Question

The average of 7 consecutive numbers which are positive integers is 10. The average of lowest and highest such numbers is : (a) 7 (b) 10 (c) 15 (d) data insufficient

Short answer

Answer: 10

Step-by-step solution

Understand the properties of consecutive integers

We know that consecutive integers have a common difference of 1. When we have an odd number of consecutive numbers, their average is the middle number. In this case, we have 7 consecutive numbers, which is odd, and their average is given as 10. So, the middle number is 10.

Determine the lowest and highest numbers

We can find the lowest and highest numbers by moving 3 steps below and 3 steps above the middle number. Since each step has a common difference of 1, the difference between the middle number and the lowest/highest number is 3.\[-\] Lowest number: Middle number - 3 = 10 - 3 = 7\[-\] Highest number: Middle number + 3 = 10 + 3 = 13

Calculate the average of the lowest and highest numbers

To find the average, we add the lowest and highest numbers together and divide by 2.\[-\] Average = (Lowest number + Highest number) / 2\[-\] Average = (7 + 13) / 2 = 20 / 2 = 10 So, the average of the lowest and highest numbers in this sequence is 10. Therefore, the correct answer is (b) 10.

Key concepts

Quantitative Aptitude

Quantitative aptitude encompasses the ability to reason and to solve numerical and mathematical problems. It is a vital skill for students tackling standardized tests and for anyone involved in mathematics or subjects requiring analytical prowess. In the context of our exercise, quantitative aptitude is demonstrated through understanding and applying the concept of consecutive integers and averages to find a solution.

Students can improve their quantitative aptitude by regularly practicing problems that involve series, sequences, and patterns. Working through consecutive integer problems, such as the one mentioned, helps in developing a methodical approach. Arithmetic operations, critical thinking, and logical deduction play crucial roles in these exercises.

Average Calculation

Calculating an average is a fundamental concept in statistics, often used to find the central value of a dataset. It is especially important when dealing with consecutive integers. To find the average of any set of numbers, you sum all the numbers and then divide by the count of numbers in the set.

When dealing with an odd number of consecutive integers, the task simplifies as the average will always be the middle number, due to symmetry. For an even number of consecutive integers, the average is the mean of the two middle numbers. Understanding the intricacies of average calculation is essential for analyzing numerical data efficiently.

Arithmetic Progression

Arithmetic progression is a sequence of numbers in which the difference of any two successive members is a constant. This difference is called the 'common difference'. If consecutive integers form a sequence, they are a perfect example of an arithmetic progression with a common difference of 1.

For instance, in the problem given, the 7 consecutive integers form an arithmetic progression. Using the properties of such progressions allows us to find specific terms within the sequence, providing a streamlined path to solving for the average of the lowest and highest terms. Mastery of arithmetic progression not only aids in solving textbook problems but also in recognizing patterns in real life.

Integer Properties

Integers are the set of whole numbers including zero, positive, and negative numbers. An important property of integers is that they exhibit an orderly and predictable pattern especially when they are consecutive. Consecutive integers follow one another in sequence, without gaps, and always differ by one unit.

In an exercise dealing with consecutive integers, recognizing that they are evenly spaced is crucial. This understanding leads to recognizing that the set's median—which is also its average in the case of an odd number of terms—will be equidistant from each end of the set. These properties are particularly handy in avoiding complicated computations and in making quick, accurate calculations, as was necessary in the exercise we discussed.