The sum of the ages of two cousins is 35. Ten years ago, the ratio of their ages was 2 : 1. What are their present ages?
20, 15
Solving Cousins Age Word Problems
This problem involves finding the present ages of two cousins based on two conditions: the sum of their present ages and the ratio of their ages ten years ago.
Let's denote the present age of the first cousin as \(C_1\) and the present age of the second cousin as \(C_2\).
Setting Up the Equations for Cousins Ages
Based on the problem statement, we can form two equations:
- Condition 1: The sum of their present ages is 35.
This gives us the equation: \(C_1 + C_2 = 35\) (Equation 1) - Condition 2: Ten years ago, the ratio of their ages was 2 : 1.
- Ten years ago, the age of the first cousin was \(C_1 - 10\).
- Ten years ago, the age of the second cousin was \(C_2 - 10\).
This gives us the equation: \(\frac{C_1 - 10}{C_2 - 10} = \frac{2}{1}\)
Cross-multiplying, we get: \(C_1 - 10 = 2(C_2 - 10)\)
Expanding this, we get: \(C_1 - 10 = 2C_2 - 20\)
Rearranging to isolate \(C_1\): \(C_1 = 2C_2 - 20 + 10\)
\(C_1 = 2C_2 - 10\) (Equation 2)
Solving the System of Equations
Now we have a system of two linear equations with two variables:
- \(C_1 + C_2 = 35\)
- \(C_1 = 2C_2 - 10\)
We can solve this system using the substitution method. Substitute the expression for \(C_1\) from Equation 2 into Equation 1:
\((2C_2 - 10) + C_2 = 35\)
Combine the terms with \(C_2\):
\(3C_2 - 10 = 35\)
Add 10 to both sides of the equation:
\(3C_2 = 35 + 10\)
\(3C_2 = 45\)
Divide by 3 to find \(C_2\):
\(C_2 = \frac{45}{3}\)
\(C_2 = 15\)
Now that we have the value for \(C_2\), substitute it back into either Equation 1 or Equation 2 to find \(C_1\). Using Equation 1:
\(C_1 + 15 = 35\)
Subtract 15 from both sides:
\(C_1 = 35 - 15\)
\(C_1 = 20\)
So, the present ages of the two cousins are 20 years and 15 years.
Verification of Cousins Ages
Let's check if these ages satisfy the original conditions:
- Sum of present ages: \(20 + 15 = 35\). This matches the first condition.
- Ages ten years ago: The first cousin was \(20 - 10 = 10\) years old. The second cousin was \(15 - 10 = 5\) years old.
- Ratio ten years ago: The ratio of their ages was \(\frac{10}{5} = \frac{2}{1}\). This matches the second condition.
Both conditions are satisfied, confirming that the present ages are indeed 20 and 15.
Summary of Cousins Present Ages
| Cousin | Present Age (Years) |
|---|---|
| First Cousin | 20 |
| Second Cousin | 15 |
Revision Table: Age Word Problems
| Concept | Explanation | Example Application |
|---|---|---|
| Representing Present Age | Use variables (e.g., x, y) for unknown present ages. | Let present age be \(x\). |
| Age in the Future | If age is \(x\) now, in \(n\) years, it will be \(x + n\). | Age after 5 years = \(x + 5\). |
| Age in the Past | If age is \(x\) now, \(n\) years ago, it was \(x - n\). | Age 10 years ago = \(x - 10\). |
| Sum of Ages | Add the ages of the individuals. | Sum of ages of two people \(x\) and \(y\) is \(x + y\). |
| Ratio of Ages | Express the relationship as a fraction. | Ratio of ages \(a\) and \(b\) is \(\frac{a}{b}\). |
Additional Information on Solving Age Problems
Age problems are a common type of word problem in algebra. They typically involve finding the current ages of people based on conditions given about their ages at different points in time (past, present, or future) or relationships between their ages.
Key steps to solve age problems:
- Identify the unknowns: Determine what ages you need to find, usually the present ages. Assign variables to these unknowns.
- Translate statements into equations: Carefully read each condition in the problem and write corresponding algebraic equations. Pay close attention to phrases like "in x years," "x years ago," "is x times," "ratio of their ages," etc.
- Solve the system of equations: Use methods like substitution or elimination to find the values of the variables.
- Check your answer: Substitute the calculated ages back into the original conditions given in the word problem to ensure they are satisfied.
Practice with various age problems helps build proficiency in setting up and solving the related equations.
Important Questions from Quant Based Puzzle
A number is subtracted from 4 times of it and then, the number obtained is added to its (the resultant’s) next number. If this gives the answer as 91, what was the original number?
When twice of a number added to 3 is multiplied by 5 and added to the number itself, it gives 158. What is the square of that number?
In a class of 72 students, the number of boys is twice the number of girls. Find the number of boys.
Two years ago, T was twice as old as P. P is thrice as old as R. In five years, P will be 29. What is the present age of T?
When a number is added to its multiple of 5 and its square, the sum of these three numbers is 91. Find the number.
