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Average of Two Numbers Theorem

Suppose that you have three numbers a, b, and c with a sum of X.

Let W be the average of the first two numbers (a and b), and let P be the average of the last two numbers (b and c).

Then:

• W = (a + b) / 2
• P = (b + c) / 2
• The average of all three numbers is (W + P) / 2

For example, if a = 5, b = 10, and c = 15:

• W = (5 + 10) / 2 = 7.5
• P = (10 + 15) / 2 = 12.5
• The average of all three numbers is (7.5 + 12.5) / 2 = 10

Average of Three Numbers Problem

Suppose you have three different bills:

• $20
• $30
• $50

You want to know the average amount of money you have across all three bills.

Solution

Let a = $20, b = $30, and c = $50. Then, the sum of these three numbers is X = $20 + $30 + $50 = $100.

Using the Average of Two Numbers Theorem, we can calculate the averages of the first two numbers and the last two numbers:

• W = (a + b) / 2 = ($20 + $30) / 2 = $25
• P = (b + c) / 2 = ($30 + $50) / 2 = $40

Finally, we can calculate the overall average of the three bills by averaging the values of W and P:

(Average amount) = (W + P) / 2 = ($25 + $40) / 2 = $32.50

Therefore, the average amount of money you have across all three bills is $32.50.

Average of Three Numbers Problem

Suppose you have three different bills:

• $20
• $30
• $50

You want to know the average amount of money you have across all three bills.

Solution

Let a = $20, b = $30, and c = $50. Then, the sum of these three numbers is X = $20 + $30 + $50 = $100.

Using the Average of Two Numbers Theorem, we can calculate the averages of the first two numbers and the last two numbers:

• W = (a + b) / 2 = ($20 + $30) / 2 = $25
• P = (b + c) / 2 = ($30 + $50) / 2 = $40

Finally, we can calculate the overall average of the three bills by averaging the values of W and P:

(Average amount) = (W + P) / 2 = ($25 + $40) / 2 = $32.50

Therefore, the average amount of money you have across all three bills is $32.50.

 The derivation and theorem are correct. Here are some short notes on the theorem:

Theorem: Let a, b, and c be three numbers with a sum of X. Let W be the average of the first two numbers (a and b), and let P be the average of the last two numbers (b and c). Then:

• The first number a is equal to (2W - P - X) / 2.
• The second number b is equal to (2W + 2P - X) / 2.
• The third number c is equal to (2P - W - X) / 2.
• The sum of the three numbers (a + b + c) is equal to X.

These formulas can be useful when working with problems involving averages and sums of multiple numbers. By knowing the average of certain subsets of the numbers, we can derive the value of each individual number. And by checking that the sum of the derived numbers is equal to the given sum X, we can verify that our calculations are correct.

It's worth noting that the formulas can also be rearranged to find W and P, given a, b, and c. For example:

• W = (a + b) / 2
• P = (b + c) / 2

Overall, the theorem provides a useful tool for solving problems involving sets of numbers with known averages and sums.

The sum of 3 numbers is X. Of these, the average of the first two numbers is W and the average of the last two numbers is P. What is the third number?

the third number is c = 2P - 2W.

c = 2P - 2W

Substituting the given values, we get:

c = 2(70) - 2(30)

Simplifying, we get:

c = 140 - 60

c = 80

Therefore, the third number is 80.

Q. if X= 0, w=30, P=70

If X=0, W=30, and P=70, we can use the formula we derived in the previous answer to find the third number:

c = 2P - 2W

Substituting the given values, we get:

c = 2(70) - 2(30)

Simplifying, we get:

c = 140 - 60

c = 80

Therefore, the third number is 80, even though the sum of the three numbers is zero. This means that the first two numbers must add up to -80, which is the negative of the third number, in order for the sum to be zero.

 5 संख्याओं का योग D है. इनमे से पहली दो संख्याओं का औसत A है तथा अंतिम दो संख्याओं का औसत G है. तीसरी संख्या क्या है?

5 संख्याओं का योग 120 है. इनमे से पहली दो संख्याओं का औसत 30 है तथा अंतिम दो संख्याओं का औसत 70 है. तीसरी संख्या क्या है? तीसरी संख्या = (120 - (30 + 30 + 70)) ÷ 2 = 10 5 संख्याओं का योग 110 है. इनमे से पहली दो संख्याओं का औसत 20 है तथा अंतिम दो संख्याओं का औसत 60 है. तीसरी संख्या क्या है? तीसरी संख्या = (110 - (20 + 20 + 60)) / 2 = 10 5 संख्याओं का योग D है. इनमे से पहली दो संख्याओं का औसत A है तथा अंतिम दो संख्याओं का औसत G है. तीसरी संख्या क्या है? तीसरी संख्या = (D - (A + A + G)) /2. 5 संख्याओं का योग X है. इनमे से पहली दो संख्याओं का औसत W है तथा अंतिम दो संख्याओं का औसत P है. तीसरी संख्या क्या है? तीसरी संख्या = (X - (W + W + P)) /2.