Six times the present age of P exceeds the present age of Q by 46 years. After 10 years, six times Q's age will be 18 years less than five times P's age at that time. What is the present age of Q (in years)?
Age Problem Calculation
This solution demonstrates how to determine the present age of Q using the information provided in the word problem involving P and Q. We will employ algebraic methods to solve the system of equations.
P and Q Present Ages Defined
To begin, let's assign variables to represent the unknown ages:
- Let \(P\) denote the present age of person P in years.
- Let \(Q\) denote the present age of person Q in years.
Formulating Age Equations from Statements
The problem provides two key pieces of information that can be converted into mathematical equations.
First Statement Equation
Statement: "Six times the present age of P exceeds the present age of Q by 46 years."
Translating this into an equation:
$6P = Q + 46 \quad (1)$
Second Statement Equation
Statement: "After 10 years, six times Q's age will be 18 years less than five times P's age at that time."
Ages after 10 years:
- P's future age: \(P + 10\)
- Q's future age: \(Q + 10\)
Translating the statement into an equation:
$6(Q + 10) = 5(P + 10) - 18$
Simplifying this equation:
$6Q + 60 = 5P + 50 - 18$
$6Q + 60 = 5P + 32$
Rearranging terms to isolate \(6Q\):
$6Q = 5P + 32 - 60$
$6Q = 5P - 28 \quad (2)$
Solving the Age Equations System
We now have a system of two linear equations:
- \(6P = Q + 46\)
- \(6Q = 5P - 28\)
We need to solve this system to find the value of \(Q\).
Step 1: Express Q in terms of P
From Equation (1), rearrange to solve for \(Q\):
\(Q = 6P - 46\)
Step 2: Substitute into Equation 2
Substitute the expression for \(Q\) into Equation (2):
$6(6P - 46) = 5P - 28$
Step 3: Solve for P's Age
Expand and solve the equation for \(P\):
$36P - 276 = 5P - 28$
Combine \(P\) terms:
$36P - 5P = 276 - 28$
$31P = 248$
Solve for \(P\):
$P = \frac{248}{31}$
$P = 8$
The present age of P is 8 years.
Step 4: Calculate Q's Age
Using the relationship from Step 1 (\(Q = 6P - 46\)) and the value \(P = 8\):
\(Q = 6(8) - 46\)
\(Q = 48 - 46\)
\(Q = 2\)
Final Answer: Q's Age
Therefore, the present age of Q is 2 years.
Important Questions from Problem on Age
The ratio of present ages of A and B is 7 : 8. After 6 years from now, the ratio of their ages will be 8 : 9. If C's present age is 10 years more than the present age of A, then the present age (in years) of C is:
Three years ago, the ratio of the age of father to that of his son was 8 ∶ 3. After 4 years, their ages will be in the ratio 11 ∶ 5. What is the present age (in years) of the father?
At present, A is younger than B by 8 years. If 4 years ago, their ages were in the ratio 1 ∶ 2, then what is the present age of B (in years)?
Eight years ago, the ratio of ages of A and B was 5 ∶ 4. The ratio of their present ages is 6 ∶ 5. What will be the sum (in years) of the ages of A and B after 7 years from now?
Ratio of the present age of a mother to that of the daughter is 7 ∶ 1. After 5 years the ratio will become 4 ∶ 1. What is the difference (in years) in their present ages?
