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as_o in_o the_o margin_n and_o the_o work_n be_v finish_v and_o i_o find_v the_o sum_n of_o the_o say_a number_n to_o amount_v to_o 132_o l._n 02_o oz._n 09_o p.w._n 22_o gr_n this_o be_v sufficient_a for_o the_o understanding_n of_o the_o follow_a example_n or_o any_o other_o that_o shall_v come_v to_o thy_o view_n the_o way_n of_o prove_v these_o or_o any_o sum_n in_o this_o rule_n be_v show_v immediate_o after_o the_o ensue_a example_n addition_n of_o english_a money_n l._n s._n d._n qr_n l._n s._n d._n qr_n 436_o 13_o 07_o 1_o 48_o 15_o 11_o 1_o 184_o 09_o 10_o 3_o 76_o 10_o 07_o 3_o 768_o 17_o 04_o 2_o 18_o 00_o 05_o 3_o 564_o 11_o 11_o 0_o 24_o 19_o 09_o 2_o 1954_o 12_o 09_o 2_o 168_o 06_o 10_o 1_o addition_n of_o troy_n weight_n l._n owned_a p.w._n gr_n l._n owned_a p.w._n gr_n 15_o 07_o 13_o 12_o 145_o 09_o 12_o 18_o 18_o 06_o 04_o 20_o 726_o 08_o 14_o 10_o 11_o 10_o 16_o 18_o 380_o 07_o 06_o 13_o 09_o 04_o 10_o 22_o 83_o 10_o 16_o 20_o 19_o 11_o 18_o 04_o 130_o 00_o 10_o 12_o 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12_o 1_o 0_o 56_o 57_o 3_o 22_o 3_o 21_o 1_o 1_o 18_o 17_o 0_o 00_o 0_o 133_o 1_o 1_o 60_o 166_o 1_o 26_o 2_o addition_n of_o dry_a measure_n chald._n qr_n bush_n pec_fw-la qr_n bush_n pec_fw-la gall_n 48_o 3_o 7_o 3_o 17_o 3_o 1_o 1_o 13_o 1_o 4_o 0_o 50_o 1_o 3_o 0_o 54_o 0_o 6_o 2_o 14_o 5_o 3_o 1_o 16_o 3_o 6_o 1_o 40_o 2_o 0_o 1_o 40_o 1_o 0_o 1_o 30_o 0_o 3_o 0_o 173_o 3_o 0_o 3_o 152_o 5_o 3_o 1_o addition_n of_o long_a measure_n yds_n qr_n na_fw-fr else_o qr_n na_fw-fr 35_o 3_o 3_o 56_o 1_o 3_o 14_o 1_o 2_o 13_o 3_o 2_o 74_o 2_o 3_o 48_o 2_o 1_o 48_o 0_o 1_o 50_o 1_o 0_o 30_o 1_o 0_o 74_o 0_o 2_o 15_o 0_o 0_o 17_o 1_o 0_o 218_o 1_o 1_o 260_o 2_o 0_o addition_n of_o land_n measure_n acre_n rood_n per._n acr._n rood_v perch_n 12_o 3_o 18_o 86_o 1_o 36_o 14_o 0_o 24_o 47_o 3_o 24_o 30_o 2_o 19_o 73_o 2_o 18_o 48_o 3_o 30_o 60_o 0_o 07_o 28_o 1_o 38_o 04_o 2_o 08_o 50_o 3_o 26_o 14_o 1_o 14_o 185_o 3_o 35_o 286_o 3_o 27_o the_o proof_n of_o addition_n 6._o addition_n be_v prove_v 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difference_n be_v 14_o i_o desire_v to_o know_v what_o be_v the_o great_a number_n and_o also_o what_o be_v the_o sum_n of_o they_o both_o first_o set_v down_o the_o least_o viz._n 40_o and_o 14_o the_o difference_n and_o add_v they_o together_o and_o their_o sum_n be_v 54_o for_o the_o great_a number_n than_o i_o set_v 40_o the_o least_o under_o 54_o the_o great_a and_o add_v they_o together_o and_o their_o sum_n be_v 94_o equal_a to_o the_o great_a and_o least_o number_n chap._n v._o of_o subtraction_n of_o whole_a number_n 1._o subtraction_n be_v the_o take_n of_o a_o lesser_a number_n out_o of_o a_o great_a of_o like_a kind_n whereby_o to_o find_v out_o a_o three_o number_n be_v or_o declare_v the_o inequality_n excess_n or_o difference_n between_o the_o number_n give_v or_o subtraction_n be_v that_o by_o which_o one_o number_n be_v take_v out_o of_o another_o number_n give_v to_o the_o end_n that_o the_o residue_n or_o remainder_n may_v be_v know_v which_o remainder_n be_v also_o 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i_o begin_v say_v seven_o time_n four_o be_v 28_o than_o i_o set_v down_o 8_o and_o carry_v 2_o then_o say_v 7_o time_n 6_o