The numbers 3, 5, 9, 15 have frequencies x+6, x+2, x−3 and x respectively. If their mean is 5.74, find the value of x. (Enter the value as next highest integer)

To find the value of \( x \), we can use the concept of a weighted mean. The formula for the mean of a set of numbers with given frequencies is:

\[
\text{Mean} = \frac{\sum ( \text{number} \times \text{frequency})}{\sum \text{frequencies}}
\]

Given numbers: \( 3, 5, 9, 15 \)

Their frequencies: \( x+6, x+2, x-3, x \)

And the mean is given as 5.74. Now, we can set up the equation:

\[
5.74 = \frac{(3(x+6) + 5(x+2) + 9(x-3) + 15x)}{(x+6) + (x+2) + (x-3) + x}
\]

### Step 1: Simplify the numerator and denominator

**Numerator:**
\[
3(x+6) + 5(x+2) + 9(x-3) + 15x
\]
Expand each term:
\[
= 3x + 18 + 5x + 10 + 9x - 27 + 15x
\]
\[
= (3x + 5x + 9x + 15x) + (18 + 10 - 27)
\]
\[
= 32x + 1
\]

**Denominator:**
\[
(x+6) + (x+2) + (x-3) + x
\]
Simplify:
\[
= x + 6 + x + 2 + x - 3 + x
\]
\[
= 4x + 5
\]

### Step 2: Set up the equation

Now substitute the simplified expressions back into the equation:
\[
5.74 = \frac{32x + 1}{4x + 5}
\]

### Step 3: Solve for \( x \)

Multiply both sides by \( 4x + 5 \) to eliminate the denominator:
\[
5.74(4x + 5) = 32x + 1
\]
Expand the left side:
\[
22.96x + 28.7 = 32x + 1
\]
Now, move all terms involving \( x \) to one side and constant terms to the other side:
\[
22.96x - 32x = 1 - 28.7
\]
\[
-9.04x = -27.7
\]
Solve for \( x \):
\[
x = \frac{-27.7}{-9.04} \approx 3.06
\]

### Step 4: Round to the next highest integer

Since \( x \approx 3.06 \), the next highest integer is \( x = 4 \).

Thus, the value of \( x \) is **4**.

Would you like further details or clarifications?

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Here are 5 related questions for you:
1. How would the mean change if all frequencies were doubled?
2. What is the formula for calculating the weighted mean?
3. How do you solve a linear equation with a fractional result like this one?
4. What is the importance of rounding in this context?
5. How does changing the value of \( x \) affect the overall distribution of the data?

**Tip:** Always simplify both the numerator and denominator separately before setting up your equation to solve for unknowns.

Discover how to solve a weighted mean problem with variable frequencies, where the numbers 3, 5, 9, and 15 have given frequencies and the mean is 5.74. This detailed solution involves algebra and linear equations, suitable for Grades 9-10 students.

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