Understanding the Age Word Problem

This problem involves finding the ages of two individuals, A and B, based on relationships given between their ages. We need to set up and solve a system of linear equations to find B's age.

Setting Up the Equations

Let's denote the age of A as $A$ and the age of B as $B$. We can translate the information given in the question into two mathematical equations:

• "If 5 times A's age is added to B's age, the sum is 95 years." This can be written as:
  $5A + B = 95$ (Equation 1)
• "If 2.2 times B's age is added to A's age, the sum is 63 years." This can be written as:
  $A + 2.2B = 63$ (Equation 2)

Solving for B's Age Step-by-Step

We have a system of two linear equations with two variables:

1. $5A + B = 95$
2. $A + 2.2B = 63$

We can use the substitution method to solve this system. First, let's isolate $B$ in Equation 1:

From Equation 1: $B = 95 - 5A$

Now, substitute this expression for $B$ into Equation 2:

$A + 2.2(95 - 5A) = 63$

Distribute the 2.2:

$A + (2.2 \times 95) - (2.2 \times 5A) = 63$

$A + 209 - 11A = 63$

Combine the terms with $A$:

$209 - 10A = 63$

Now, we need to solve for $A$. Subtract 209 from both sides:

$-10A = 63 - 209$

$-10A = -146$

Divide by -10:

$A = \frac{-146}{-10}$

$A = 14.6$

So, A's age is 14.6 years. Now we can find B's age using the expression we found earlier ($B = 95 - 5A$):

$B = 95 - 5(14.6)$

$B = 95 - 73$

$B = 22$

Verification

Let's check if these ages satisfy the original conditions:

• Condition 1: $5A + B = 5(14.6) + 22 = 73 + 22 = 95$. (Matches)
• Condition 2: $A + 2.2B = 14.6 + 2.2(22) = 14.6 + 48.4 = 63$. (Matches)

Both conditions are satisfied.

Conclusion

Based on the calculations, B's age is 22 years.