Middle School Math & Grammar/ProOneSoftware-MiddleSchoolMath&Grammar-Win31.iso/math/chapter2.3b

Text File | 1993-10-22 | 7.5 KB | 358 lines

 356
àï2.3ïAdding Fractions.
äïPlease Add the following Fractions that have the same
êêdenominators.
âSê2è3êï2 + 3êï5
#êè 1)ï─ + ─è =è ─────è =è ─
êê 7è7êè 7êè 7

êêï5è 7ë1êï5 + 7 + 1êï13êï1
#êè 2)ï── + ──ï+ ──è =è ─────────è =è ──ïorï1 ──
êê 12è12è 12êë 12êë12ê 12
éS To Add the fractions 5/12, 7/12, and 1/12, you should write
the common denominator down once and then Add the numerators.
êêê 5è 7è 1êï5 + 7 + 1
#êêê── + ── + ──è =è ─────────
êêê12è12è12êë12

At this point, Adding the Whole Numbers on top is the same as in
Chapter 1.êêï5 + 7 + 1êï13
#êêêë─────────è =è ──
êêêê 12êë 12

The answer can be left in this form or changed to the mixed number,
êêêêê 1
#êêêêë1 ──.
êêêêê12
 1êêêè2ë1
#êêêè Addë─ï+ï─
êêêêë5ë5

êè 3êêï3
#ë A)ï──êëB)ï─êë C)ï5êë D)ïå
êè10êêï5
ü

êêê2ë1êï2 + 1êï3
#êêê─ï+ï─è =è ─────è =è ─
êêê5ë5êè 5êè 5
Ç B
 2êêêè7ë11
#êêêè Addë─ï+ï──
êêêêë3ë 3

êè13êêï17
#ë A)ï──êëB)ï──êëC)ï6êë D)ïå
êè 3êêè3
ü

êê 7ë11êï7 + 11êï18
#êê ─ï+ï──è =è ──────è =è ──è =è 6
êê 3ë 3êë3êë3
Ç C
 3êêê 5ë3ë1
#êêêïAddë─ï+ï─ï+ï─ .
êêêêè8ë8ë8

êè9ê1êï7êêè11ë 3
#ë A)ï─ïor 1 ─ëB)ï─êë C)ï── or 1 ─ëD)ïå
êè8ê8êï8êêè 8ë 8
ü

êë5ë3ë1êï5 + 3 + 1êï9ê 1
#êë─ï+ï─ï+ï─è =è ─────────è =è ─ïorï1 ─
êë8ë8ë8êë 8êë 8ê 8
Ç A
 4êêê 3ë 9ë11
#êêê Addë──ï+ï──ï+ï── .
êêêêï16ë16ë16

êè17êêï23
#ë A)ï──êëB)ï──êëC)ï2êë D)ïå
êè16êêï16
ü

êè3ë 9ë11êï3 + 9 + 11êï23êï7
#êï──ï+ï──ï+ï──è =è ──────────è =è ──ïorï1 ──
êï16ë16ë16êë 16êë 16ê 16
Ç B
 5êêêè4ë5ë9
#êêêè Addë─ï+ï─ï+ï─ .
êêêêë7ë7ë7

ê 9ë 1êè23ë 2êè18ë 4
#è A)ï─ or 1 ─ë B)ï── or 3 ─ë C)ï── or 2 ─ë D)ïå
ê 7ë 7êè 7ë 7êè 7ë 7
ü

êè4ë5ë9êï4 + 5 + 9êï18ê 4
#êè─ï+ï─ï+ï─è =è ─────────è =è ──ïorï2 ─
êè7ë7ë7êë 7êê7ê 7
Ç C
äïPlease Add the following Fractions that have unlike
êêdenominators.
âèAddè 1ë5êè1ë 5êè1ë 5è2
#êêï──ï+ï─è =è ───ï+ï─è =è ───ï+ï─ ∙ ─è=
êêï14ë7êï2∙7ë7êï2∙7ë7è2

