Sure! Let's break down the problem step-by-step.
1. Understand the survey results and given values:
- Total number of youths surveyed: 80
- Fraction of youths who liked only iPhone: [tex]\(\frac{1}{3}\)[/tex]
- Fraction of youths who liked only Android: [tex]\(\frac{2}{5}\)[/tex]
- Number of youths who liked none of the phones: 40
2. Calculate the number of youths who liked only iPhone:
[tex]\[
\text{Youths who liked only iPhone} = \frac{1}{3} \times 80 = \frac{80}{3} \approx 26.67
\][/tex]
3. Calculate the number of youths who liked only Android:
[tex]\[
\text{Youths who liked only Android} = \frac{2}{5} \times 80 = \frac{160}{5} = 32
\][/tex]
4. Determine the number of youths who did not like iPhone at all:
This includes youths who liked only Android and those who liked none of the phones.
[tex]\[
\text{Youths who did not like iPhone at all} = \text{Youths who liked only Android} + \text{Youths who liked none}
\][/tex]
[tex]\[
= 32 + 40 = 72
\][/tex]
5. Determine the number of youths who liked iPhone (either only iPhone or both):
[tex]\[
\text{Youths who liked iPhone or both} = \text{Total youths} - \text{Youths who did not like iPhone at all}
\][/tex]
[tex]\[
= 80 - 72 = 8
\][/tex]
6. Calculate the number of youths who liked both types of phones:
We know that the number of youths who liked iPhone (either only iPhone or both) is 8. From this, we subtract the number of youths who liked only iPhone.
[tex]\[
\text{Youths who liked both types of phones} = \text{Youths who liked iPhone or both} - \text{Youths who liked only iPhone}
\][/tex]
[tex]\[
= 8 - 26.67 \approx -18.67
\][/tex]
This final result indicates a logical inconsistency with the provided fractions and results. Therefore, we see that according to the given data and calculations, the number of youths who liked both phones is approximately [tex]\(-18.67\)[/tex]. This negative result implies an error in the given data or interpretation of fractions which require a re-examination.
