Recently I was in a Shul for Friday Mincha. As soon as there were 10 adults they began Ashrei. In the 2nd Bracha of the Chazan repeating Shemona Esrei they realized that one of those that they counted as an adult was first going to be Bar Mitzva in a few weeks. Thus there were really only 9 adults participating in this Minyan. What is the proper way to proceed in such an instance?
According to what I would infer from
Shulchan Arukh, Orach Chayyim 55:4:
יֵשׁ מַתִּירִין לוֹמַר דָּבָר שֶׁבַּקְּדֻשָּׁה בְּתִשְׁעָה וְצֵרוּף קָטָן, שֶׁהוּא יוֹתֵר מִבֶּן שֵׁשׁ וְיוֹדֵעַ לְמִי מִתְפַּלְּלִין, וְלֹא נִרְאִין דִּבְרֵיהֶם לִגְדוֹלֵי הַפּוֹסְקִים, וְהוּא הַדִּין דְּעֶבֶד וְאִשָּׁה אֵין מִצְטָרְפִין: הַגָּה: וַאֲפִלּוּ עַל יְדֵי חֻמָּשׁ שֶׁבְּיָדוֹ אֵין לְצָרְפוֹ, מִיהוּ יֵשׁ נוֹהֲגִין לְהָקֵל בִּשְׁעַת הַדַּחַק. (הָרֹא''שׁ וּמָרְדְּכַי וְהַגָּהוֹת מַיְמוֹנִי פֶּרֶק ח' מֵהִלְכוֹת תְּפִלָּה).
according to some opinions you may stil be able to continue. The beginning of this pargaraph states that that there are some who permit saying parts of the prayers with 9 people and a minor age 6+ who understands who he is praying to. EVen though the majority of people disagree with this opinion, one may be lenient bish'at hadchak (when there is no other choice.)
Ba'er Hetev commentary says that nowadays, even a "Sh'at Hadchal* would not allow the young boy to be part of a minyan.
Inferring from a previous par. in OC 55, it says that if in the middle of Shemoneh Esreh, you are left without a minyan, you may continue. This may be an irrelevant point, of course, if you rely on the She'at Hadchak leniency. If you could, and it were a She'at Hadchak, then the boy would count as part of the minyan to start, and there is no concern.
I think you need to inquire a rav as to whether your sit. is a "emergency", and what opinion the rav will follow.