The term half does not always refer to 50%. When someone is called a half-wit, the intent is not that he has half the intelligence of a regular human. The term is merely use to convey the speaker’s opinion that that the subject of the insult has less than 100% of normal intelligence. When a store has a half-off sale, very often the consumer discovers that in actuality the products are only discounted 20%. If you complain to the manager he will explain that the half-off moniker refers to the never-used and theoretical manufacturer’s suggested retail price (MSRP). Finally, on Pesach we supposedly break the middle matzah in half, but who is able to produce two perfect halves? Machine matzah can be divided on the perfectly placed perforations, but with hand-matzah you have no such luck.
On chol ha’moed and the last days of yom tov of Pesach, we recite half-Hallel. Half-Hallel is more like 85% of Hallel. We just cut out 22 pesukim from the complete Hallel. Once again the term “half” is not used literally. Most likely the term “half-Hallel” originates from the fact that the two paragraphs that are skipped are pretty much the first half of two chapters in Tehillim. When we recite half-Hallel we don’t say the first 11 pesukim of kepittel 115 and 116, but we do recite the remainder. The import of half-Hallel is that it is the Hallel that contains some half-chapters, not that it is half of the original Hallel.
Why do we recite half-Hallel on the last days of yom tov on Pesach? The answer given in the Talmud (Arachin 10a–b) is that after the first day of Pesach, the Mussaf sacrifices are identical each day, whereas on Sukkos a unique Mussaf sacrifice is offered each day. The Mussaf sacrifices are indicative of the character of the yom tov. By mandating the same sacrifice for each day of Pesach, the Torah is teaching us that the character of each day remains the same throughout the yom tov. However, the varied sacrifices of Sukkos reveal to us that each day of the holiday is different from the others.
Therefore, in honor of the special meaning of each day of Sukkos, represented by different sacrifices, a complete Hallel is recited. The latter days of Pesach do not have this special characteristic; consequently, only half-Hallel is recited. (This answer was explained by Rabbi Ari Z. Zivotofsky, Ph.D., in Jewish Action.)
When the Mishnah Berurah explains why only half-Hallel is recited, he offers a totally different reason. His source is a Midrash quoted in the Beis Yosef (O.C. 490). That Midrash references a passage found in Sanhedrin (39b). The Gemara says that the angels wished to recite Shirah after the Egyptians drowned in the Yam Suf but Hashem stopped them. Hashem said, “My creations are drowning in the sea and you are reciting Shirah?” The Midrash explains that just as the angels did not recite Hallel on the seventh day of Pesach (at night), we also do not recite Hallel on the seventh day. The first obvious problem with this Midrash is that it fails to explain why only half-Hallel is recited on chol ha’moed. After all, the Egyptians drowned on the seventh day, not on chol ha’moed. Some Acharonim explain that it would be inappropriate to recite full Hallel on chol ha’moed and only half-Hallel on the seventh day, which is a full-fledged yom tov. That would seemingly lend more importance to chol ha’moed than to yom tov. Once we instituted that only half-Hallel should be recited on the seventh day, perforce Hallel had to be reduced on chol ha’moed as well.
However, why do we need an alternative explanation for the recitation of only half-Hallel in addition to the one already mentioned in the Gemara Arachin? The Kollel Iyun HaDaf offered the following answer in their publication Thoughts on the Daily Daf on Arachin. There are two basic reasons for reciting Hallel. First, we recite Hallel on festival days in order to praise Hashem as we celebrate His festival. Second, we recite Hallel in order to commemorate a miraculous salvation from danger.
The Gemara in Arachin is asking why we do not recite the full Hallel on all of the days of Pesach because of the first reason, since each day is called a mo’ed. The Gemara answers that the ensuing days of Pesach are not considered independent mo’adim, as we see from the fact that each day does not have its own unique korban (as does each day of Sukkos). The Midrash, however, is addressing a different question: why do we not recite the complete Hallel on the last day of Pesach because of the miraculous salvation (Kriyas Yam Suf) that occurred on that day (the second reason)? The Midrash answers that since some of Hashem’s creations were destroyed by this miracle, it is not fitting to recite Hallel to commemorate such a salvation. This clearly answers our first question as well. The reason mentioned in the Midrash was only meant to explain why we don’t complete Hallel on the seventh day of Pesach, not chol ha’moed. The complete Hallel is not recited on chol ha’moed because there was no significant miracle that occurred on those days and their yom tov status does not require Hallel as explained in the Gemara in Arachin.
Some questions still remain. If we are not supposed to recite Hallel or Shirah on the day that the Egyptians drowned, why did the B’nei Yisrael during the year of Yetzias Mitzrayim sing Az Yashir on the seventh day? (The Gemara in Pesachim states according to Rabbi Elazar that they recited Hallel as well!) Furthermore, if we really are not supposed to sing Shirah on the seventh day of Pesach, we should eliminate Hallel altogether, not just subtract a few pesukim. Lastly, the Gemara states in Megillah that Mordechai kicked Haman as he was stepping on him to get onto the royal horse. Haman asked Mordechai, “Doesn’t your Torah say, ‘Don’t rejoice at the downfall of your enemy’?” Mordechai responded, “That is only for one’s Jewish enemies.” So why should we curtail our recital of Hallel at all because of the drowning of the gentile Egyptians?
We are forced to conclude that in fact we really should recite the full Hallel on the seventh day in commemoration of the great miracle Hashem performed for us. After Kriyas Yam Suf, the Jews recited Shirah. This is because we as B’nei Yisrael have a special dispensation to rejoice at the downfall of our non-Jewish enemies. No such exception, however, is extended to the malachim, who are not members of the Jewish nation. They could not sing Shirah because Hashem’s creations were drowning. To commemorate the occurrence that the malachim did not sing Shirah and to remind us of the accompanying moral lesson, we curtail our recitation of Hallel on the seventh day somewhat by eliminating some pesukim. We are not mourning the death of the Egyptians who wanted to kill us; rather we are signifying that the malachim did not sing Shirah and that Hashem shows compassion for all of His creations. Therefore, half-Hallel on Pesach should be recited with complete joy and exuberance and not in a half-hearted manner. You may then go home and enjoy your well-done matzos that are not half-baked.

Rabbi Sebrow leads a daf yomi chaburah at Eitz Chayim of Dogwood Park in West Hempstead and is a rebbi at Yeshiva Tiferes Torah in Kew Gardens for post-high-school boys. He can be contacted at ASebrow@gmail.com.