In the spirit of the song "Echad - mi yodeya", please post interesting and significant Jewish facts about the number ⅓.
Lazy gematria may be worthy of discission, but perhaps not of a full-fledged answer.
Check out mi-yodeya-series for the previous three hundred forty nine entries in this ongoing series.
Please include sources for your information wherever possible, as with all other answers on this site.
I did not find this number asked anywhere.
The Gemoro in Kiddushin 30a writes:
לעולם ישלש אדם שנותיו שליש במקרא שליש במשנה שליש בתלמוד
A person should always divide his years into three parts: A third for Torah, a third for Mishna, and a third for Talmud.
The Zman Tefila for Shacharis is (four hours) or a third of the day.
The Gemoro in Bava Metzia 42a writes:
וא"ר יצחק לעולם ישליש אדם את מעותיו שליש בקרקע ושליש בפרקמטיא ושליש תחת ידו
And Rabbi Yitzcḥak says: A person should always divide his money into three; he should bury one-third in the ground, and invest one-third in business [bifrakmatya], and keep one-third in his possession.
The Gemoro in Shabbos 20a talks about how bishul takes effect from the moment that a solid food becomes edible. The Gemoro talks about Ben Drosai who was a robber who didn't have time for his food to be completely cooked:
גְּמָ׳ וְכַמָּה? אָמַר רַבִּי אֶלְעָזָר אָמַר רַב: כְּדֵי שֶׁיִּצּוֹלוּ מִבְּעוֹד יוֹם כְּמַאֲכָל בֶּן דְּרוֹסַאי. אִיתְּמַר נָמֵי, אָמַר רַב אַסִּי אָמַר רַבִּי יוֹחָנָן: כׇּל שֶׁהוּא כְּמַאֲכָל בֶּן דְּרוֹסַאי, אֵין בּוֹ מִשּׁוּם בִּישּׁוּלֵי גוֹיִם. תַּנְיָא חֲנַנְיָא אוֹמֵר: כׇּל שֶׁהוּא כְּמַאֲכָל בֶּן דְּרוֹסַאי — מוּתָּר לְהַשְׁהוֹתוֹ עַל גַּבֵּי כִּירָה וְאַף עַל פִּי שֶׁאֵין גְּרוּפָה וּקְטוּמָה.
GEMARA: We learned in the mishna that one may only roast meat and other food items if there remains sufficient time so that they could be roasted while it is still day. The Gemara asks: And how much do they need to be roasted in order to be considered sufficient, so that it will be permitted to complete their cooking afterward? Rabbi Elazar said that Rav said: So that they will be roasted while it is still day like the food of ben Drosai, which was partially roasted. Ben Drosai was a robber and pursued by all. He could not wait for his food to roast completely, so he sufficed with a partial roasting. It was also stated by another of the Sages, as Rav Asi said that Rabbi Yoḥanan said: Anything that is already cooked like the food of ben Drosai by a Jew, no longer has a problem of the cooking of gentiles. If a gentile completed cooking this food, it is, nevertheless, permitted to eat, even though, as a rule, it is prohibited to eat food cooked by gentiles. It was taught in a baraita, Ḥananya says: With regard to anything that is already cooked like the food of ben Drosai, it is permitted to keep it on the stove on Shabbat and even though this stove is not swept of coals and the burning coals are not covered with ashes. Since the food was already cooked to that extent, there is no concern that he will come to stoke the coals. (Sefaria translation and added annotation)
Rashi there states:
בן דרוסאי - לסטים היה ומבשל בישולו שליש
Ben Drosai - he was a robber and he would cook his food a third
I.e. since he was in such a hurry his food would only be one-third cooked.
The mishna in Ma'asrot (1:3), when talking about how ripe different things need to be before ma'aser can be taken from them, says that ma'aser can be taken from olives and grains once they are a third of their full ripeness:
הֶחָרוּבִין, מִשֶּׁיִּנָּקֵדוּ. וְכָל הַשְּׁחוֹרִים, מִשֶּׁיִּנָּקְדוּ. הָאֲגָסִים וְהַקְּרֻסְטוֹמֵלִין וְהַפָּרִישִׁין וְהָעֻזְרָדִים, מִשֶּׁיִּקָּרֵחוּ. וְכָל הַלְּבָנִים, מִשֶּׁיִּקָּרֵחוּ. הַתִּלְתָּן, מִשֶּׁתְּצַמֵּחַ. הַתְּבוּאָה וְהַזֵּיתִים, מִשֶּׁיַּכְנִיסוּ שְׁלִישׁ:
Carobs [are liable to] tithes after they form dark spots... Grain and olives after they are one-third ripe.
The Rambam says:
The day and the night consist of twenty-four hours. It is enough that a man should sleep a third of it, which is eight hours. [Mishneh Torah, Human Dispositions 4:4]
Sforno on Genesis 4:15:
God decreed therefore that when someone killed half the human race, or at least a third of the human race, the appropriate vengeance would involve a twice sevenfold penalty.
הידור מצוה עד שליש בב"ק ט
There's a Mitzvah to 'beautify' or enhance a Mitzvah like Tallis, Tzitzis & Arba Minim (Examples are from Rashi there) up to a third over original costs.
