Two times the present age of P exceeds the present age of Q by 41 years. After 7 years, nine times Q's age will be 24 years less than six times P's age at that time. What is the present age (in years) of Q?
Age Calculation for P and Q: Step-by-Step Solution
This problem involves finding the present age of Q based on two conditions relating the ages of P and Q, both currently and in the future. We can solve this using algebra by setting up and solving a system of linear equations.
Setting Up the Algebraic Equations
First, let's define variables for the present ages:
- Let $P$ represent the present age of P (in years).
- Let $Q$ represent the present age of Q (in years).
Now, let's translate the given conditions into equations:
- "Two times the present age of P exceeds the present age of Q by 41 years." This translates to: $2P = Q + 41 \quad (Equation \, 1)$
- "After 7 years, nine times Q's age will be 24 years less than six times P's age at that time." P's age after 7 years will be $P + 7$. Q's age after 7 years will be $Q + 7$. The condition translates to: $9(Q + 7) = 6(P + 7) - 24 \quad (Equation \, 2)$
Solving the System of Equations
We need to solve these two equations to find the value of $Q$.
Step 1: Simplify Equation 2
$9(Q + 7) = 6(P + 7) - 24$ $9Q + 63 = 6P + 42 - 24$ $9Q + 63 = 6P + 18$Step 2: Express P in terms of Q using Equation 1
From Equation 1 ($2P = Q + 41$), we can isolate $P$: $P = \frac{Q + 41}{2}$Step 3: Substitute the expression for P into the simplified Equation 2
Substitute $P = \frac{Q + 41}{2}$ into $9Q + 63 = 6P + 18$: $9Q + 63 = 6 \left( \frac{Q + 41}{2} \right) + 18$ $9Q + 63 = 3(Q + 41) + 18$Step 4: Solve for Q
Distribute the 3 and simplify: $9Q + 63 = 3Q + 123 + 18$ $9Q + 63 = 3Q + 141$ Now, gather the $Q$ terms on one side and the constants on the other: $9Q - 3Q = 141 - 63$ $6Q = 78$ Divide by 6 to find $Q$: $Q = \frac{78}{6}$ $Q = 13$Verification
Let's check if $Q = 13$ works with the original conditions.
- From Equation 1: $2P = 13 + 41 \implies 2P = 54 \implies P = 27$. So, P is 27 years old and Q is 13 years old.
- After 7 years: P will be $27 + 7 = 34$ and Q will be $13 + 7 = 20$.
- Check condition 2: Nine times Q's future age is $9 \times 20 = 180$. Six times P's future age is $6 \times 34 = 204$. Is $180$ equal to $204 - 24$? Yes, $180 = 180$.
Both conditions are satisfied.
Conclusion
The present age of Q is 13 years.
Important Questions from Age
- Five times the present age of P exceeds the present age of Q by 29 years. After 8 years, five times Q's age will be 17 years less than seven times P's age at that time. What is the present age (in years) of Q?
- 7 years ago, the age of a father was 13 years more than twice his son's age. After how many years, from now, will he be twice his son's age?
If 2 times the mother's age is 26 years more than 4 times her daughter's age, and 3 times the daughter's age is 7 years less than the mother's age, then what is the difference (in years) between the ages of the mother and the daughter?
If 2 times a son's age is added to his father's age, the sum is 36 years. If 2.5 times the father's age is added to the son's age, the sum is 72 years. What is the father's age (in years)?
- Six times the present age of P exceeds the present age of Q by 11 years. After 10 years, three times Q's age will be 27 years less than six times P's age at that time. What is the present age (in years) of Q?
