Finding the Integers

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 $1 : 3 : 10$. Let's assume, without loss of generality, that $x \ge y$. The difference is $x - y$, the sum is $x + y$, and the product is $x \times y$.

Setting Up Equations

We can represent the given proportion using a constant $k$. This means:

• The difference, $x - y$, is proportional to 1. So, $x - y = 1k = k$. (Equation 1)
• The sum, $x + y$, is proportional to 3. So, $x + y = 3k$. (Equation 2)
• The product, $x \times y$, is proportional to 10. So, $x \times y = 10k$. (Equation 3)

Solving for the Unknown Constant

We now have a system of three equations. We can solve the first two equations simultaneously to find $x$ and $y$ in terms of $k$.

Add Equation 1 and Equation 2:

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

$2x = 4k$

$x = \frac{4k}{2}$

$x = 2k$

Now substitute the value of $x$ into Equation 2:

$(2k) + y = 3k$

$y = 3k - 2k$

$y = k$

Now substitute the expressions for $x$ and $y$ in terms of $k$ into Equation 3:

$x \times y = 10k$

$(2k) \times (k) = 10k$

$2k^2 = 10k$

To solve for $k$, rearrange the equation:

$2k^2 - 10k = 0$

Factor out $2k$:

$2k(k - 5) = 0$

This gives two possible solutions for $k$:

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

If $k = 0$, then $x = 2(0) = 0$ and $y = 0$. The difference, sum, and product would all be 0, and the ratio $0:0:0$ is undefined, so $k=0$ is not a valid solution.

If $k = 5$, this is a valid solution.

Determining the Integers

Using $k = 5$, we can find the values of $x$ and $y$:

$x = 2k = 2(5) = 10$

$y = k = 5$

So, the two integers are 10 and 5.

Verifying the Ratio

Let's check if the difference, sum, and product of 10 and 5 are in the ratio $1 : 3 : 10$.

• Difference: $10 - 5 = 5$
• Sum: $10 + 5 = 15$
• Product: $10 \times 5 = 50$

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

Dividing all parts of the ratio by 5, we get:

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

This matches the given proportion.

Thus, the two integers are 5 and 10.