What is the solution?
-*geniuises is not the answer to all question but the question to all answer*
Start by noticing that algebraically, your statement translates to :
$\displaystyle 2 < \frac{88+n}{19+n} < 7$ (Do you see why?)
We'll solve the left side first:
$\displaystyle 2 < \frac{88+n}{19+n}$
$\displaystyle 2(19+n) < 88+n$ (Since n is positive, 19+n is also positive, and we don't need to worry about direction of the inequality.)
$\displaystyle 38+2n < 88+n$
$\displaystyle n < 50$
On the right side:
$\displaystyle \frac{88+n}{19+n} < 7$
$\displaystyle 88+n < 7(19+n)$
$\displaystyle 88+n < 133+7n$
$\displaystyle -45 < 6n$
$\displaystyle -7.5 < n$
So $\displaystyle -7.5 < n < 50$. Since you said n is a positive integer, we'll adjust to $\displaystyle 0 < n < 50$, of which there are 49 of them (0 and 50 aren't included).
Hello, hotgal24!
Find the number of possible positive integers $\displaystyle n$ such that
there are exactly two positive integers between $\displaystyle \frac{88+n}{19+n}$ and $\displaystyle \frac{88}{19}$
Since $\displaystyle \frac{88}{19} \:\approx\:4.63$, the two positive integers are 3 and 4
. . and: .$\displaystyle \frac{88+n}{19+n} \:>\:2 \quad\Rightarrow\quad n \:<\:50$
Therefore: .$\displaystyle n \:=\:1,2,3, \hdots 49$
. . There are 49 possible values of $\displaystyle n.$