Ram sacrifices had 1/3 of a Hin of oil in the meal offering and 1/3 of a Hin as a libation. From Bamidbar 15:6-7
אוֹ לָאַיִל תַּעֲשֶׂה מִנְחָה, סֹלֶת שְׁנֵי עֶשְׂרֹנִים, בְּלוּלָה בַשֶּׁמֶן, שְׁלִשִׁית הַהִין. וְיַיִן לַנֶּסֶךְ, שְׁלִשִׁית הַהִין--תַּקְרִיב רֵיחַ-נִיחֹחַ, לַיהוָה.
In the case of a ram, you shall present as a meal offering: two-tenths of a measure of choice flour with a third of a hin of oil mixed in; and a third of a hin of wine as a libation—as an offering of pleasing odor to the LORD.
וישאהו במוט בשנים. מִמַּשְׁמַע שֶׁנֶּאֱמַר "וַיִּשָּׂאֻהוּ בַמּוֹט" אֵינִי יוֹדֵעַ שֶׁהוּא בִשְׁנַיִם? מַה תַּ"ל "בִּשְׁנָיִם"? בִּשְׁנֵי מוֹטוֹת. הָא כֵיצַד? שְׁמוֹנָה נָטְלוּ אֶשְׁכּוֹל, אֶחָד נָטַל תְּאֵנָה וְאֶחָד רִמּוֹן; יְהוֹשֻׁעַ וְכָלֵב לֹא נָטְלוּ כְלוּם, לְפִי שֶׁכָּל עַצְמָם לְהוֹצִיא דִּבָּה נִתְכַּוְּנוּ — כְּשֵׁם שֶׁפִּרְיָהּ מְשֻׁנֶּה כָּךְ עַמָּהּ מְשֻׁנֶּה; וְאִם חָפֵץ אַתָּה לֵידַע כַּמָּה מַשּׂאוֹי אֶחָד מֵהֶם, צֵא וּלְמַד מֵאֲבָנִים שֶׁהֵקִימוּ בַּגִּלְגָּל — הֵרִימוּ לָהֶם אִיש אֶבֶן אַחַת מִן הַיַּרְדֵן עַל שִׁכְמוֹ וֶהֱקִימוּהָ בַּגִּלְגָּל (יהושע ד') — וּשְׁקָלוּם רַבּוֹתֵינוּ מִשְׁקַל כָּל אַחַת אַרְבָּעִים סְאָה, וּגְמִירֵי טוּנָא דְּמִידְלֵי אֱינַשׁ עַל כַּתְפֵּיהּ אֵינוֹ אֶלָּא שְׁלִישׁ מַשּׂאוֹי כַּמַּשּׂאוֹי שֶׁמְּסַיְּעִין אוֹתוֹ לְהָרִים (סוטה ל"ד): וישאהו במוט בשנים AND THEY BARE IT UPON A POLE BETWEEN TWO — From what is implied in the statement “And they bare it upon a pole”, do I not know that it was carried by two men”? why then does it add the word בשנים? It means, not as you assume, “by two men” but “upon two poles”. How was that done? Eight of the spies bore the cluster of grapes. Besides these, one spy took the fig and one the pomegranate, but Joshua and Caleb did not take anything (although all of them were bidden to do so; cf. v. 20), because the very essence of their (the other spies’) intention was only to bring an evil report: “Just as its fruit is extraordinary in size, so is its people extraordinary in size”. — Now if you wish to know how much was the burden of each of them, go and learn from the weight of the stones which they erected at Gilgal; for it is related (Joshua 4:5 and 20) that each man lifted by himself one stone from out of the Jordan on to his shoulder, and set it up at Gilgal; — and our Rabbis ascertained the weight of each stone to be forty Seahs, and it is an accepted fact that a load which a person can by himself lift up on to his shoulder is only the third of the load he can carry if people assist him in lifting it up (Sotah 34b).
Here you have 1, 1/2, 1/3 and 1/4:
And in our days, when it is common to lease a bath house for a year, a half-year, a third, or a quarter, as a field (is leased), it is permitted to rent a bath house to a gentile. [Bartenura on Mishnah Avodah Zarah 1:9]
It is a biblical estimate for 1/π - the ratio between the diameter of a circle and its circumference:
וַיַּעַשׂ אֶת-הַיָּם, מוּצָק: עֶשֶׂר בָּאַמָּה מִשְּׂפָתוֹ עַד-שְׂפָתוֹ עָגֹל סָבִיב, וְחָמֵשׁ בָּאַמָּה קוֹמָתוֹ, וקוה (וְקָו) שְׁלֹשִׁים בָּאַמָּה, יָסֹב אֹתוֹ סָבִיב.
Kings I chapter 7, verse 23.
"And he [Hiram] made a molten sea, ten cubits from the one rim to the other it was round all about, and...a line of thirty cubits did compass it round about...." — First Kings, chapter 7, verses 23 and 26
d=10, C=30; d/C = 1/3 according to this verse. In reality, 1/π is closer to 10/31.41, so it should have said "thirty-one cubits" at least.
וְזֵיתוֹם הַמִּצְרִי וְכוּ׳. מַאי זֵיתוֹם הַמִּצְרִי? תָּנָא רַב יוֹסֵף: תְּלָתָא שְׂעָרֵי, תְּלָתָא קוּרְטְמֵי, וּתְלָתָא מִלְחָא
And Egyptian Zeitom - What is an Egyptian Zeitom? Rav Yosef taught from a baraita: It is one-third barley, one-third safflower, and one-third salt.
