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Greatness of the Mincha Prayer
Mincha is very important because whereas Shacharit is recited in the morning before one gets involved in his daily routine and Arvit is recited at night, after one finishes with his day's work, mincha is recited in the middle of the day, which requires a person to put his personal affairs aside in order to pay attention to G-d. 
Starting without a Minyan
Ashrei should not be recited without ten people present. If ashrei is recited without a minyan,they should recite a chapter of tehillim before reciting the kaddish. 
Unless there’s a very great need (such as if there’s no time before Shekiyah or if there’s a small minyan and those in the minyan won’t answer Amen to the Chazarat HaShatz) this abbreviated Chazara may not be used. 
According to Ashkenazim, if there’s not enough time to complete Mincha with Chazarat HaShatz before Shekiyah, then the Tzibbur should do an abbreviated Chazara by having the Sheliach Tzibbur say the first three Brachot out loud including Kedusha, and then everyone else should start from the beginning. If there’s less time, while the Shaliach Tzibbur says that first three Brachot aloud everyone except one should read along word by word and then everyone should continue silently from there. 
According to Sephardim, if there’s not enough time to complete Mincha with Chazarat HaShatz before Tzet HaKochavim (13.5 minutes after Shekiah), then the Tzibbur should do an abbreviated Chazara by having the Sheliach Tzibbur say the first three Brachot out loud including Kedusha, while everyone else should read along word by word with the Sheliach Tzibbur. 
Some have the custom to recite Petach Eliyahu before Mincha. If one doesn't have a minyan at the end of Petach Eliyahu they can recite Kaddish Al Yisrael after Ketoret before Ashrei.
Eating before Mincha
According to Ashkenazim, there is what to rely on to be lenient to eat a meal before Mincha except for having a big meal (like the meal of a wedding or Brit Milah) from the beginning of the tenth hour of the day (a half hour before Mincha Ketana) before praying Mincha for which there is no leniency to rely on. Nonetheless, it’s preferable to be strict not to have a big meal from the beginning of the seventh hour (midday) before praying Mincha. 
According to Sephardim, it is preferable not to have a big (like the meal of a wedding or Brit Milah) or small meal (greater than a KeBaytzah of bread) from the beginning of the seventh hour (midday), however, the minhag is to be lenient and hold like the same halacha as Ashkenazim (see above) and there is what to rely on. 
This prohibition to eat before Mincha only applies to having a small meal which is more than a KeBaytzah of bread, however, a snack such as a KeBaytzah of bread or less or a lot of fruit is permissible. 
↑ Kitzur Shulchan Aruch 69:1. The gemara in berachot 6b tells us that mincha is extremely important because Eliyahu Hanavi was answered during the mincha prayer in Melachim Alef 18:36
↑Rama 124:2, Mishna Brurah 124:6, Sh”t Az Nidbaru 12:33, Sh”t Yachave Daat 3:16, Piskei Teshuvot 232:2, Rabbi Ari Enkin
↑Mishna Brurah 124:8, 232:4, and 233:14. Kitzur Shulchan Aruch 69:6. Rav Soloveitchik quoted by Rav Herschel Schachter in Nefesh Harav, pg. 126 maintains that the tzibbur should begin Shemoneh Esrei with the shaliach tzibbur in any event that the tzibbur does an abbreviated chazara.
↑Rabbi Yonatan Mazuz writes that if there's no minyan after Petach Eliyahu they can delay the Kaddish until after Ketoret. Rabbi Michal Sigron clarifies that Petach Eliyahu can be said by an individual since it is learning torah and not a tefillah.
The Mishna (Shabbat 9b) writes that one may not start a meal close to the time of Mincha and if one started one may continue. The Gemara 9b initially posits that this prohibition must only apply from Mincha Katana because why would it start from Mincha Gedolah isn’t there a lot of time from Mincha Gedolah (and one will certainly have time to pray). Then the Gemara says that the fact that the Mishna states that if one started one doesn’t have to stop the meal contradicts Rabbi Yehoshua Ben Levi who says one may not taste food before Mincha. Rather, says the Gemara, the Mishna was talking about having a big meal (before Mincha Gedolah) and this doesn’t contradict Rabbi Yehoshua Ben Levi who forbad eating after Mincha Ketana. Then the Gemara quotes a second opinion, Rav Acha Bar Yacov who says that the Mishna means that one can’t have even a small meal before Mincha Gedolah.
The Rishonim (early authorities) discuss what is the halacha based on the Gemara Brachot 28b which states that the halacha doesn’t follow Rabbi Yehoshua Ben Levi.
