Question

The average length of first 3 fingers is 3 inches and the average length of the other 2 fingers i.e, thumb and the index finger is \(2.8\) inches. If the length of the index fingers is 3 inches then the length of the thumb is : (a) 2 inches (b) \(2.6\) inches (c) 3 inches (d) none of these

Short answer

Answer: (b) 2.6 inches

Step-by-step solution

Calculate total lengths of the three fingers and the thumb+index finger

To find the total length of the first 3 fingers, we have to multiply the average length by the number of fingers. Similarly, we multiply the average length of the thumb and index finger by 2 to find their total length. Total length of first 3 fingers = \(3 \cdot 3 = 9\) inches \\ Total length of thumb and index finger = \(2.8 \cdot 2 = 5.6\) inches

Calculate total length of all 5 fingers

To find the total length of all 5 fingers, we need to add the total lengths of the first 3 fingers and the thumb and index finger. Total length of 5 fingers = Total length of first 3 fingers + Total length of thumb and index finger \\ Total length of 5 fingers = \(9 + 5.6 = 14.6\) inches

Calculate the length of the thumb

Since we are given the length of the index finger as 3 inches, we can subtract its length from the total length of thumb and index finger to find the length of the thumb. Length of thumb = Total length of thumb and index finger - Length of index finger \\ Length of thumb = \(5.6 - 3 = 2.6\) inches The correct answer is (b) 2.6 inches.

Key concepts

Arithmetic Mean

The arithmetic mean, commonly known as the average, is a useful statistical concept that helps us understand the central tendency of a data set. To calculate it, you sum up all the values and then divide by the total number of values. In our exercise, we're given the average of the length of certain fingers.

• The average length of the first three fingers is said to be 3 inches. Thus, by multiplying the average by the number of items (here, three fingers), we achieve the total length, resulting in 9 inches.
• For the thumb and index finger, the average is 2.8 inches, and as there are two fingers, the total length is 5.6 inches.

This kind of calculation can be applied in numerous situations, wherever determining an overall mean value is required, making it an invaluable tool in data analysis. Remember, while calculating the arithmetic mean, it's crucial to have all terms in similar units to ensure accuracy.

Problem Solving

Problem solving is about using a logical approach to tackle issues and find solutions. The exercise at hand provides an elegant example of how mathematical tools can untangle real-world questions. Let's break down the process:

• Firstly, we identified what is given: average lengths and one specific length, the index finger which measures 3 inches.
• Secondly, we calculated the missing values step by step. First calculating the total length of the first group of fingers, then of the combined thumb and index finger set, and finally combining these to reach a total length of all fingers.
• The final step showcased a direct application of given data to isolate the length of a single component, the thumb in this case, using subtraction.

This exercise demonstrates the importance of organizing information and systematically solving each part to achieve the desired result. By ensuring all steps are clear and logical, problem solving becomes more manageable.

Mathematical Reasoning

Mathematical reasoning is the cognitive process that allows us to make sense of mathematical concepts and forms the basis of problem-solving. It involves making connections, recognizing patterns, and applying logical thinking. In this exercise, we relied heavily on reasoning to conclude.

• We used reasoning to understand that the total length of specific groups of fingers could help us deduce individual lengths.
• The reasoning also led us to consider the known quantities and manipulate them formulaically, subtracting the known length of the index finger from the total of the thumb and index to find just the thumb's length.
• Finally, mathematical reasoning helped in judging and matching our result with the provided options, reinforcing the solution was correct with respect to the given choices.

Through mathematical reasoning, abstract problems are analyzed and validated, facilitating learning and application in diverse mathematical situations.