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Estimating with Fractions Suppose you need to go somewhere or to buy
something. Do you need a tool to calculate the time
needed to go there? Do you need to know the exact
time? If not, what will you do?
As in integers, the computational results of fractionscan be estimated by rounding off or using
compatible numbers.
a. Rounding to 0,21 , or 1
22 ..22
at are you go ng to learn? To est ima te
f r a c t i o n s To a d d f ra c t i o n s To sub tra c t
fra c tions . To multiply fra c tions To d ivide fra c tions.
Key Terms:Ad d , subt rac t , multip ly,divide , est ima te,prod uc t, quo tient , likefra c tion, un likefra c tion, p rop er,imp rop er fra c tion
We can estimate the result of addition and
subtraction by rounding each fraction to 0,21 , or 1.
If I have87 of a bar and you give me
94 more of a
bar, estimate how long I have altogether.
EXAMPLE 1
Solution:
Since87
is close to 1 and94
is close to21
, then87
+94
approximates 1 +21 = 1
21 . So, I have approximately
121 bars.
Problem 1 Find two fractions which their sum is close but not
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equal to 1.
4
1 3
2 8
7 8
5
21
101
43
105
83
81
125
109
127
103
31
107
b. Rounding to the nearest integer
If an addition, subtraction, multiplication, or
division involves a mixed number, we can estimate a
logical result by rounding each operand to the
nearest integer.
Estimate the result of 861
- 421
EXAMPLE 2
Solution :
861
can be rounded to 8 and 421
can be rounded to 5.
Therefore, 8 5 = 3 approximates 861
- 421
Write a story problem based on the expression of
Example 2.
c. Using compatible numbers
To estimate the quotient of mixed numbers,
compatible numbers are frequently used, especially
Problem 2
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for mental calculation.
Estimate the result of 434
1 : 58
7
Well, it willbe easier if Idivide 42 by
EXAMPLE 3
Solution:
4341
: 587
can be estimated by using compatible
numbers. Round 4341 to 42 and 5
87
to 6.
Then 42 : 6 = 7 approximates 4341 : 5
87 .
Adding and Subtracting Fractions
Erna and Wati bought sliced bread, which
consists of 8 equal pieces. While they sat down in
the backyard, Erna ate81 part of the bread, and
Wati ate 83 of it. How many parts of the bread did
they eat?
To answer this question, let us carry out the Mini
Lab on the next page.
In this Mini Lab, we know that
81 +
83 =
84 or
21 .
Therefore, Wati and Erna have eaten21 of the
bread. Do the activities in the Mini Lab to find
61 + 6
3 and 51 + 5
3 , and so on.
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Based on the examples, we have the procedure to
add like fractions.
Apply the procedure to solve the following
example.
Add 53 to 5
4 .
53
+ 54
= 543 +
Work in PairsMaterials and Tools: millimeter blocks, ruler,two colors)
Add
drawing pencils (at least
81 to
83 .
Draw bar on the millimeter blocks as shown in theof this bar illustrates the one-eight.
following figure. Each part
Color one part of the bar to represent81 . Using di
parts of the bar to illustrate
fferent color, shade three
83 .
Discuss the following questions.
a. How many parts of the bar that are colored?b. What fraction represents the number of uncoloredc. If you color the two other parts of the bar, what
number of uncolored parts?
parts?fraction illustrates the
Mini-Lab
Adding Like FractionsTo add like fractions, add the numerators, whilethe denominators remain unchanged
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Subtracting Like Fractions
Tini found 85 parts of a cake on the table. Then she
ate81 part of the cake. What part of the cake
remained?
815
81
85 = =
84
or21
Therefore, the cake that was left was 21 .
Based on the example, write the procedure to
subtract like fractions, then use the procedure to
solve the problem below.
There were three-fourths of a watermelon on the
table. Then you ate one-fourth. How many parts of
the watermelon remained?
53 +
54 = 5
7
= 521
521
64748
Subtracting Like Fractions.....
Problem 3
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(Relation with the Real World)
Mr. Salmeto had65 drum of kerosene. His
neighbour bought this kerosene so that the amountof kerosene in the drum was now
63 drum. How
many drums of kerosene did Mr. Salmetos
neighbour buy?
You could do the problem in the following way.