be_v 42_o and_o 2_o that_o i_o carry_v be_v 44_o that_o be_v 4_o and_o go_z 4_o then_o 7_o time_n 7_o be_v 49_o and_o 4_o that_o i_o carry_v be_v 53_o which_o i_o set_v down_o because_o i_o have_v not_o another_o figure_n to_o multiply_v thus_o have_v i_o do_v with_o the_o 7_o then_o i_o begin_v with_o the_o 2_o say_v 2_o time_n 4_o be_v 8_o which_o i_o set_v down_o under_o the_o 4_o the_o second_o figure_n or_o place_n of_o ten_o in_o the_o line_n above_o it_o as_o you_o may_v see_v in_o the_o margin_n then_o i_o proceed_v say_v 2_o time_n 6_o be_v 12_o that_o be_v 2_o and_o carry_v one_o than_o two_o time_n seven_o be_v fourteen_o and_o one_o that_o i_o carry_v be_v fifteen_o which_o i_o set_v down_o because_o it_o be_v the_o product_n of_o the_o last_o figure_n so_o that_o the_o product_n of_o 764_o by_o 7_o be_v 5348_o and_o by_o 2_o be_v â—528_n which_o be_v place_v the_o one_o under_o the_o other_o as_o before_o be_v direct_v and_o as_o you_o see_v in_o the_o margin_n and_o a_o line_n draw_v under_o they_o and_o they_o add_v together_o respective_o make_v 20628_o the_o true_a product_n require_v be_v equal_a to_o 27_o time_n 764._o another_o example_n may_v be_v this_o let_v it_o be_v require_v to_o multiply_v 5486_o by_o 465_o i_o dispose_v of_o the_o multiplicand_a and_o multiplier_n according_a to_o rule_n and_o begin_v multiply_v the_o first_o figure_n of_o the_o multiplier_n which_o be_v five_o into_o the_o whole_a multiplycand_n and_o the_o product_n be_v 27430_o then_o i_o proceed_v and_o multiply_v the_o second_o figure_n 6_o of_o the_o multiplier_n into_o the_o multiplicand_a and_o find_v the_o product_n to_o amount_v to_o 32916_o which_o be_v subscribe_v under_o the_o other_o product_v respectiu_o then_o do_v i_o multiply_v the_o three_o and_o last_o figure_n 4_o of_o the_o multiplier_n into_o the_o multiplicand_a and_o the_o product_n be_v 21944_o which_o be_v likewise_o place_v under_o the_o second_o line_n respective_o then_o i_o draw_v a_o line_n under_o the_o say_a product_n be_v place_v the_o one_o under_o the_o other_o according_a to_o rule_n and_o add_v they_o together_o and_o the_o sum_n be_v 2550990_o the_o true_a product_n seek_v be_v equal_a to_o 5486_o time_n 465_o or_o 465_o time_n 5486._o more_o example_n in_o this_o rule_n be_v these_o follow_v 7._o although_o the_o foorme_a rule_n be_v sufficient_a for_o all_o case_n in_o multiplication_n yet_o because_o in_o the_o work_n of_o multiplication_n many_o time_n great_a labour_n may_v be_v save_v 4.3_o i_o shall_v acquaint_v the_o learner_n therewith_o viz._n if_o the_o multiplicand_a or_o multiplier_n or_o both_o of_o they_o end_n with_o cipher_n then_o in_o your_o multiply_v you_o may_v neglect_v the_o cipher_n and_o multiply_v only_o the_o significant_a figure_n and_o to_o the_o product_n of_o those_o significant_a figure_n add_v so_o many_o cipher_n as_o the_o number_n give_v to_o be_v multiply_v do_v end_n with_o that_o be_v annex_v they_o on_o the_o right_a hand_n of_o the_o say_a product_n so_o shall_v that_o give_v you_o the_o true_a product_n require_v as_o if_o i_o be_v to_o multiply_v 32000_o by_o 4300_o i_o set_v they_o down_o in_o order_n to_o be_v multiply_v as_o you_o see_v in_o the_o margin_n but_o neglect_v the_o cipher_n in_o both_o number_n i_o only_o multiply_v 32_o by_o 43_o and_o