5$

If $k = 5$, then $x = 2(5) = 10$ and $y = 5$.

The two integers are 10 and 5.

Verifying the Solution

Let's check if the integers 10 and 5 satisfy the given ratio:

• Difference ($x \ge y$): $10 - 5 = 5$
• Sum: $10 + 5 = 15$
• Product: $10 \times 5 = 50$

The ratio of difference : sum : product is $5 : 15 : 50$.

To simplify this ratio, we can divide all terms by the greatest common divisor, which is 5:

$\frac{5}{5} : \frac{15}{5} : \frac{50}{5} = 1 : 3 : 10$

This matches the ratio given in the problem. Therefore, the two integers are 10 and 5.

Checking the Options

The problem asks for the two integers. We found the integers are 5 and 10. Let's look at the given options:

The pair of integers (5, 10) is the one that satisfies the given condition that their difference, sum, and product are in the proportion 1 : 3 : 10.