êêê 1ë10êï1 + 10êï11
#êêê──ï+ï──è =è ──────è =è ──
êêê14ë14êè 14êè 14
éS To Add the fractions,ï1/14ïandï5/7 , it is first necessary
to express the denominators in prime factored form like we did in
Section 1.7 of Chapter 1.
êêêè1ë5êè1ë 5
#êêêï──ï+ï─è =è ───ï+ï─
êêêï14ë7êï2∙7ë7

At this point you can see that the second fraction is missing a factor
of "2" in its denominator.ïIt is necessary to multiply both the top
and the bottom of this fraction by "2".
êêê1ë 5êè1ë 5è2
#êêë ───ï+ï─è =è ───ï+ï─ ∙ ─
êêë 2∙7ë7êï2∙7ë7è2

Now, both denominators have all the same factors and you can multiply
them out.êë1ë 5è2êè1ë10
#êêë ───ï+ï─ ∙ ─è =è ──ï+ï──
êêë 2∙7ë7è2êï14ë14

Since the two fractions have the same denominators, you can write the
denominator down once and add the numerators like we did in the earlier
problems.êë 1ë10êï1 + 10êï11
#êêê──ï+ï──è =è ──────è =è ──
êêê14ë14êè 14êè 14

Thus, the sum ofï1/14ïandï5/7ïisï11/14.
 6êêêë 2ë1
#êêêêAddë─ï+ï─
êêêêê 3ë6

êè3êêè5êêè1
#ë A)ï─êë B)ï─êë C)ï─êë D)ïå
êè9êêè6êêè3
üë 2ë1êï2ë 1êè2è2ë 1
#êë─ï+ï─è =è ─ï+ï───è =è ─ ∙ ─ï+ï───è=
êë3ë6êï3ë2∙3êï2è3ë2∙3

êêê4ë1êï4 + 1êï5
#êêê─ï+ï─è =è ─────è =è ─
êêê6ë6êè 6êè 6
Ç B
 7êêêë4ë 2
#êêêë Addë─ï+ï──
êêêêê9ë15

êè26êêè6êêï1
#ë A)ï──êëB)ï──êëC)ï─êë D)ïå
êè45êêï24êêï4
üë 4ë 2êè4ê2êè5è 4ë 3è 2
#êë─ï+ï──è =è ───ï+ï───è =è ─ ∙ ───ï+ï─ ∙ ───è=
êë9ë15êï3∙3ë3∙5êï5è3∙3ë3è3∙5

êêë 20ë 6êï20 + 6êï26
#êêë ──ï+ï──è =è ──────è =è ──
êêë 45ë45êè 45êè 45
Ç A
 8êêêë3ë 4
#êêêëAddë──ï+ï──
êêêêë 40ë25

êè 7êêè7êêè47
#ë A)ï──êëB)ï──êëC)ï───êè D)ïå
êè60êêï65êêï200
üêê 3ë 4êë3êï4
#êêë ──ï+ï──è =è ───────ï+ï───ë=
êêë 40ë25êï2∙2∙2∙5ë5∙5

è3ë 5ë 4è 2∙2∙2êè15ë 32êï15 + 32ë 47
#─────── ∙ ─ï+ï─── ∙ ─────è =è ───ï+ï───è =è ───────ï=ï───
2∙2∙2∙5è5ë5∙5è2∙2∙2êï200ë200êè 200ê200
Ç C
 9êêêè 1ë 5
#êêêëAddë─ï+ï──
êêêêë 3ë19

êè 6êêï34êêï2
#ë A)ï──êëB)ï──êëC)ï─êë D)ïå
êè21êêï57êêï7
üêê 1ë 5êï19è1ë 5è3
#êêê─ï+ï──è =è ── ∙ ─ï+ï── ∙ ─è =
êêê3ë19êï19è3ë19è3