the_o product_v i_o find_v to_o be_v 1376_o to_o which_o i_o annex_v the_o 5_o cipher_n that_o be_v in_o the_o multiplicand_a and_o multiplier_n and_o than_o it_o make_v 137600000_o for_o the_o true_a product_n of_o 32000_o by_o 4300._o 8._o if_o in_o the_o multiplier_n cipher_n be_v place_v between_o significant_a figure_n then_o multiply_v only_o by_o the_o significant_a figure_n neglect_v the_o cypher_n arithm._n but_o here_o special_a notice_n be_v to_o be_v take_v of_o the_o true_a place_n of_o the_o first_o figure_n after_o the_o neglect_n of_o such_o cypher_n or_o cipher_n and_o therefore_o you_o must_v observe_v in_o what_o place_n of_o the_o multiplier_n the_o figure_n you_o multiply_v by_o stand_v and_o set_v the_o first_o figure_n of_o that_o product_n under_o the_o same_o place_n of_o the_o product_n of_o the_o first_o figure_n of_o your_o multiplier_n as_o for_o example_n let_v it_o be_v require_v to_o multiply_v 371568_o by_o 40007_o first_o i_o multiply_v the_o multiplicand_v by_o seven_o and_o the_o product_n be_v 2600976_o then_o neglect_v the_o cipher_n i_o multiply_v by_o 4_o and_o that_o product_n be_v 1486272_o now_o i_o consider_v that_o four_o be_v the_o five_o figure_n in_o the_o multiplier_n therefore_o i_o place_v two_o the_o first_o figure_n of_o the_o product_n by_o four_o under_o the_o five_o place_n of_o the_o first_o product_n by_o seven_o and_o the_o rest_n in_o order_n and_o have_v add_v they_o together_o the_o total_a product_n be_v find_v to_o be_v 14865320976._o other_o example_n in_o this_o rule_n be_v these_o follow_v 9_o if_o you_o be_v to_o multiply_v any_o number_n by_o a_o unit_fw-la with_o cipher_n viz._n by_o 10_o 100_o 1000_o etc._n etc._n then_o prefix_v so_o many_o cipher_n before_o the_o multiplicand_a and_o that_o numbeâ—_n when_o the_o cipher_n be_v prefix_v be_v the_o proâ—duct_n require_v as_o if_o you_o will_v multiply_v 428_o by_o 100_o annex_v two_o cipher_n to_o 428_o and_o it_o be_v 42800_o if_o it_o be_v require_v tâ—_n multiply_v 102_o by_o 10000_o annex_v 4_o cypherâ—_n and_o it_o give_v 1020000_o for_o the_o product_v require_v the_o proof_n of_o multiplication_n 10._o multiplication_n be_v prove_v by_o division_n and_o to_o speak_v truth_n all_o other_o way_n be_v false_a and_o therefore_o it_o will_v be_v most_o convenient_a in_o the_o first_o place_n â—ris_n to_o learn_v division_n and_o by_o that_o to_o prove_v multiplication_n there_o be_v a_o way_n at_o this_o day_n general_o use_v in_o school_n to_o prove_v multiplication_n which_o be_v this_o first_o add_v all_o the_o figure_n in_o the_o multiplicand_a together_o as_o if_o they_o be_v simple_a number_n cast_v away_o the_o nine_o as_o often_o as_o it_o come_v to_o so_o much_o and_o note_v the_o remainder_n at_o last_o which_o in_o this_o case_n can_v be_v so_o much_o as_o 9_o cast_v likewise_o the_o nine_o out_o of_o the_o multiplier_n as_o you_o do_v out_o of_o the_o multiplicand_a and_o note_v that_o remainder_n then_o multiply_v the_o remainder_n the_o one_o by_o the_o other_o and_o cast_v the_o nine_o out_o of_o that_o product_n observe_v the_o remainder_n and_o last_o cast_v the_o nine_o out_o of_o the_o total_a product_n and_o if_o this_o remainder_n be_v equal_a to_o the_o remainder_n last_o find_v than_o they_o conclude_v the_o work_n to_o be_v right_o