Understanding the Context

What Is the Equation 10t + 25b = 155?

The equation 10t + 25b = 155 is a linear Diophantine equation involving two variables:

• t and b represent unknown quantities.
  
• The coefficients 10 and 25 multiply these variables.
  
• The constant 155 is the total value on the right-hand side.

This type of equation appears in various fields such as economics, budgeting, scalability modeling, and even in physics when balancing weighted averages or ratios.

Key Insights

Why Does This Equation Matter?

Integer solutions to linear equations like 10t + 25b = 155 help in:

• Optimizing resource allocation where two variables cost different amounts.
  
• Modeling real-life scenarios like splitting payments between two parties (e.g., splitting a bill or funding).
  
• Teaching foundational algebra skills and Diophantine problem-solving.

Step-by-Step Solution to Solve 10t + 25b = 155

Step 1: Simplify the Equation

Check if the equation is reducible by finding the greatest common divisor (GCD) of 10 and 25.

• GCD(10, 25) = 5, which divides 155 exactly:
  155 ÷ 5 = 31.
  This confirms integer solutions exist.

Final Thoughts

Divide the entire equation by 5:
  10t + 25b = 155
 → 2t + 5b = 31

Step 2: Solve for One Variable

Solve for t in terms of b:
 2t = 31 - 5b
 t = (31 - 5b) / 2

Step 3: Find Integer Solutions

Since t and b must be integers (e.g., counting items), 31 - 5b must be even.
Try integer values of b such that (31 - 5b) is divisible by 2:

• b = 1 → t = (31 - 5×1) / 2 = 26/2 = 13 ✔
  
• b = 3 → t = (31 - 15)/2 = 16/2 = 8 ✔
  
• b = 5 → t = (31 - 25)/2 = 6/2 = 3 ✔
  
• b = 7 → t = (31 - 35)/2 = negative → invalid

Valid integer solutions include:

• (t, b) = (13, 1)
  
• (8, 3)
  
• (3, 5)