If the frequencies of the first four numbers out of 1,2,4,6,8 are 2,3,3,2 respectively, then the frequency of 8,if their AM is 5 is:

• 7
• 6
• 4
• 5

To determine the frequency of the number 8 given the frequencies of the first four numbers and the information about the arithmetic mean (AM), we can use a systematic approach. Let's break this down step by step.

Understanding the Problem

You have the numbers 1, 2, 4, 6, and 8, with the following frequencies:

• 1 appears 2 times
• 2 appears 3 times
• 4 appears 3 times
• 6 appears 2 times

The arithmetic mean (AM) of these numbers is given as 5. We need to find the frequency of the number 8. Let's denote its frequency as x.

Calculating Total Frequency

The total frequency (N) of the numbers can be calculated by adding up the individual frequencies:

• Total frequency = 2 (for 1) + 3 (for 2) + 3 (for 4) + 2 (for 6) + x (for 8)

This simplifies to:

N = 10 + x

Using the Arithmetic Mean Formula

The formula for the arithmetic mean (AM) is:

AM = (Sum of all values) / (Total frequency)

We know the AM is 5, so we can set up the equation:

5 = (Sum of frequencies × AM) / N

To find the sum of all values, we need to calculate it using the frequencies we have:

• Sum = 1×2 + 2×3 + 4×3 + 6×2 + 8×x
• Sum = 2 + 6 + 12 + 12 + 8x

This simplifies to:

Sum = 32 + 8x

Setting Up the Equation

Now, substitute the sum and total frequency into the AM formula:

5 = (32 + 8x) / (10 + x)

We can cross-multiply to eliminate the fraction:

5(10 + x) = 32 + 8x

This expands to:

50 + 5x = 32 + 8x

Solving for x

Now, rearranging the equation gives us:

• 50 - 32 = 8x - 5x
• 18 = 3x

Dividing both sides by 3 gives:

x = 6

Final Answer

The frequency of the number 8 is 6. Therefore, if the frequencies of the numbers 1, 2, 4, 6, and 8 are 2, 3, 3, 2, and 6 respectively, the arithmetic mean of these numbers will indeed be 5 as stated in the problem.