perform_v but_o there_o may_v be_v give_v a_o thousand_o nay_o infinite_a false_a product_n in_o a_o multiplication_n which_o after_o this_o manner_n may_v be_v prove_v to_o be_v true_a and_o therefore_o this_o way_n of_o prove_v do_v not_o deserve_v any_o example_n but_o we_o shall_v defer_v the_o proof_n of_o this_o rule_n till_o we_o come_v to_o prove_v division_n and_o then_o we_o shall_v prove_v they_o both_o together_o 11._o the_o general_a effect_n of_o multiplication_n be_v contain_v in_o the_o definition_n of_o the_o same_o which_o be_v to_o find_v out_o a_o three_o number_n so_o often_o contain_v one_o of_o the_o two_o give_v number_n as_o the_o other_o contain_v unit_fw-la the_o second_o effect_n be_v by_o have_v the_o length_n and_o breadth_n of_o any_o thing_n as_o a_o pararellogram_n or_o long_o plain_a to_o find_v the_o superficial_a content_n of_o the_o same_o and_o by_o have_v the_o superficial_a content_n of_o the_o base_a and_o the_o length_n to_o find_v the_o solidity_n of_o any_o parallelepipedon_n cylinder_n be_v other_o solid_a figure_n the_o three_o effect_n be_v by_o the_o content_n price_n value_n buy_v sell_v expense_n wage_n exchange_n simple_a interest_n gain_n or_o loss_n of_o any_o one_o thing_n be_v it_o money_n merchandise_n etc._n etc._n to_o find_v out_o the_o value_n price_n expense_n buy_v sell_v exchange_n or_o interest_n of_o any_o number_n of_o thing_n of_o like_a name_n nature_n and_o kind_a the_o four_o effect_n be_v not_o much_o unlike_o the_o other_o by_o the_o content_n value_n or_o price_n of_o one_o part_n of_o any_o thing_n denominate_v to_o find_v out_o the_o content_a value_n or_o price_n of_o the_o whole_a thing_n all_o the_o part_n into_o which_o the_o whole_a be_v divide_v multiply_v the_o price_n of_o one_o of_o those_o part_n the_o five_o effect_n be_v to_o aid_v to_o compound_v and_o to_o maâ—e_n 〈◊〉_d rule_n as_o chief_o the_o rule_n of_o 〈◊〉_d call_v thâ—_n golden_a rule_n 〈…〉_z
follow_v quest._n 1._o if_o 22_o thing_n cost_v 66_o shilling_n what_o will_v 1_o such_o like_a thing_n cost_v facit_fw-la 3_o shilling_n for_o if_o you_o divide_v 66_o by_o 22_o the_o quotient_n be_v 3_o for_o the_o answer_n so_o if_o 36_o yard_n or_o ell_n of_o any_o thing_n be_v buy_v or_o sell_v for_o 108_o l._n how_o much_o shall_v 1_o yard_n or_o ell_n be_v buy_v or_o sell_v for_o facit_fw-la 3_o l._n for_o if_o you_o divide_v 108_o l._n by_o 36_o yard_n the_o quotient_n will_v be_v 3_o l._n the_o price_n of_o the_o integer_fw-la quest._n 2._o if_o the_o expense_n charge_n or_o wage_n of_o 7_o year_n amount_n to_o 868_o l._n what_o be_v the_o expense_n charge_n or_o wage_n of_o one_o year_n facit_fw-la 124_o l._n for_o if_o you_o divide_v 868_o the_o wage_n of_o 7_o year_n by_o 7_o the_o number_n of_o year_n the_o quotient_n will_v be_v 124_o l._n for_o the_o answer_n see_v the_o work_n quest._n 3._o if_o the_o content_a of_o a_o superficial_a foot_n be_v 144_o inch_n and_o the_o breadth_n of_o a_o board_n be_v 9_o inch_n how_o many_o inch_n of_o that_o board_n in_o length_n will_v make_v such_o a_o foot_n facit_fw-la 16_o inch_n for_o by_o divide_v 144_o the_o number_n of_o square_a inch_n in_o a_o square_a foot_n by_o 9_o the_o inch_n in_o the_o breadth_n of_o the_o board_n the_o quotient_a be_v 16_o for_o the_o number_n of_o inch_n