êêë 19ë15êï19 + 15êï34
#êêë ──ï+ï──è =è ───────è =è ──
êêë 57ë57êë57êè 57
Ç B
 10êêêè 1ë1
#êêêë Addë─ï+ï─
êêêêê3ë4

êè 7êêï2êêè7
#ë A)ï──êëB)ï─êë C)ï─êë D)ïå
êè12êêï7êêè8
üêê 1ë1êï4è1ë1è3
#êêê─ï+ï─è =è ─ ∙ ─ï+ï─ ∙ ─è =
êêê3ë4êï4è3ë4è3

êêê 4ë 3êï4 + 3êè7
#êêê──ï+ï──è =è ─────è =è ──
êêê12ë12êè 12êè12
Ç A
 11êêêè 3ë5
#êêêë Addë─ï+ï─
êêêêê7ë8

êè 8êêï9êêè59ê3
#ë A)ï──êëB)ï─êë C)ï── or 1 ──è D)ïå
êè15êêï8êêè56ë 56
üêê 3ë5êï3ê5
#êêê─ï+ï─è =è ─ï+ï─────è =
êêê7ë8êï7ë2∙2∙2

ï2∙2∙2è3ê5ë7êï24ë35è 24 + 35ê59ê3
#ï───── ∙ ─ï+ï───── ∙ -è =è ──ï+ï──è ───────è=è── or 1 ──
ï2∙2∙2è7ë2∙2∙2è7êï56ë56ê56êï56ë 56
Ç C
 12êêê 1ë5ë 5
#êêêèAddë─ï+ï─ï+ï──
êêêêè 3ë6ë12

êè19ê7ê 11êêï1
#ë A)ï── or 1 ──è B)ï──êëC)ï─êë D)ïå
êè12ë 12ê 21êêï2
üêï1ë5ë 5êï1ê5ê5
#êê ─ï+ï─ï+ï──è =è ─ï+ï───ï+ï─────è=
êê 3ë6ë12êï3ë2∙3ë2∙2∙3

2∙2è1ë2è 5ê 5êë4ë10ë 5ê19ê7
#─── ∙ ─ï+ï─ ∙ ───ï+ï─────è =è ──ï+ï──ï+ï──è=è── or 1 ──
2∙2è3ë2è2∙3ë2∙2∙3êï12ë12ë12ê12ë 12
Ç A
 13ë 2êêêï13êêë3
#êê ───êêèA)ï────êë B)ï───
êêï3êêêï15êêë8
ëAddê 1
#êë+ï───êêê 3
#êêï5êêè C)ï───êêD)ïå
#êë ───────êêë 5
üïIt is possible to Add fractions in a column by first finding the
least common multiple of the two denominators like we did in Section 2.1
of this chapter.ïThe least common multiple of 3 and 5 is "15".ïWe can
begin to change the original problem by finding the missing numerators
like we did is Section 2.2 of this chapter.
#êêë╪╩╪êè ╪╪╪╪êè ╪╔╚╪
#êêë 3ë╫ê15ë ╫ë 15

#êêë╪╔╪êè ╪╪╪╪êè ╪╪╦╪
#êê ╙ ╪╪╪═╪╪╪ï╫è╙ ╪╪╔═╪╪è ╫è╙╪╪╔═╪╪

#êêêêêêë ╪╔╦╪
êêêêêêê15
Once the missing numerators are found, the two fractions are added to
give a sum ofï13/15.
Ç A
 14ë 2êêêè5êêë1
#êê ───êêèA)ï────êë B)ï───
êêï7êêêï14êêë2
ëAddê 3
#êë+ï───êêêï9
#êê 14êêè C)ï────êë D)ïå
#êë────────êêë 14
#üêë ╪╩╪êè ╪╪╪╪êè ╪╪╠╪
#êêë 7ë╫ê14ë ╫ë 14
#êêë╪╦╪êè ╪╪╪╪êè ╪╪╦╪
#êê ╙ ╪╪╔╠╪╪╪ï╫è╙ ╪╪╔╠╪╪è ╫è╙╪╪╔╠╪╪

