The start time of Mincha-Davening Mincha Gedola:[1]

One who Davens Mincha from 6 ½ Zmaniyos hours into the day [i.e. half an hour after the Halachic midday], and onwards, fulfills his obligation. [This time period is commonly referred to as Mincha Gedola.[2]] [However] the main time to Daven Mincha is from the 9th ½ hour into the day and onwards [which is commonly referred to as Mincha Ketana].[3] [Accordingly, one is not to initially Daven Mincha prior to the 9 ½ hour into the day, and is hence not to Daven during Mincha Gedola.[4] Furthermore, some Poskim[5] rule it is better to Daven Mincha past Plag Hamincha, which is 10 ¾ hours into the day, than to Daven beforehand. Furthermore, some are particular to Daven Mincha specifically near sunset.[6] However, other Poskim[7] rule one may even initially Daven Mincha past 6 ½ hours into the day, and it is even a Mitzvah Min Hamuvchar to do so. Practically, [one is to initially Daven Mincha past 9/12 hours into the day although] the custom is to even initially Daven Mincha prior to Plag Hamincha.[8] However, in a time of need, one may Daven Mincha Gedola after 6 ½ hours into the day. For example, if one desires to eat a meal in the time of Mincha Gedola, he may even initially Daven Mincha beforehand. Certainly, if one desires to travel, he may Daven Mincha Gedola prior to traveling.[9] Likewise, if one will not be able to Daven with a Minyan by Mincha Ketana, he may Daven with a Minyan by Mincha Gedola.[10] Likewise, one may even initially Daven Mincha by Mincha Gedola if doing so will schedule one’s day in a way that as a result he will be able to increase his time for Torah learning. Accordingly, many Yeshivos even initially Daven Mincha by Mincha Gedola for scheduling purposes.[11] Practically, despite the above ruling, many people even initially Daven Mincha by Mincha Gedola throughout the year[12], and so was the custom of the Rebbe to Daven Mincha at 3:15 throughout the year even when it fell before Mincha Ketana had arrived.[13]]

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[1] Michaber 233:1; Beis Yosef 233 in his opinion of Rambam Tefila 3:2; The concept of Mincha Gedola and Ketana is mentioned in Brachos 26b; See Piskeiy Teshuvos 233:2

[2] See Beis Yosef ibid; Rambam ibid; Brachos ibid

[3] The reason: The main Biblical time for the slaughtering of the afternoon Tamid is from 6 ½ hours into the day, however, in actuality they would only offer it from 9 ½ hours into the day. The reason for this is because it is forbidden to offer private sacrifices once the afternoon Tamid is offered. Hence, in order to give time to sacrifice the other Karbanos that are brought that day, the offering of the afternoon Tamid was always delayed until past 9 ½ hours. Due to this, the main time of Mincha is from 9 ½ hours into the day. The only exception to this was when Erev Pesach falls on Erev Shabbos, in which they would slaughter the afternoon Tamid by 6 ½ hours into the day [and thus Bedieved, if one Davens Mincha from that time and onwards he is Yotzei]. [M”A 233:2; Beis Yosef 233; Rambam ibid; Brachos ibid; Kaf Hachaim 233:2]

[4] Implication of Michaber ibid; M”A 233:2; Taz 233:1 that so is implication of Michaber ibid and Rambam; Radbaz 676; Birkeiy Yosef 233:4; Shaareiy Teshuvah 233:1; Maaseh Rav 65 of Gr”a; Or Tzadikim 25; Ashel Avraham Butchach 233; Ben Ish Chaiy Vayakhel 3; Kaf Hachaim 233:1 and 3; Divrei Yatziv 1:98; Piskeiy Teshuvos 233:2

[5] Radbaz 676 in name of Rabbeinu Chananel; Birkeiy Yosef 233:4; Shaareiy Teshuvah 233:1; Kaf Hachaim 233:3

[6] Or Tzadikim 25 that so was custom of Arizal to Daven Mincha near sunset, brought in Shaareiy Teshuvah 235:2; Rebbe Yochanon in Brachos 29b that it is a Mitzvah to Daven with sunset; Hagahos Maimanis Tefila 3:2 in name of Rabbeinu Chananel;

[7] Beis Yosef 233 in implication of Tur 233 and Rosh Brachos Klal 4:9 that one may even initially Daven past 6 ½ hours into the day, brought in Taz ibid [The Tur ibid writes “The time of Mincha is from 6 ½ hours into the day” and does not differentiate between Lechatchila or Bedieved and between Mincha Ketana or Geodla]; Ritva Yuma 28b that it is a Mitzvah Min Hamuvchar to Daven then; Semak 1; Chinuch Eikev; Abudarham; Rokeiach; Sefer Hayashar 622; Hamanhig Tefila 82;  Aruch Hashulchan 233:12 in name of Rav Shlomo Parchon in Shoresh Mincha that one is specifically to Daven by Mincha Gedola and that lack of doing so is what has caused poverty in Europe; M”B 233:1 in name of Rishonim; Igara Deparka 250 that so is implied from Zohar; Custom of Bnei Yissachar, brought in Divrei Torah 1 104; Mishmeres Shalom Kudinov 21 that so is implied from Zohar; Shulchan Hatahor 233:1

[8] Taz ibid

[9] M”A 233:2; Kaf Hachaim 233:1

[10] M”B 233:1

[11] Yechaveh Daas 4:19; Yisrael Vihazmanim 13 in name of Satmar Rav; Piskeiy Teshuvos 233:2

[12] So was accustomed the Chazon Ish; The Brisker Rav and his father Rav Chaim of Brisk

[13] Rabbi Groner related that the Rebbe Davened Mincha at 3:15throughout the year, as this was the Seder of the Yeshiva.