in_o length_n of_o that_o board_n to_o make_v a_o superficial_a foot_n quest._n 3._o if_o the_o content_a of_o a_o acre_n of_o ground_n be_v 160_o square_a perch_n and_o the_o length_n of_o a_o furlong_n propound_v be_v 80_o perch_n how_o many_o perch_n will_v there_o go_v in_o breadth_n to_o make_v a_o acre_n facit_fw-la 2_o perch_n for_o if_o you_o divide_v 160_o the_o number_n of_o perch_n in_o a_o acre_n by_o 80_o the_o length_n of_o the_o furlong_n in_o perch_n the_o quotient_a be_v 2_o perch_n and_o so_o many_o in_o breadth_n of_o that_o furlong_n will_v make_v a_o acre_n quest._n 5._o if_o there_o be_v 893_o man_n to_o be_v make_v up_o into_o a_o battle_n the_o front_n consist_v of_o 47_o man_n what_o number_n must_v there_o be_v in_o the_o file_n facit_fw-la 19_o deep_a in_o the_o file_n for_o if_o you_o divide_v 893_o the_o number_n of_o man_n by_o 47_o the_o number_n in_o front_n the_o quotient_n will_v be_v 19_o the_o file_n in_o depth_n the_o work_n follow_v quest._n 6._o there_o be_v a_o table_n who_o superficial_a content_n be_v 72_o foot_n and_o the_o breadth_n of_o it_o at_o the_o end_n be_v 3_o foot_n now_o i_o demand_v what_o be_v the_o length_n of_o this_o table_n facit_fw-la 24_o foot_n long_o for_o if_o you_o divide_v 72_o the_o content_a of_o the_o table_n in_o foot_n by_o 3_o the_o breadth_n of_o it_o the_o quotient_a be_v 24_o foot_n for_o the_o length_n thereof_o which_o be_v require_v see_v the_o operation_n as_o follow_v the_o proof_n of_o multiplication_n and_o division_n multiplication_n and_o division_n interchangeable_o prove_v each_o other_o for_o if_o you_o will_v prove_v a_o sum_n in_o division_n whether_o the_o operation_n be_v right_o or_o no_o multiply_v the_o quotient_a by_o the_o divisor_n and_o if_o any_o thing_n remain_v after_o the_o division_n be_v end_v add_v it_o to_o the_o product_n which_o product_n if_o your_o sum_n be_v right_o divide_v will_v be_v equal_a to_o the_o dividend_n and_o contrariwise_o if_o you_o will_v prove_v a_o sum_n in_o multiplication_n divide_v the_o product_n by_o the_o multipliar_a and_o if_o the_o work_n be_v right_o perform_v the_o quotient_n will_v be_v equal_a to_o the_o multiplicand_a see_v the_o example_n where_o the_o work_n be_v do_v and_o undo_v let_v 7654_o be_v give_v to_o be_v multiply_v by_o 3242_o the_o product_n will_v be_v 24814268_o as_o by_o the_o work_n appear_v and_o than_o if_o you_o divide_v the_o say_a product_n 24814268_o by_o 3242_o the_o multipliar_a the_o quotient_n will_v be_v 7654_o equal_a to_o the_o give_v multiplicand_v in_o like_a manner_n to_o prove_v a_o sum_n or_o number_n in_o division_n if_o 24814268_o be_v divide_v by_o 3242_o the_o quotient_n will_v be_v find_v to_o be_v 7654_o then_o for_o proof_n if_o you_o multiply_v 7654_o the_o quotient_a by_o 3242_o the_o divisor_n the_o product_n will_v amount_v to_o 24814268_o equal_a to_o the_o dividend_n or_o you_o may_v prove_v the_o last_o or_o any_o other_o example_n in_o multiplication_n thus_o viz._n divide_v the_o product_n by_o the_o multiplicand_a and_o the_o quotient_n will_v be_v equal_a to_o the_o multipliar_a see_v the_o work_n from_o whence_o arise_v this_o corollary_n that_o any_o operation_n in_o division_n may_v be_v prove_v by_o division_n for_o if_o after_o your_o division_n be_v end_v you_o divide_v the_o