Basic maths: Interview question getting proportion from averages

Question

It's probably simple but I was given this question in a video interview recently and I spent ages coming up with two different answers.

(please let me know if this is the wrong place for this, I'm not entirely sure where is best suited for this type of question)

Question as follows:

A mobile app is on both iPhone and Android. Overall, there are 600,000 app users who log into the app an average of 11 times a month. The average iPhone user logs in 7 times a month. The average Android user logs in 13 times a month. What proportion of users access the app with an iPhone?

My two answers:
Answer one
$a$: proportion of iphone users
$$\frac{7}{20\cdot a} = 11$$ $$a=\frac{11 \cdot 20}{7}$$ $$=\frac{220}{7}$$ $$31.43% $$

Answer two

Answer: $33.33\%$

Are either of these right or have I completely lost the plot?

Answer 1

Let $x$ be the number of iPhone users, and let $y$ be the number of Android users, measured in units of $100000$ users.

Then we have the system \begin{align*} x+y&=6&&\text{[this equation counts users, in units of $100000$]}\\[4pt] 7x+13y&=6\cdot 11&&\text{[this equation counts logins, in units of $100000$]}\\[4pt] \end{align*} which yields $x=2,\;y=4$, hence the fraction of iPhone users is $$\frac{200000}{600000}=\frac{1}{3}$$

Answer 2

Let $x$ be the proportion of users that access the app with an iPhone.

Then the number of iPhone users is $600{,}000x$ and the number of Android users is $600{,}000(1-x)$.

Let $L_i$ be the number of all logins from iPhone users, and $L_a$ the total number of logins from Android users.

We have more info:

• The average iPhone user logs in $7$ times per month. This means that $\frac{L_i}{600{,}000x} = 7$
• The average Android user logs in $13$ times per month. This means that $\frac{L_a}{600{,}000(1-x)} = 13$
• The average user logs in $11$ times per month. This means that $\frac{L_a+L_b}{600{,}000} = 11$.

You have $3$ equations for $3$ variables, so it should be possible to solve this. Good luck!

Answer 3

You don't need the number of users. Suppose a proportion $p$ of users access via an iPhone. Then $11=7p+13(1-p)=13-6p$, and $p=\frac{1}{3}$.