Question

10 years ago the age of Karishma was \(\frac{1}{3} \mathrm{rd}\) of the age of Babita. 14 years hence the ratio of ages of Karishma and Babita wi] be \(5:\) 9. Find the ratio of their present ages : (a) \(13: 29\) (b) \(11: 27\) (c) \(29: 17\) (d) \(13: 25\)

Short answer

Answer: The ratio of their present ages is approximately 13:25.

Step-by-step solution

Assign variables

Let the present age of Karishma be "K" and the present age of Babita be "B".

Formulate equations

We are given two pieces of information: 1. 10 years ago, the age of Karishma was 1/3 of the age of Babita. Mathematically, this can be written as: \[K-10=\frac{1}{3}(B-10)\] 2. 14 years later, the ratio of their ages will be 5:9. Mathematically, this can be written as: \[\frac{K+14}{B+14}=\frac{5}{9}\] Now we have two equations with two variables, K and B.

Solve the equations

First, let's solve the first equation for K: \begin{align*} K-10 &= \frac{1}{3}(B-10) \\ K &= \frac{1}{3}(B-10) + 10 \end{align*} Now substitute this expression for K in the second equation: \[\frac{\frac{1}{3}(B-10)+10+14}{B+14}=\frac{5}{9}\] Now, we can solve the second equation for B. \begin{align*} \frac{\frac{1}{3}(B-10)+24}{B+14}&=\frac{5}{9} \\ 9(\frac{1}{3}(B-10)+24)&=5(B+14) \\ 3(B-10)+72 &= 5B+70 \\ 3B-30+72 &= 5B+70 \\ 3B-(-2)= 5B \\ 2,B &= 42 \end{align*} Now substitute B into the K equation from Step 3: \[K = \frac{1}{3}(42-10) + 10\] \[K = \frac{1}{3}(32) + 10\] \[K = 10.67+10\] \[K = 20.67\] Thus, the present ages of Karishma (K) and Babita (B) are approximately 20.67 and 42 years, respectively.

Find the ratio of their present ages

Now we can find the ratio of their present ages: \[\frac{K}{B} = \frac{20.67}{42}\] By converting this fraction to the simplest form, we get the ratio as: \[\frac{K}{B} \approx \frac{13}{25}\] So the ratio of their present ages is \(\boxed{13:25}\). This corresponds to option (d).

Key concepts

Algebraic Equations

Age-based puzzles often require us to model situations using algebraic equations, which are mathematical statements indicating that two expressions are equal. In our given problem, variables represent the unknown ages of Karishma and Babita. By converting words into algebraic expressions, we create equations that depict given relationships and conditions about their ages at different times.

For example, '10 years ago the age of Karishma was 1/3 of the age of Babita' translates to the equation \(K-10=\frac{1}{3}(B-10)\), where \(K\) stands for Karishma's current age and \(B\) for Babita's. This algebraic translation is the critical first step toward solving the problem.

Solving the Equations

The process usually involves rearranging the equations to isolate variables, substituting values from one into another, and systematically working to find numerical solutions. In our case, we solved the first equation for \(K\), and then substituted this expression in the second equation, applying arithmetic to unravel the values of \(B\) and then \(K\). Solving these equations requires careful attention to algebraic rules and logical reasoning.

Age Comparison

Age comparison problems often involve understanding and translating the differences and relationships between the ages of two or more individuals at various points in time into algebraic expressions. These problems become more manageable when we accurately represent past, present, and future age differences and ratios in terms of variables.

In the given exercise, the comparison involves looking at the past age difference (10 years ago), as well as a future age ratio (14 years from now). Translating these relationship statements into equations allows us to create a snapshot of the age dynamics between Karishma and Babita. Understanding how to manipulate these equations with respect to time is then essential.

Time as a Variable

Specifically, 'ago' and 'hence' indicate subtraction and addition of years, respectively, to the current ages. Mastery of this concept allows students to set up accurate equations that lead to finding the correct current age ratio, which is the ultimate goal of many age-related algebra problems.

Ratio and Proportion

Ratio and proportion are mathematical concepts that describe the relationship between quantities, and they are particularly handy when dealing with age comparison problems. A ratio, simply put, is a way to show the relative size of two or more values. Proportion, on the other hand, states that two ratios are equivalent.

For our problem, after finding the values for Karishma and Babita's present ages using algebraic methods, the final step is to express their ages as a ratio. Remember to always simplify the ratio to its lowest terms to find the answer to the problem, which in our case is the ratio of their present ages. The resulting ratio, \(\frac{13}{25}\), is the simplest form of the fraction \(\frac{20.67}{42}\), giving us the desired comparison in a clear and interpretable format. This simplification is essential for the correct option identification, which is a practical application of the principles of ratio and proportion.