The Baal HaMoer (Shabbat 3b s.v. Matni Lo Yeshev) says that we hold like the first thought of the Gemara that the Mishna forbids having a big meal before Mincha Ketana, however, it’s permissible to have a big or small meal before Mincha Gedolah. Additionally, if one started having a meal before Mincha Ketana one may continue the meal. This is quoted (as an individual opinion) by the Ran Shabbat 4a s.v. Aval and Bet Yosef 232:2. The Rashba (Shabbat 9b s.v. VeYeInyan Pesak Halacha) initially agrees with the Baal HaMoar but then concludes that it should also be prohibited to have a big meal before Mincha Gedolah.
The Ri (quoted by Tur 232:2) says that we hold like the first answer of the Gemara that there’s only a prohibition for having a big meal before Gedolah and there’s no prohibition to have a small meal neither before Mincha Gedolah or Mincha Ketana. This is also the opinion of Rabbenu Tam according to the Rosh (Brachot 4:11).
The Rabbenu Tam (quoted by the Tur 232:2) says that we hold like the first answer of the Gemara and the primary prohibition is having a big meal before Mincha Gedolah. However, there’s a second prohibition of having a small meal before Mincha Ketana. Additionally, if one began before Mincha Gedolah one may continue however if one before before Mincha Ketana one should stop. This is also the opinion of Tosfot (Shabbat 9b s.v. BeTisporet).
The Rif (Shabbat 4a) rules like Rav Acha Bar Yacov that one may not have a small or big meal from before Mincha Gedolah. Rambam (Tefillah 6:5) holds like the Rif. The Rosh (Brachot 4:11, Shabbat 1:18) quotes the opinions of Tosfot and Rif and the Tur in C”M 5 writes that the Rosh agrees to the Rif, while the Bet Yosef C”M 5 questions this. The Rosh (Shabbat 1:18) writes that the Rabbenu Yonah agrees with the Rif. The Rashba (Shabbat 9b s.v. VeYeInyan Pesak Halacha) writes that his Rebbe (the Ramban) agreed with the Rif. S”A 232:2 rules like the Rif.
The Rama 232:2 writes that one may be more lenient than S”A to have a small meal before Mincha Gedolah and before Mincha Ketana but one should still be strict like the Ri not to have a big meal even before Mincha Gedolah and the Minhag is even more lenient to permit any meal except for a big meal before Mincha Katana (which is a combination of the leniency of the Baal HaMoar and the leniency of the Ri). [The Yalkut Yosef (vol 3 pg 635) writes that this is also the opinion of the Mordechai. However, the Mordechai in Shabbat (Siman 225) seems to hold like Rabbenu Tam.]
The achronim discuss whether the leniency to permit having a small meal even before Mincha Ketana allows one to have a small meal at any time or only until the actual time of Mincha Ketana (nine and a half hour and not the beginning of the ninth hour). The Magen Avraham 232:15 is strict, however, the Mahariv is lenient. The Mishna Brurah 232:26 is lenient.
The Rama 232:2 defines a big meal as a meal of wedding or Brit Milah. This is based on the Tosfot (Shabbat 9b s.v. BeTisporet) who says that a big meal like those of an engagement, wedding, or Brit Milah. The Bet Yosef 232:2 quotes the Hagahot Maimonot (Tefillah 6:7) who writes that a Shabbat or Yom Tov meal isn’t considered a big meal but only a meal where many people gather together such as a wedding or Brit Milah. The Kol Bo (Siman 11 pg 8a) agrees to the Hagahot Maimonot. The Mishna Brurah 232:24 rules that a Shabbat and Yom Tov meal isn’t considered a big meal, however, a wedding, Brit Milah or Pidyon HaBen meal (where many people gather together) is considered a big meal.
The Bet Yosef 232:2 quotes the Ran (Shabbat 4a s.v. Hay) who says that this prohibition which begins close to the time of Mincha starts a half hour before the time of Mincha. The Bet Yosef writes that this is also the opinion of the Rashbam (Pesachim 99b). [The Mordechai in Shabbat (Siman 225) also says a half hour.]
The Yalkut Yosef (vol 3, pg 634-6) writes that it’s preferable to hold like Shulchan Aruch not to eat a small or big meal after the sixth hour. However, the minhag for centuries was to be lenient and there is what to rely on. Nonetheless, even according to the lenient opinions there is no room to be lenient to have a big meal after the tenth hour.
Piskei Teshuvot 232:3 writes that if one always goes to minyan and there’s a set time, it’s permissible to eat, even after 9½ hours except having a feast such as a wedding or Brit Milah after 9 hours.
↑ Tur and S”A 232:3 define the meal that is forbidden as having bread more than a KeBaytzah. [See also Kesef Mishna (Tefillah 5:6) who gives another amount for this prohibition.]