dividend_n by_o the_o quotient_a the_o new_a quotient_n thence_o arise_v will_v be_v equal_a to_o the_o divisor_n of_o the_o first_o operation_n for_o trial_n whereof_o let_v the_o last_o example_n be_v again_o repeat_v for_o proof_n whereof_o divide_v again_o 24814268_o by_o the_o quotient_a 7654_o and_o the_o quotient_a thence_o will_v be_v equal_a to_o the_o first_o divisor_n 3242_o see_v the_o work_n but_o in_o prove_v division_n by_o division_n the_o learner_n be_v to_o observe_v this_o follow_a caution_n that_o if_o after_o his_o division_n be_v end_v there_o be_v any_o remainder_n before_o you_o go_v about_o to_o prove_v your_o work_n subtract_v that_o remainder_n out_o of_o your_o dividend_n and_o then_o work_v as_o before_o as_o in_o the_o follow_a example_n where_o it_o be_v require_v to_o divide_v 43876_o by_o 765_o the_o quotient_a here_o be_v 57_o and_o the_o remainder_n be_v 271_o see_v the_o work_n follow_v now_o to_o prove_v this_o work_n subtract_v the_o remainder_n 271_o out_o of_o the_o dividend_n 43876_o and_o there_o remain_v 43605_o for_o a_o new_a dividend_n to_o be_v divide_v by_o the_o former_a quotient_a 57_o and_o the_o quotient_a thence_o arise_v be_v 765_o equal_a to_o the_o give_v divisor_n which_o prove_v the_o operation_n to_o be_v right_o thus_o have_v we_o go_v through_o the_o four_o species_n of_o arithmetic_n viz._n addition_n 1._o subtraction_n multiplication_n and_o division_n upon_o which_o all_o the_o follow_a rule_n and_o all_o other_o operation_n whatsoever_o that_o be_v possible_a to_o be_v wrought_v by_o number_n have_v their_o immediate_a dependence_n and_o by_o they_o be_v resolve_v therefore_o before_o the_o learner_n make_v a_o further_a step_n in_o this_o art_n let_v he_o be_v well_o acquaint_v with_o what_o have_v be_v deliver_v in_o the_o forego_n chapter_n chap._n viii_o of_o reduction_n 1._o reduction_n be_v that_o which_o bring_v together_o 2_o or_o more_o number_n of_o different_a denomination_n into_o one_o denomination_n or_o it_o serve_v to_o change_v or_o alter_v number_n 13.152_o money_n weight_n measure_n or_o time_n from_o one_o denomination_n to_o another_o and_o likewise_o to_o abridge_v fraction_n to_o their_o low_a term_n all_o which_o it_o do_v so_o precise_o that_o the_o first_o proportion_n remain_v without_o the_o least_o jot_n of_o error_n or_o wrong_n commit_v so_o that_o it_o belong_v as_o well_o to_o fraction_n as_o integer_n of_o which_o in_o its_o proper_a place_n reduction_n be_v general_o perform_v either_o by_o multiplication_n or_o division_n from_o whence_o we_o may_v gather_v that_o 2_o reduction_n be_v either_o descend_v or_o ascend_v 3._o reduction_n descend_v be_v when_o it_o be_v require_v to_o reduce_v a_o sum_n or_o number_n of_o a_o great_a denomination_n into_o a_o lesser_a which_o number_n when_o it_o be_v so_o reduce_v shall_v be_v equal_a in_o value_n to_o the_o number_n first_o give_v in_o the_o great_a denomination_n as_o if_o it_o be_v require_v to_o know_v how_o many_o shilling_n 4._o penny_n or_o farthing_n be_v equal_a in_o value_n to_o a_o hundred_o pound_n or_o how_o many_o ounce_n be_v contain_v in_o 45_o hundred_o weight_n or_o how_o many_o day_n hour_n or_o minute_n there_o be_v in_o 240_o year_n etc._n etc._n and_o this_o 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