ใบความรท 6 โจทยปญหาระบบสมการเชงเสน1

การแกโจทยปญหาทางคณตศาสตรทด าเนนการอยางเปนระบบ จ าเปนตองด าเนนการตามขนตอน เพอใหไดมาซงค าตอบทถกตอง โดยสรปม 4 ขนตอน ประกอบดวย 1) ขนท าความเขาใจปญหา 2) ขนวางแผนแกปญหา 3) ขนด าเนนการตามแผน และ 4) ขนตรวจสอบค าตอบ จะตองด าเนนการใหครบทกขนตอนซงจะไดค าตอบทถกตองและกระบวนการท างานทเปนระบบ และในใบความรน กลาวถงการแกโจทยปญหาเกยวกบอตราสวนและรอยละ จ าเปนอยางทนกเรยนจะตองมความรพนฐานทเพยงพอ

อตราสวน

รอยละ

ใบความรท 3.2 โจทยปญหาระบบสมการเชงเสน (ตอ)

รอยละ

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สโรชาตองการผสมน าเกลอชนดทมความเขมขน 80% กบน าเกลอชนดทมความเขมขน 40% ใหไดน าเกลอทมความเขมขน 60% จ านวน 20 ลตร สโรชาจะตองใชน าเกลอแตละชนดเปนจ านวนเทาไร วธท า โจทยตองการ สโรชาจะตองใชน าเกลอแตละชนดเปนจ านวนเทาไร โจทยก าหนด สโรชาตองการผสมน าเกลอชนดทมความเขมขน 80% กบน าเกลอชนดทมความเขมขน 40% ใหไดน าเกลอทมความเขมขน 60% จ านวน 20 ลตร

ให x แทนจ านวนน าเกลอชนดทมความเขมขน 80% y แทนจ านวนน าเกลอชนดทมความเขมขน 40%

สโรชาตองการผสมน าเกลอชนดทมความเขมขน 80% กบน าเกลอชนดทมความเขมขน 40%

ไดน าเกลอทมความเขมขน 60% )yx(10060

y10040

x10080

หรอ 0yx

จ านวน 20 ลตร 20yx 0yx

20yx

200x2

จาก 20x2 สมการท 1 + สมการท 2

10x น า 2 มาหารทงสองขาง

และ 20yx สมการท 2 20y10 แทน 10x 10y น า10 ลบทงสองขาง น าเกลอชนดทมความเขมขน 80% จ านวน 10 ลตร และน าเกลอชนดทมความเขมขน 40% จ านวน 10 ลตร

ตรวจสอบ ผสมน าเกลอชนดทมความเขมขน 80% กบน าเกลอชนดทมความเขมขน 40% อยางละ 10 ลตรไดน าเกลอทมความเขมขน 60% จรง น าเกลอชนดทมความเขมขน 80% ม 10 ลตรน าเกลอชนดทมความเขมขน 60% ม 10 ลตรรวมเปน 20 ลตร จรง

ดงนน

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ซอถานมาสองชนดราคาถงละ 18 บาท และ 27 บาท เอามาคละกนขายไป ถงละ 24 บาท ไดก าไร 20% อยากทราบอตราสวนในการผสม

วธท า โจทยตองการ อตราสวนในการผสม โจทยก าหนด ซอถานมาสองชนดราคาถงละ 18 บาท และ 27 บาท เอามาคละกนขายไปถงละ 24 บาท ไดก าไร 20%

ให x แทนจ านวนถานชนดราคาถงละ 18 บาท y แทนจ านวนถานชนดราคาถงละ 27 บาท

คดเปนตนทนทงหมด y27x18 คละกนขายไปถงละ 24 บาท ไดก าไร 20%

คดเปนตนทนหลงผสม )yx(20)yx(24120100

ตนทนกอนผสมถานเทากบตนทนหลงผสมถาน จะได )yx(20y27x18

y20x20y27x18 สมบตการแจกแจง y20x2y27 น า 18x

มาบวกทงสองขาง

x2y7 น า 20y มาบวกทงสองขาง

xy27 น า 2

มาหารทงสองขาง

yx

27 น า y

มาหารทงสองขาง

อตราสวนในการผสมถานชนดราคาถงละ 18 บาท ตอถานชนดราคาถงละ 27 บาท เปน 7 : 2

ตรวจสอบ ใหผสมถานชนดราคาถงละ 18 บาท จ านวน 7 ถง และถานชนดราคาถงละ 27 บาท จ านวน 2 ถง คดเปนตนทน 180)2(27)7(18 บาท ผสมแลวไดถานทงหมด 9 ถง น าไปขาย

