Question

The ratio of age of \(A\) and \(B\) is \(8: 9\) and the age of \(B\) is \(2 / 3\) of \(C\) 's age and age of \(C\) is \(\frac{9}{13}\) times the age of \(D .\) If the age of \(B\) is 18 years then the age of \(C\) is : (a) 36 years (b) 39 years (c) 27 years (d) 54 years

Short answer

Answer: (c) 27 years

Step-by-step solution

Find the age of A

Given the ratio of the ages of A and B as 8:9, set up a proportion: $$\frac{A}{B} = \frac{8}{9}$$ We are given that B's age is 18 years, so substitute this into the proportion and solve for A's age: $$\frac{A}{18} = \frac{8}{9}$$ Multiply both sides by 18 to get the age of A: $$A = 18 * \frac{8}{9}$$ $$A = 16$$ So, the age of A is 16 years.

Find the age of C

We know that the age of B is 2/3 the age of C. We can use this information to set up an equation and solve for the age of C: $$B = \frac{2}{3}C$$ Since the age of B is 18 years, substitute this into the equation and solve for C's age: $$18 = \frac{2}{3}C$$ To find the value of C, multiply both sides by \(\frac{3}{2}\): $$C = 18 * \frac{3}{2}$$ $$C = 27$$ So, the age of C is 27 years. The correct option is (c) 27 years.

Key concepts

Arithmetic Ratios

In arithmetics, ratios are a way to compare two or more quantities by division, expressing how many times one number contains another. The ratio of two numbers is written in the format of 'A:B' or sometimes as a fraction \(\frac{A}{B}\). When dealing with age ratio problems, we are commonly comparing the ages of individuals to understand their relationships in terms of 'how much older' or 'how much younger' one is compared to the other.

Understanding and simplifying ratios is critical, as it allows us to solve for unknown variables by using a proportion — an equation that equates two ratios. For instance, in our example, the proportion \(\frac{A}{B} = \frac{8}{9}\) is set up based on the given age ratio of A to B. Simplifying or cross-multiplying these ratios will enable us to find the comparative ages for the individuals involved.

Proportion Calculations

Proportions are equations that show two ratios are equivalent. When calculating proportions, you often solve for a missing term by cross-multiplying. This involves multiplying the numerator of one ratio by the denominator of the other, and vice versa, setting the products equal to each other. It's a fundamental mathematical skill that comes in handy for many real-world situations, such as scaling recipes or determining distances on maps.

In age ratio and proportion calculations, like the one for finding the age of A, we can solve for the unknown by multiplying both sides by the known term, in this case, B's age. Proportion calculations are very systematic and require only simple algebraic manipulation to arrive at the correct answer.

Age Comparison Mathematics

Comparing ages using mathematical concepts such as ratios and proportions often involves understanding the direct and inverse relationships between ages. In age comparison mathematics, we can determine the relative age differences and express one age in terms of another. For instance, when we say B's age is \(\frac{2}{3}\) of C's age, we establish a direct relationship — as C's age increases or decreases, so does B’s.

Age comparison problems might require one to convert these relationships into equations that can be solved algebraically. In our example, knowing B's age directly leads to calculating C’s age, as their ages are linked by a given ratio. It’s a real-life application of maths that helps in understanding and solving everyday problems involving age and other quantities.

Solving Age-Related Equations

Solving equations related to ages requires an understanding of algebraic expressions and the ability to manipulate them to find the unknowns. The process typically starts by translating the word problem into mathematical equations using the given ratios or proportions. Once the equations are formed, they can be solved through various algebraic operations such as addition, subtraction, multiplication, and division.

In the example given, we solve the age-related equations step by step, beginning with establishing A's age using the ratio and B's age. Then, we move on to determining C's age by using the proportion that relates B's and C's age. This step-by-step approach simplifies complex problems by breaking them down into manageable parts.