Question

The ratio of age of \(A\) and \(B\) is \(x: y\). If \(A\) 's age is increased by 3 years and B's age is increased by 2 years then new ratio of their ages becomes \(24: 25\) Given that the sum of their actual ages is 93 years. Find the actual ratio of their ages. (a) \(21: 22\) (b) \(42: 45\) (c) \(45: 48\) (d) can't be determined

Short answer

Answer: The actual ratio of their ages is 45:48.

Step-by-step solution

Write given information as mathematical equations

We have the following information: 1. Ratio of A's and B's ages: \(A = kx\) and \(B = ky\), where k is the constant of proportionality. 2. After increasing A's age by 3 years and B's age by 2 years, the new ratio is 24:25. So, \((A+3):(B+2) = 24:25\). 3. The sum of their actual ages is 93 years. So, \(A + B = 93\)

Solve the system of equations for x and y

We will now solve these equations for x and y: We have \(A = kx\) and \(B = ky\). From the given ratio after increasing ages, we have \(\frac{A + 3}{B + 2} = \frac{24}{25}\). Plugging A and B from the first two equations, we get \(\frac{kx + 3}{ky + 2} = \frac{24}{25}\). From the given sum of actual ages, we have \(A + B = 93\) or \(kx + ky = 93\). Now, we have: 1. \(\frac{kx + 3}{ky + 2} = \frac{24}{25}\) 2. \(kx + ky = 93\)

Check which of the given options matches with our calculated values

We can cross-multiply the first equation and keep the second equation as it is: 1. \(25(kx + 3) = 24(ky + 2)\) 2. \(kx + ky = 93\) Expanding the equation 1, we get \(25kx + 75 = 24ky + 48\) From equation 2, \(kx = 93 - ky\) Substitute the \(kx\) in the first equation, \(25(93 - ky) + 75 = 24ky + 48\) Solving this equation for ky, \(ky = 45\) Since \(kx = 93 - ky\), \(kx = 48\) Therefore, the actual ratio of ages is \(48:45\) or \(\boxed{(c) 45: 48}\).

Key concepts

Ratio and Proportion

Ratio and proportion are fundamental concepts used to express the relative size of two or more quantities. In this problem, the ages of two people, A and B, are given in a ratio. A ratio compares values, showing how much of one thing there is compared to another.

• A's age to B's age is expressed as \( x:y \), meaning for every \( x \) units of A's age, B's age corresponds to \( y \) units.
• To solve for actual ages, we use a constant of proportionality \( k \), which scales the ratio to their actual ages: \( A = kx \) and \( B = ky \).

Understanding ratios helps convert these relationships into equations useful for solving age-related problems.
It gives a structured approach to solve complex real-world problems in simple terms.

Equation Solving

Solving equations is an essential step in finding unknown values in age ratio problems. Once the ages of A and B are expressed as equations, we use known values and relationships to determine the actual ages.

• Start by setting up the initial equations based on the problem statement, such as \( A = kx \) and \( B = ky \).
• Use any given conditions, like the change in ratio or their combined age, and translate them into algebraic equations.
• For example, the condition that their ages change to a 24:25 ratio leads to the equation \( \frac{kx + 3}{ky + 2} = \frac{24}{25} \).
• Simultaneously, use the sum of their ages: \( kx + ky = 93 \).

Solve these equations systematically by substituting known values and simplifying. This practice hones algebraic skills and teaches problem-solving logic.

Word Problems

Word problems like this age ratio challenge require careful translation from words to mathematical equations. Here are steps to make this process run smoothly:

• Identify clues in the wording, such as "increased by" or "sum of", which indicate operations like addition or forming equations.
• Extract the relationship between entities (e.g., A's and B's ages) and directly translate these into equations.
• Transform situational changes into algebraic expressions, such as forming new ratios when ages are adjusted.
• Clarify what is known (e.g., the ratio or sum) and what must be found (e.g., the constant \( k \) or actual ratio).

Practicing word problems enhances your ability to decode complex situations into solvable mathematical statements, improving comprehension and skill in problem-solving.

Mathematical Reasoning

Mathematical reasoning involves using logical processes to find solutions and make sense of equations within age ratio problems. This involves:

• Understanding underlying principles, such as how proportions equate relationships in numbers.
• Applying logic to simplify and solve equations step by step, ensuring each manipulation maintains the equation's integrity.
• Justifying each step with reasoning, like cross-multiplying to solve proportions or substituting to simplify.
• Checking results for consistency with the original problem statement, confirming calculated ratios match given conditions.

Ultimately, mathematical reasoning allows you to navigate problems systematically, ensuring confidence in your solution method and accuracy in your results.