ถงละ 24 บาท ไดเงนทงหมด 216 บาท ไดก าไร 36 บาท คดเปนไดก าไร %2010018036

ดงนน

36

แบบฝกท 10 โจทยปญหาระบบสมการเชงเสน5 1. จงแกโจทยปญหาระบบสมการเชงเสนตอไปน (ตวอยางท 1) 1) เหลาชนดหนงมแอลกอฮอล 30% ผสมกบเหลาทมแอลกอฮอล 20% ตองการใหไดเหลาทมแอลกอฮอล 28% จ านวน 50 ขวด จะตองใชเหลาทมแอลกอฮอล 30% กขวด วธท า ขนท 1 ............................................................................................................................... ............................................................................................................................................................. ............................................................................................................................................................. ขนท 2................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 3................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 4................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. .............................................................................................................................................................

แบบฝกทกษะท 3.2 โจทยปญหาระบบสมการเชงเสน

37

2) สถตยตองการผสมน าเกลอชนดทมความเขมขน 80% กบน าเกลอชนดทมความเขมขน 40% ใหไดน าเกลอทมความเขมขน 60% จ านวน 40 ลตร สถตจะตองใชน าเกลอแตละชนดเปนจ านวนเทาไร วธท า ขนท 1 ............................................................................................................................... ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 2................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 3................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 4................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. .............................................................................................................................................................

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3) น าเชอมสองชนดมน าตาลอย 4% และ 8% ตองน าน าเชอมชนดแรกมากลตร เมอรวมกบน าเชอมชนดทสองแลวไดน าเชอม 40 ลตร และมน าตาลอย 5% วธท า ขนท 1 ............................................................................................................................... ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 2................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 3................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 4................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. .............................................................................................................................................................

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4) ทกษณมน ายากรดบอรกเขมขน 30% และน ายากรดบอรกเขมขน 60% ทกษณจะตองผสมน ายาทงสองชนดอยางละกออนซ จงจะไดน ายากรดบอรกเขมขน 50% จ านวน 60 ออนซ วธท า ขนท 1 ............................................................................................................................... ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 2................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 3................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 4................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. .............................................................................................................................................................

40

5) แมคาซอของเลนชนดท 1 มา 20 ชน และชนดท 2 มา 25 ชน เปนเงน 4,500 บาท ถาขายของเลนชนดท 1 ไดก าไร 15 % ขายของเลนชนดท 2 ไดก าไร 20 % ไดก าไรทงหมด 750 บาท อยากทราบวาแมคาซอของเลนชนดท 1 มาราคาชนละเทาไร วธท า ขนท 1 ............................................................................................................................... ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 2................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 3................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 4................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. .............................................................................................................................................................

41

6) รตนมเงนเกบอย 120,000 บาท สวนหนงน าไปฝากธนาคารเพอรบดอกเบย 2 % อกสวนหนงน าไปลงทนเพอรบเงนปนผล 4 % สนปมรายไดรวมกน 4,000 บาท อยากทราบวารตนน าเงนไปฝากธนาคารจ านวนเทาใด วธท า ขนท 1 ............................................................................................................................... ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 2................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 3................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 4................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. .............................................................................................................................................................

42

7) แมคาซอสมมา 25 กโลกรม กบองน 30 กโลกรม เปนเงน 2,460 บาท ถาขายสมไปทงหมดในราคาทไดก าไร 10% และขายองนไปทงหมดในราคาทขาดทน 5% ปรากฏวายงมก าไร 102 บาท อยากทราบวาแมคาซอสมราคากโลกรมละเทาใด วธท า ขนท 1 ............................................................................................................................... ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 2................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 3................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 4................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. .............................................................................................................................................................

