Answer :
* The problem translates to the equation $8(4 + n) = 112$.
* Dividing both sides by 8 gives $4 + n = 14$.
* Subtracting 4 from both sides yields $n = 10$.
* The number is $\boxed{10}$.
### Explanation
1. Understanding the Problem
Let's break down the problem. We are told that 'the product of 8 and the sum of 4 and a number is 112'. Let's use $n$ to represent the unknown number. The sum of 4 and the number is $4 + n$. The product of 8 and this sum is $8(4 + n)$. We know this product equals 112. So, the equation we need to solve is $8(4 + n) = 112$.
2. Identifying the Correct Equation
Now, let's identify the correct equation from the given options. The options are:
1. $8 \times 4 + n = 112$
2. $8 + 4n = 112$
3. $8(4 + n) = 112$
Our equation $8(4 + n) = 112$ matches option 3.
3. Dividing Both Sides by 8
Next, we solve the equation $8(4 + n) = 112$ for $n$. To do this, we first divide both sides of the equation by 8:
$$\frac{8(4 + n)}{8} = \frac{112}{8}$$
$$4 + n = 14$$
4. Subtracting 4 from Both Sides
Now, subtract 4 from both sides of the equation:
$$4 + n - 4 = 14 - 4$$
$$n = 10$$
5. Finding the Number
Therefore, the number is 10.
### Examples
Imagine you're planning a rectangular garden. You know the total area you want the garden to be (112 square feet), and you know one side will have a width that is 8 times the length of the other side plus 4 feet. This problem helps you determine the exact length of the other side needed to achieve your desired garden area. Understanding how to set up and solve such equations is crucial in various real-world applications, from construction to landscape design.
* Dividing both sides by 8 gives $4 + n = 14$.
* Subtracting 4 from both sides yields $n = 10$.
* The number is $\boxed{10}$.
### Explanation
1. Understanding the Problem
Let's break down the problem. We are told that 'the product of 8 and the sum of 4 and a number is 112'. Let's use $n$ to represent the unknown number. The sum of 4 and the number is $4 + n$. The product of 8 and this sum is $8(4 + n)$. We know this product equals 112. So, the equation we need to solve is $8(4 + n) = 112$.
2. Identifying the Correct Equation
Now, let's identify the correct equation from the given options. The options are:
1. $8 \times 4 + n = 112$
2. $8 + 4n = 112$
3. $8(4 + n) = 112$
Our equation $8(4 + n) = 112$ matches option 3.
3. Dividing Both Sides by 8
Next, we solve the equation $8(4 + n) = 112$ for $n$. To do this, we first divide both sides of the equation by 8:
$$\frac{8(4 + n)}{8} = \frac{112}{8}$$
$$4 + n = 14$$
4. Subtracting 4 from Both Sides
Now, subtract 4 from both sides of the equation:
$$4 + n - 4 = 14 - 4$$
$$n = 10$$
5. Finding the Number
Therefore, the number is 10.
### Examples
Imagine you're planning a rectangular garden. You know the total area you want the garden to be (112 square feet), and you know one side will have a width that is 8 times the length of the other side plus 4 feet. This problem helps you determine the exact length of the other side needed to achieve your desired garden area. Understanding how to set up and solve such equations is crucial in various real-world applications, from construction to landscape design.
