How to Solve a Math Puzzle Involving Averages

Math puzzles can be both intriguing and challenging, especially when they involve averages and series of numbers. This article will guide you through solving a specific math puzzle step-by-step using three different methods, and delve into the problem-solving techniques involved.

The Problem Statement

The puzzle at hand is as follows: Given seven numbers, the average of the first four numbers is 4, the average of the last four numbers is also 4, and the average of all seven numbers is 3. The task is to find the fourth number.

Understanding the Problem

Let's denote the seven numbers as a, b, c, d, e, f, g. We need to determine the value of d, the fourth number.

Solving the Puzzle: Method 1 - Direct Arithmetic Approach

H2: Method 1 - Direct Arithmetic Approach

According to the given information:

The average of the first four numbers (a, b, c, d) is 4. This implies: Sum of the first four numbers a b c d 4 * 4 16 The average of the last four numbers (d, e, f, g) is 4. This implies: Sum of the last four numbers d e f g 4 * 4 16 The average of all seven numbers is 3. This implies: Total sum of all seven numbers a b c d e f g 7 * 3 21

Now, adding the sums of the first four and the last four numbers, we have:

(a b c d) (d e f g) 21

This simplifies to:

a b c d d e f g 21

a b c e f g 2d 21

Since a b c d e f g 21, substituting the sum of the first four and the last four numbers, we get:

16 16 21 2d

Solving for d:

32 21 2d

2d 11

d 11 / 2 12

Solving the Puzzle: Method 2 - Using Total Sum

H2: Method 2 - Using Total Sum

Alternatively, we can use the total sums directly:

Total sum of all seven numbers a b c d e f g 21 Total sum of the first four numbers a b c d 24 Total sum of the last four numbers d e f g 20

The sum of the first four and the last four numbers (a b c e f g 2d) should equal 21:

(24 20) - 21 12

Solving the Puzzle: Method 3 - Using Algebraic Equations

H2: Method 3 - Using Algebraic Equations

Let x the 4th number, A sum of the 1st 3 numbers, and B sum of the last 3 numbers. Then:

Sum of the 1st 4 numbers A x 4 * 4 16 Sum of the last 4 numbers x B 4 * 4 16 Adding these two equations, we get: (A x) (x B) 32 2x A B 32 Given the average of all 7 numbers is 3, their total sum is 21: A x B x 21 A B 2x 21 Subtracting the second equation from the first, we get: 2x A B - (A B 2x) 32 - 21 0 11 - 2x 2x 11 x 11 / 2 x 12

Thus, the fourth number is 12.

Problem-Solving Techniques

When approaching such problems, it is important to write down what you know in mathematical notation. In this scenario:

Taking the seven numbers as a, b, c, d, e, f, g, write down the average of the first four (a, b, c, d), the average of the last four (d, e, f, g), and the average of all seven. Utilize the information to form equations and solve for the unknowns.

Conclusion

Solving math puzzles, especially those involving averages, requires a systematic approach and careful application of problem-solving techniques. By breaking down the problem and utilizing algebraic methods or direct arithmetic, we can arrive at the correct solution. The key is to clearly outline the given information and translate it into mathematical expressions, which can then be manipulated to find the unknown.