Solving Age Ratio Problems: A Comprehensive Guide
Age ratio problems are a common challenge in mathematics and algebra. In this article, we will explore a particular age ratio problem that involves finding the present age of two individuals, Samir and Ashok, given their ages one year ago and one year hence.
Problem Statement
One year ago, the ratio between Samir's age and Ashok's age was 4:3. One year hence, the ratio of their ages will be 7:6. The question is to determine the sum of their present ages.
Step-by-Step Solution
Let us denote the present ages of Samir and Ashok as (S) and (A) respectively.
Step 1: Formulate Equations
From the given information, we can set up the following equations:
(frac{S-1}{A-1} frac{4}{3})
(frac{S 1}{A 1} frac{7}{6})
Step 2: Solve for Variables
We can solve these equations step-by-step:
Equation 1:
[ 3S - 3 4A - 4 Rightarrow 4A - 3S 1 ]Equation 2:
[ 6S 6 7A 7 Rightarrow 6S - 7A 1 ]Step 3: Combine the Equations
Multiply Equation 1 by 6 and Equation 2 by 3:
Equation 1 (multiplied by 6):
[ 24S - 18A 6 ]Equation 2 (multiplied by 3):
[ 18S - 21A 3 ]Subtract the second equation from the first:
[ (24S - 18A) - (18S - 21A) 6 - 3 ][ 6S 3A 3 ]
[ 2S A 1 ]
From this, we can solve for (A):
[ A 1 - 2S ]
Now, substitute (A) back into Equation 1:
(4(1 - 2S) - 3S 1 )
( 4 - 8S - 3S 1 )
( 4 - 11S 1 )
( -11S -3 )
( S frac{3}{11} )
Now, solve for (A):
( A 1 - 2 left( frac{3}{11} right) )
( A 1 - frac{6}{11} )
( A frac{11}{11} - frac{6}{11} )
( A frac{5}{11} )
These values seem incorrect. Let's re-evaluate the arithmetic:
From the correct steps:
From Equation 1: (4A - 3S 1 )
From Equation 2: (6S - 7A 1 )
Multiplying Equation 1 by 6 and Equation 2 by 4:
Equation 1 (multiplied by 6):
[ 24A - 18S 6 ]Equation 2 (multiplied by 4):
[ 24S - 28A 4 ]Combining these:
[ 24A - 18S 24S - 28A 6 - 4 ][ 6S - 4A 2 ]
(6S 4A 2 )
(3S 2A 1 )
From Equation 1: (4A - 3S 1 )
(4A 3S 1 )
(A frac{3S 1}{4} )
Substitute back:
(frac{3S 1}{4} ) and solve:
(A 3 ) and (S frac{11}{3} )
(S A 3 frac{11}{3} frac{9 11}{3} frac{20}{3} 6frac{2}{3})
Conclusion
The sum of their present ages is (6frac{2}{3}) years.
Key Takeaways
Problems involving age ratios can be solved by setting up and solving equations. Present age is a crucial variable in solving age-related problems. Algebraic manipulation is essential for deriving accurate solutions.By mastering these steps, you can confidently solve a variety of age ratio problems and other similar algebraic equations.