#êêêêêêë ╪╪╧╪êï╪╔╪
#êêêêêêê14ë╫ë2
Ç B
 15ë 5êêêï17êêë15
#êê ───êêèA)ï────êë B)ï────
êêï6êêêï18êêë18
ëAddê 2
#êë+ï───êêê 19ê 1
#êêï9êêè C)ï──── or 1 ──ëD)ïå
#êë────────êêë 18ê18
#üêêè ╪═╪êè ╪╔═╪
#êêêè 6ë╫ê18
#êêêè╪╩╪êè ╪╪╠╪
#êêë ╙ ╪╪╪╤╪╪╪ï╫è╙ ╪╪╔╨╪╪
êêêêêêêï1
#êêêêêè╪╔╤╪ë 1 ──
êêêêêè 18èorè 18
Ç C
 16ë 4êêêï41êêë11
#êê ───êêèA)ï────êë B)ï────
êêï9êêêï45êêë24
ëAddê 7
#êë+ ────êêê 12
#êê 15êêè C)ï────êë D)ïå
#êë────────êêë 32
#üêêê ╪╠╪êè ╪╩╚╪
#êêêê 9ë╫ê45
#êêêê╪╧╪êè ╪╩╔╪
#êêêè╙ ╪╪╔═╪╪╪ï╫è╙ ╪╪╠═╪╪

#êêêêêê╪╠╔╪
êêêêêê 45
Ç A
äïPlease add the following Mixed Numbers.
â
êAdd
êêï1ê1êë4ê 3êë7
#êê2 ─ï+ï5 ─è =è 2 ──ï+ï5 ──è =è 7 ──
êêï3ê4êè 12ê12êè 12
éS To Add the Mixed Numbers,ï2 1/3ïandï5 1/4 , it is first
necessary to changeï1/3ïandï1/4ïtoï4/12ïandï3/12ïby finding the
least common denominator, 12, and finding the missing numerators.ïThen
it is possible to add the Whole Number parts and the fractional parts.
êêêê 1êê4
#êêêë 2 ─ë=ë2 ──
êêêê 3êë 12
êêêê 1êê3
#êêêë 5 ─ë=ë5 ──
#êêêï╙ë4êë 12
#êêêè──────────è ───────────
êêêêêêï7
#êêêêêë 7 ──
êêêêêê 12
 17ê 1êêêê2êê 12
#êê 4ï─êêë A)ï12 ──ê B)ï10 ──
ë Addêï3êêêë 15êêï5
êêè 4
#êê 7 ──êêêë 3
#êêè15êêë C)ï11 ─êïD)ïå
#êë ─────────êêêè5
ü
êAdd
ê 1ê 4êë5ê 4êë 9êè3
#ë 4 ─ï+ï7 ──è =è 4 ──ï+ï7 ──è =è 11 ──è=è11 ─
ê 3ê15êè 15ê15êë15êè5
Ç C
 18ê 3êêêë 12êê 33
#êê 7ï─êêë A)ï12 ──ê B)ï12 ──
ë Addêï8êêêë 28êê 40
êêè 9
#êê 5 ──êêêë 3
#êêè20êêë C)ï12 ─êïD)ïå
#êë ─────────êêêè7
ü
èAddê3ê 9êè 15ê18êë33
#êè 7 ─ï+ï5 ──è =è 7 ──ï+ï5 ──è =è 12 ──
êë 8ê20êè 40ê40êë40
Ç B
 19ê 2êêêë 13êêï9
#êê 6ï─êêë A)ï21 ──ê B)ï22 ──
ë Addêï7êêêë 35êê 35
êêè 3
#êê15 ──êêêë 21
#êêè35êêë C)ï23 ──ê D)ïå
#êë ─────────êêêè35
ü
è Addë 2êï3êè 10êï3êë13
#êè 6 ─ï+ï15 ──è =è 6 ──ï+ï15 ──è =è 21 ──
êë 7ê 35êè 35ê 35êë35
Ç A