To solve the problem, we need to find the exact value of [tex]\( N \)[/tex] by first determining the value of [tex]\( n \)[/tex]. Here’s the step-by-step solution:
1. Given Information:
- Mean [tex]\( \bar{X} = 40 \)[/tex]
- Sum of the product of frequency and mid-point values [tex]\( \Sigma fm = 1400 \)[/tex]
- Number of terms [tex]\( N = 20 + 3n \)[/tex]
2. Formula for Mean:
The formula for the mean in grouped data is:
[tex]\[
\bar{X} = \frac{\Sigma fm}{N}
\][/tex]
3. Substitute Given Values:
We can substitute the given values into the mean formula:
[tex]\[
40 = \frac{1400}{20 + 3n}
\][/tex]
4. Solving for [tex]\( n \)[/tex]:
- Multiply both sides of the equation by [tex]\( (20 + 3n) \)[/tex] to remove the denominator:
[tex]\[
40 \times (20 + 3n) = 1400
\][/tex]
- Simplify the left-hand side:
[tex]\[
800 + 120n = 1400
\][/tex]
- Isolate the term with [tex]\( n \)[/tex]:
[tex]\[
120n = 1400 - 800
\][/tex]
- Simplify the right-hand side:
[tex]\[
120n = 600
\][/tex]
- Solve for [tex]\( n \)[/tex]:
[tex]\[
n = \frac{600}{120}
\][/tex]
[tex]\[
n = 5
\][/tex]
5. Find the Exact Value of [tex]\( N \)[/tex]:
- Substitute [tex]\( n = 5 \)[/tex] back into the equation for [tex]\( N \)[/tex]:
[tex]\[
N = 20 + 3n
\][/tex]
[tex]\[
N = 20 + 3 \times 5
\][/tex]
[tex]\[
N = 20 + 15
\][/tex]
[tex]\[
N = 35
\][/tex]
6. Conclusion:
The exact value of [tex]\( N \)[/tex] is 35.
Therefore, the solution correctly determines that [tex]\( n = 5 \)[/tex] and the number of terms [tex]\( N = 35 \)[/tex].
