, 2026

In the second bowl, the ratio of the number of blueberries to the
number of raspberries is $2:5$, and
so the number of blueberries is a positive integer multiple of $2$, and the number of raspberries is the
same positive integer multiple of $5$.

For example, there could be $2\times1=2$ blueberries and $5\times1=5$ raspberries, or $2\times2=4$ blueberries and $5\times2=10$ raspberries, and so on.

Thus, the number of raspberries in the second bowl has units digit $5$ or $0$.

There are a total of $89$
raspberries, and so the number of raspberries in the first bowl must
have units digit $4$ (when the number
of raspberries in the second bowl has units digit $5$), or it must have units digit $9$ (when the number of raspberries in the
second bowl has units digit $0$).

In the first bowl, the ratio of the number of blueberries to the number
of raspberries is $3:7$, and so the
number of raspberries is a positive integer multiple of $7$.

The positive integer multiples of $7$
less than $89$ that have units digit
$4$ or $9$ are $14$, $49$ and $84$.

If the number of raspberries in the first bowl is $14$, the number of raspberries in the
second bowl is $89-14=75$.

If there are $14$ raspberries in the
first bowl, there are $3\times\dfrac{14}{7}=6$ blueberries in the
first bowl ($6:14=3:7$).

If there are $75$ raspberries in the
second bowl, there are $2\times\dfrac{75}{5}=30$ blueberries in
the second bowl ($30:75=2:5$).

In this case, the total number of blueberries is $6+30=36$.

If the number of raspberries in the first bowl is $49$, the number of raspberries in the
second bowl is $89-49=40$.

If there are $49$ raspberries in the
first bowl, there are $3\times\dfrac{49}{7}=21$ blueberries in
the first bowl ($21:49=3:7$).

If there are $40$ raspberries in the
second bowl, there are $2\times\dfrac{40}{5}=16$ blueberries in
the second bowl ($16:40=2:5$).

In this case, the total number of blueberries is $21+16=37$.

If the number of raspberries in the first bowl is $84$, the number of raspberries in the
second bowl is $89-84=5$.

If there are $84$ raspberries in the
first bowl, there are $3\times\dfrac{84}{7}=36$ blueberries in
the first bowl ($36:84=3:7$).

If there are $5$ raspberries in the
second bowl, there are $2$
blueberries in the second bowl.

In this case, the total number of blueberries is $36+2=38$.

Thus, the smallest possible number of blueberries is $36$.