Solving the Length of Service Problem

This problem asks us to find the current length of service for person P, given information about the combined length of service of P, Q, and R at different points in time.

Understanding the Given Information

We are given two main pieces of information:

1. After 6 years from now, the sum of the years of service for P, Q, and R will be 56 years.
2. Before 3 years from now, the sum of the years of service for Q and R was 17 years.

Setting up Variables

Let's define variables for the current length of service for each person:

• Current length of service for P = $\text{P}_c$ years
• Current length of service for Q = $\text{Q}_c$ years
• Current length of service for R = $\text{R}_c$ years

Translating Information into Equations

Now, let's write these statements as mathematical equations:

Statement 1: After 6 years

After 6 years, each person's length of service will increase by 6 years.

• P's service after 6 years = $\text{P}_c + 6$
• Q's service after 6 years = $\text{Q}_c + 6$
• R's service after 6 years = $\text{R}_c + 6$

The sum of their services after 6 years is 56.

Equation 1: $(\text{P}_c + 6) + (\text{Q}_c + 6) + (\text{R}_c + 6) = 56$

Simplifying Equation 1:

$\text{P}_c + \text{Q}_c + \text{R}_c + 18 = 56$

$\text{P}_c + \text{Q}_c + \text{R}_c = 56 - 18$

$\text{P}_c + \text{Q}_c + \text{R}_c = 38$

Statement 2: Before 3 years

Before 3 years, each person's length of service was 3 years less than their current service.

• Q's service before 3 years = $\text{Q}_c - 3$
• R's service before 3 years = $\text{R}_c - 3$

The sum of Q's and R's services before 3 years was 17.

Equation 2: $(\text{Q}_c - 3) + (\text{R}_c - 3) = 17$

Simplifying Equation 2:

$\text{Q}_c + \text{R}_c - 6 = 17$

$\text{Q}_c + \text{R}_c = 17 + 6$

$\text{Q}_c + \text{R}_c = 23$

Solving the System of Equations

We now have a system of two equations:

1. $\text{P}_c + \text{Q}_c + \text{R}_c = 38$
2. $\text{Q}_c + \text{R}_c = 23$

Notice that Equation 2 gives us the combined current length of service for Q and R ($\text{Q}_c + \text{R}_c$). We can substitute this value into Equation 1.

Substitute $(\text{Q}_c + \text{R}_c) = 23$ into Equation 1:

$\text{P}_c + (\text{Q}_c + \text{R}_c) = 38$

$\text{P}_c + 23 = 38$

Now, solve for $\text{P}_c$:

$\text{P}_c = 38 - 23$

$\text{P}_c = 15$

Current Length of Service for P

The current length of service for P is 15 years.

The table helps visualize how the service years change over time and how the sums relate to the current service lengths.

Conclusion

By setting up equations based on the information given about the sum of lengths of service at different points in time, we found that the current length of service for P is 15 years.

Revision Table - Length of Service Math Problem

Additional Information - Solving Word Problems

Solving word problems often involves a systematic approach:

• Read Carefully: Understand exactly what information is given and what is being asked. Pay attention to keywords indicating time (e.g., 'after', 'before', 'current') and operations (e.g., 'sum', 'difference').
• Assign Variables: Use letters to represent the unknown quantities.
• Formulate Equations: Translate the relationships described in the problem into mathematical equations using the variables.
• Solve Equations: Use appropriate algebraic techniques to solve for the unknown variables.
• Check Your Answer: Plug your solution back into the original word problem statements to see if they hold true. This helps catch errors.
• State the Answer Clearly: Make sure your final answer directly addresses the question asked in the problem.