אורח חיים קפ״וOrach Chayim 186

Se’if 1א׳

נָשִׁים חַיָּבוֹת בְּבִרְכַּת הַמָּזוֹן, וְסָפֵק הוּא אִם הֵן חַיָּבוֹת מִדְּאוֹרַיְיתָא וּמוֹצִיאוֹת אֶת הָאֲנָשִׁים, אוֹ אִם אֵינָן חַיָּבוֹת אֶלָּא מִדְּרַבָּנָן וְאֵינָן מוֹצִיאוֹת אֶלָּא לְמִי שֶׁאֵין חִיּוּבוֹ אֶלָּא מִדְּרַבָּנָן. וְעַיֵּן לְקַמָּן סי' קצ"ט בְּעִנְיַן זִמּוּן שֶׁלָּהֶם.

If Women and Children are obligated in Grace after Meals Women are obligated to recite Grace after Meals. A doubt exists whether they are obligated from the Torah, and thus are able to fulfill the obligation of men (who listen to their recitation), or if their obligation is merely Rabbinical in origin, and thus can only fulfill the obligation for someone whose obligation is likewise merely Rabbinical. [See siman 199 concerning the laws of zimmun for women.]

Se’if 2ב׳

קָטָן חַיָּב מִדְּרַבָּנָן, כְּדֵי לְחַנְּכוֹ; וְהַהִיא דְּבֵן מְבָרֵךְ לְאָבִיו, כְּשֶׁלֹּא אָכַל הָאָב כְּדֵי שְׂבִיעָה, שֶׁאֵינוֹ חַיָּב אֶלָּא מִדְּרַבָּנָן.

Children are obligation by Rabbinical law (to recite Grace after Meals), in order to educate them. This teaching that a son may recite (the Grace) on behalf of his father only applies when the father ate an amount less that the point of satisfaction, thus making his obligation merely Rabbinical in origin.