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8) แมคาซอผาลกไม 20 เมตร และผาไหม 25 เมตร เปนเงน 4,500 บาท ถาขายผาลกไมไดก าไร 15 % ขายผาไหมไดก าไร 20 % ท าใหไดก าไรทงหมด 750 บาท แลวแมคาซอผาไหมเมตรละเทาไร วธท า ขนท 1 ............................................................................................................................... ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 2................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 3................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 4................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. .............................................................................................................................................................

44

2. จงแกโจทยปญหาระบบสมการเชงเสนตอไปน (ตวอยางท 2) 1) ตองผสมน าตาลทรายชนดกโลกรมละ 12 บาท และกโลกรมละ 14 บาท ในอตราสวนเทาไรจงจะขายน าตาลทรายผสมไปราคากโลกรมละ 15 บาท แลวไดก าไร 20% วธท า ขนท 1 ............................................................................................................................... ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 2................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 3................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 4................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. .............................................................................................................................................................

45

2) ตองการผสมน าตาลทรายสองชนดราคากโลกรมละ 10 บาท และกโลกรมละ 15 บาท แลวขายน าตาลทรายผสมไปกโลกรมละ16 บาท จะไดก าไร20% จงหาอตราสวนในการผสมน าตาลทรายทงสองชนดน วธท า ขนท 1 ............................................................................................................................... ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 2................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 3................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 4................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. .............................................................................................................................................................

46

3) ผสมแปงสาลชนดราคากโลกรมละ 12 บาท กบแปงมนราคากโลกรมละ 13.50 บาท โดยใชอตราสวนในการผสมเทาไร จงจะขายแปงผสมในราคากโลกรมละ 15 บาท แลวไดก าไร 20% วธท า ขนท 1 ............................................................................................................................... ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 2................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 3................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 4................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. .............................................................................................................................................................

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4) ซอมะเขอมา 2 ชนด ราคากโลกรมละ 35 บาท และ 20 บาท เอามาคละกน แลวขายไปกโลกรมละ 25 บาท ไดเงนเทาทนพอด อตราการผสมของมะเขอเทศทงสองชนดเปนเทาใด วธท า ขนท 1 ............................................................................................................................... ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 2................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 3................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 4................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. .............................................................................................................................................................

48

5) ตองการผสมน ามนสองชนดราคาลตรละ 28 บาท แลราคาลตรละ 35 บาท แลวขายน ามนผสมไปลตรละ36 บาท จะไดก าไร20% จงหาอตราสวนในการผสมน ามนทงสองชนดน วธท า ขนท 1 ............................................................................................................................... ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 2................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 3................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 4................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. .............................................................................................................................................................

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6) ผสมแปงสาลชนดราคากโลกรมละ 4 บาท กบแปงมนราคากโลกรมละ 5.50 บาท โดยใชอตราสวนในการผสมเทาไร จงจะขายแปงผสมในราคากโลกรมละ 6 บาท แลวไดก าไร 20% วธท า ขนท 1 ............................................................................................................................... ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 2................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 3................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 4................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. .............................................................................................................................................................

50

7) ผสมแปงสาลชนดราคากโลกรมละ 6.50 บาท กบแปงมนราคากโลกรมละ 7 บาท โดยใชอตราสวนในการผสมเทาไร จงจะขายแปงผสมในราคากโลกรมละ 8 บาท แลวไดก าไร 20% วธท า ขนท 1 ............................................................................................................................... ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 2................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 3................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 4................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. .............................................................................................................................................................

51

8) แมคาซอกาแฟชนดกโลกรมละ 170 บาท กบกาแฟชนดราคากโลกรมละ 150 บาท มาผสมขาย จงหาวาแมคาจะตองใชกาแฟชนดละกกโลกรม เพอใหกาแฟผสม 30 กโลกรม และขายกโลกรมละ 200 บาท แลวไดก าไร 20% วธท า ขนท 1 ............................................................................................................................... ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 2................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 3................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ขนท 4................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. .............................................................................................................................................................