Let's break down this age problem step by step. We are given information about the ratio of ages of two people, A and B, at two different points in time: one year ago and seven years from now. We need to find the present age of B.

Understanding the Age Ratios

Age ratio problems involve setting up equations based on the given ratios at different time periods. The key is to represent the ages at each point in time relative to their present ages.

• Let the present age of A be $A_{present}$ years.
• Let the present age of B be $B_{present}$ years.

Setting Up Equations from the Given Information

We have two pieces of information about age ratios:

1. One year ago: The ratio of ages of A and B was 4 : 3.
   • Age of A one year ago = $A_{present} - 1$
   • Age of B one year ago = $B_{present} - 1$
   • The ratio is $\frac{A_{present} - 1}{B_{present} - 1} = \frac{4}{3}$
   • Cross-multiplying gives: $3(A_{present} - 1) = 4(B_{present} - 1)$
   • $3A_{present} - 3 = 4B_{present} - 4$
   • Rearranging into a linear equation: $3A_{present} - 4B_{present} = -4 + 3$
   • Equation 1: $3A_{present} - 4B_{present} = -1$
2. After 7 years from now: The ratio of their ages will be 9 : 7.
   • Age of A after 7 years = $A_{present} + 7$
   • Age of B after 7 years = $B_{present} + 7$
   • The ratio is $\frac{A_{present} + 7}{B_{present} + 7} = \frac{9}{7}$
   • Cross-multiplying gives: $7(A_{present} + 7) = 9(B_{present} + 7)$
   • $7A_{present} + 49 = 9B_{present} + 63$
   • Rearranging into a linear equation: $7A_{present} - 9B_{present} = 63 - 49$
   • Equation 2: $7A_{present} - 9B_{present} = 14$

Solving the System of Linear Equations

We now have a system of two linear equations with two variables, $A_{present}$ and $B_{present}$:

1) $3A_{present} - 4B_{present} = -1$

2) $7A_{present} - 9B_{present} = 14$

We can solve this system using the elimination method. Let's eliminate $A_{present}$.

• Multiply Equation 1 by 7: $7 \times (3A_{present} - 4B_{present}) = 7 \times (-1) \implies 21A_{present} - 28B_{present} = -7$ (Equation 3)
• Multiply Equation 2 by 3: $3 \times (7A_{present} - 9B_{present}) = 3 \times 14 \implies 21A_{present} - 27B_{present} = 42$ (Equation 4)

Now, subtract Equation 3 from Equation 4:

$(21A_{present} - 27B_{present}) - (21A_{present} - 28B_{present}) = 42 - (-7)$

$21A_{present} - 27B_{present} - 21A_{present} + 28B_{present} = 42 + 7$

$-27B_{present} + 28B_{present} = 49$

$B_{present} = 49$

We have found the present age of B.

Verifying the Solution

Let's check if this present age of B ($B_{present}=49$) works with the given ratios. We can find the present age of A using Equation 1:

$3A_{present} - 4(49) = -1$

$3A_{present} - 196 = -1$

$3A_{present} = -1 + 196$

$3A_{present} = 195$

$A_{present} = \frac{195}{3}$

$A_{present} = 65$

So, the present ages are A=65 and B=49.

• One year ago: A's age = $65-1 = 64$, B's age = $49-1 = 48$. The ratio is $\frac{64}{48}$. Dividing both by 16, we get $\frac{4}{3}$. This matches the given ratio of 4:3.
• After 7 years: A's age = $65+7 = 72$, B's age = $49+7 = 56$. The ratio is $\frac{72}{56}$. Dividing both by 8, we get $\frac{9}{7}$. This matches the given ratio of 9:7.

The calculated ages satisfy both conditions. Therefore, the present age of B is 49 years.

Final Answer

The present age of B is 49 years.

Revision Table: Age Ratio Problem

Additional Information: Solving Age Problems

Age problems often involve relationships between ages at different points in time. Here are some common strategies:

• Always start by defining variables for the present ages. This is the most common reference point.
• Read the problem carefully to identify the time periods (e.g., 'x years ago', 'in y years').
• Express the ages of all individuals at each specified time period in terms of their present ages. If the present age is $P$, then the age 'x years ago' is $P-x$, and the age 'in y years' is $P+y$.
• Translate the given relationships (ratios, sums, differences, products) into algebraic equations.
• If there are multiple unknowns, you will need a system of equations. Solve the system to find the values of the variables.
• Finally, make sure to answer the specific question asked, which might be a present age, a future age, or a past age.
• Always check your answer by plugging the values back into the original problem statement to ensure all conditions are met.