Question

A sum of money doubles itself in 12 years. In how many years would it treble itself? (a) 36 years (b) 18 years (c) 24 years (d) 15 years

Short answer

Answer: Approximately 18 years.

Step-by-step solution

Identify the relationship between the time and the amount of money doubled.

Since the amount of money doubles every 12 years, we know that the amount of money at any given time can be calculated as follows: Amount at time t = Initial amount * (2^(t/12)), where t is in years.

Determine the equation for the money to triple itself.

We need to find the time (t) when the amount of money triples. To do this, we can write another equation relating the amount of money at time t to the initial amount: Amount at time t = Initial amount * 3

Set up the equation.

Now we can set up the equation by equating the two expressions for the amount of money at time t: Initial amount * (2^(t/12)) = Initial amount * 3

Solve for the time t.

First, we can simplify the equation by canceling out the "Initial amount" which appears on both sides of the equation: 2^(t/12) = 3 Now, to solve for t, we need to determine the power to which 2 should be raised to obtain the result 3. To do this, we can use logarithms. We will use the property log(a^b) = b*log(a): t/12 * log(2) = log(3) Now, we can solve for t by multiplying both sides by 12/log(2): t = 12 * log(3) / log(2)

Calculate the time t.

By calculating the value of t, we get approximately: t ≈ 12 * (1.585) = 19.02 years

Choose the closest answer.

Looking at the options given in the problem, we can see that the closest answer is (b) 18 years. However, we should note that 18 years is not an exact solution, but it is the closest answer among the given choices.

Key concepts

Exponential Growth

Exponential growth occurs when a quantity increases at a rate proportional to its current value. Such growth is common in many natural processes, as well as in finance, where money can grow exponentially through compound interest.

In mathematical terms, exponential growth is described by the equation: \( A(t) = A_0 \cdot e^{rt} \), where \( A(t) \) represents the amount at time \( t \), \( A_0 \) is the initial amount, \( r \) is the growth rate, and \( e \) is Euler's number (approximately 2.71828). In the context of the given problem, the money doubling over a certain period can be translated to an exponential growth scenario where the base is 2, as it represents the doubling factor.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions and are used to find the time it takes for exponentially growing quantities to reach a certain level. In other words, if exponential functions tell us how much there is after a certain time, logarithmic functions tell us how much time it takes to reach a certain amount.

The general form of a logarithmic function is given by: \( y = \log_b(x) \), which answers the question “To what power must \( b \) be raised, to yield \( x \)?”. When solving for time in financial problems involving exponential growth, logarithms are invaluable, as seen in the provided solution, where logarithms were used to determine how long it takes for the money to triple.

Problem Solving in Quantitative Aptitude

Problem-solving in quantitative aptitude involves a systematic approach to tackle mathematical and numerical challenges. This process typically includes understanding the problem, identifying the relevant mathematical principles, setting up equations, and calculating the solution.

When faced with a problem, such as determining the time for money to triple given a doubling period, it's important to first translate the verbal description into a mathematical model, then use quantitative techniques such as logarithms or exponential functions to solve for the unknown variable. This structured approach ensures that the solutions are accurate and derived logically.

Mathematical Equation Solving

Solving equations is a fundamental aspect of mathematics. The key to solving equations is to isolate the variable in question, making it the subject of the formula. In our problem, we had the equation \( 2^{(t/12)} = 3 \) to solve for \( t \). By employing logarithmic functions, we can solve for \( t \) effectively.

To isolate \( t \), we apply logarithms to both sides of the equation, utilizing the laws of logarithms to simplify and solve for the variable. This technique sheds light on why understanding the properties and operations of exponential and logarithmic functions is crucial for effectively handling mathematical equation solving in real-world contexts.