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In a survey of some youths, it was found that:

- One third of them liked the iPhone only,
- [tex]\frac{2}{5}[/tex] of them liked the Android phone,
- 80 of them did not like the iPhone at all, and
- 40 of them liked none of the phones.

a) Find the number of youths who liked both types of phones.



Answer :

GinnyAnswer
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.

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