Understanding the Sum of Three Numbers Problem

This problem asks us to find the value of the second number among three numbers. We are given the sum of the three numbers and how the first and third numbers relate to the second number.

Let's break down the information provided:

• The sum of the three numbers is 2.
• The first number is \(\frac{1}{2}\) times the second number.
• The third number is \(\frac{1}{4}\) times the second number.

Our goal is to find the specific value of the second number.

Setting Up Algebraic Equations

To solve this, we can use algebra. Let's represent the unknown numbers with variables:

• Let the first number be \(n_1\).
• Let the second number be \(n_2\).
• Let the third number be \(n_3\).

Now, we can write the given information as equations:

1. The sum of the three numbers is 2: $$n_1 + n_2 + n_3 = 2$$
2. The first number is \(\frac{1}{2}\) times the second number: $$n_1 = \frac{1}{2} n_2$$
3. The third number is \(\frac{1}{4}\) times the second number: $$n_3 = \frac{1}{4} n_2$$

Step-by-Step Solution for Finding the Second Number

We have three equations and three unknowns. Notice that equations (2) and (3) express \(n_1\) and \(n_3\) in terms of \(n_2\). We can substitute these expressions into equation (1) to get an equation with only one variable, \(n_2\).

Substitute \(n_1 = \frac{1}{2} n_2\) and \(n_3 = \frac{1}{4} n_2\) into \(n_1 + n_2 + n_3 = 2\):

$$\left(\frac{1}{2} n_2\right) + n_2 + \left(\frac{1}{4} n_2\right) = 2$$

Now, we need to combine the terms involving \(n_2\). To do this, we can find a common denominator for the fractions, which is 4.

Rewrite the equation with a common denominator:

$$\frac{2}{4} n_2 + \frac{4}{4} n_2 + \frac{1}{4} n_2 = 2$$

Now, add the coefficients of \(n_2\):

$$\left(\frac{2}{4} + \frac{4}{4} + \frac{1}{4}\right) n_2 = 2$$

$$\left(\frac{2 + 4 + 1}{4}\right) n_2 = 2$$

$$\frac{7}{4} n_2 = 2$$

To find \(n_2\), we need to isolate it. We can multiply both sides of the equation by the reciprocal of \(\frac{7}{4}\), which is \(\frac{4}{7}\).

$$n_2 = 2 \times \frac{4}{7}$$

$$n_2 = \frac{8}{7}$$

So, the second number is \(\frac{8}{7}\).

We can check our answer by finding the first and third numbers:

• \(n_1 = \frac{1}{2} n_2 = \frac{1}{2} \times \frac{8}{7} = \frac{8}{14} = \frac{4}{7}\)
• \(n_3 = \frac{1}{4} n_2 = \frac{1}{4} \times \frac{8}{7} = \frac{8}{28} = \frac{2}{7}\)

Now, let's check if their sum is 2:

$$n_1 + n_2 + n_3 = \frac{4}{7} + \frac{8}{7} + \frac{2}{7} = \frac{4 + 8 + 2}{7} = \frac{14}{7} = 2$$

The sum is indeed 2, so our value for the second number, \(\frac{8}{7}\), is correct.

Revision Table: Key Concepts

Additional Information: Solving Word Problems

Solving word problems involving numbers often follows a systematic approach:

1. Read Carefully: Understand what the problem is asking and identify all the given information.
2. Assign Variables: Choose letters to represent the unknown quantities. Clearly define what each variable stands for.
3. Write Equations: Translate the relationships described in the problem into mathematical equations using the variables.
4. Solve the System: Use algebraic methods (like substitution or elimination for multiple equations) to find the values of the variables.
5. Check Your Answer: Plug the values you found back into the original equations or the problem statement to ensure they make sense and satisfy all conditions.

In this specific problem, since the first and third numbers were defined directly in terms of the second number, the substitution method was a very efficient way to solve it.