65 - ..... =
63
Solution:
65 -
62 =
63 , so we know that Mr. Salmetos
neighbour had bought62 or
31 drum of kerosene.
Adding and Subtracting Unlike
Fractions
Ani read a novel. Two days ago, she finished41
of the book. Today she continued reading3
2of
the book. What parts of the novel has Ani read?
To answer this question, you need to add unlike
fractions and may use fraction bars.
How many parts of the novel has Ani read?
What numbershould be added
to63 to get
65 ?
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Add 41 to
32 .
41 +
32 =
1211 or 4
1 +32 =
123 +
128 =
1283 + =
1211
Thus, Ani has read1211 parts of the book.
Solve31
21 using fraction bars.
EXAMPLE 6
61
623
62
63
31
21 ===
Based on Example 6 and Example 7, it is clear
that to add or subtract unlike fractions, you have
to convert the fractions to like fractions first by
using the Least Common Multiple ( LC M ) of the
denominators.
ba r fo r 63 o r 2
1
ba r fo r 62
o r 31
.
Sub trac t: 62
63 .
Find the b ar forthe sum .
ba r fo r 41
. ba r fo r 3
2 .
EXAMPLE 7
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Find the sum of72
41 + .
Since the LCM of 4 and 7 is 28, then
28x4x1
41 7 ==
7
7
28x7x2
72 8==
4
4
and41 + =
72
28287 8
+ =28
15
Adding Mixed Numbers
Before you proceed to adding mixed numbers, do
the following activities.
One of the methods to add mixed numbers is to
calculate the whole parts and the fraction parts
separately. Sometimes the sum of the fraction
parts is an improper fraction. If so, first convertthe improper fraction to a mixed number.
EXAMPLE 8
Repeat step 1 for some other pairs of ropes. Check each of the expression by measuring the whole
lengths for each pair of ropes.
Take a pair of ropes. Estimate the sum of the length of thetwo ropes, and then add them. Write down an expression forthis addition.
Take some pieces of rope4
12 cm,2
11 cm,10
14 cm,8
13 cm, and
435 cm in length.
Put two ropes in such a way that the end of the first ropetouches the end of the second rope.
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Find the result of 213
4315 + .
4315 = 4
315
213 = 22
213
= 4
23
= 4518
= 18 + 411
= 4119
Subtracting Mixed Numbers
Sometimes, you need to change the form of
fractions before you subtract them.
Calculate 316 - 2
14
316 - 2
14 = 626 - 6
34
= 685 - 6
34
= 651
Thus, 316 - 2
14 = 651 .
EXAMPLE 9
The LC M is 4. Exp ress it a slike frac tions.
A d d th e w h o le p a r tsa nd the frac tion pa rts.
Co nvert the fra c tion.
41
141
44
45 =+=
Add the w hole p ar ts.
EXAMPLE 10
Expre ss it a s like f rac tio ns
Conver t the f rac t ion p ar t .
62 = 5 +6
621 =
685 .
Sub tra c t the who le p a rtsen the f rac t ion p ar tsth
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Multiplying and Dividing Fractions
Before operating fractions, do the following activity.
a. Multiplying a Fraction by a Fraction
You can use a model (fraction bars) to multiply a
fraction by a fraction.
Mr. Arif has a large garden. He wants to plant
one-half of his garden with medicinal herbs.
Wild gingers will be planted on one-third of it .
What part of the garden will be planted wildgingers?
Solution:
The shadowed part is 61 of the whole bar. This
part represents the part of the garden that will be
Activities1. Take a piece of paper and cut the paper into two equal
parts. What fraction represents each part?2. Take one of the parts. Then, cut it i nto two equal parts.
What fraction represents each part?3. Take one of the parts. What fraction represents each part?
EXAMPLE 11
The g arde n p lant ed me d ic inal herbs is21 o f
Co lour one-half of the
for me d ic inal herbs into
Sha d e
the whole ga rde n ;b a r.
Div ide the g arden
three eq ual pa rts.
p a r t .31 of the co loured
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Do you still remember the meaning of 4 x 2? This
meaning also applies in multiplying whole
numbers and a fraction.
Consider the following illustration:
When multiplying a whole number with a
fraction, you may convert the whole number to a
fraction having 1 as the denominator, then
multiply the fractions.
Relation with Real World
Suppose you have 16 marbles. Three-eighths of
the whole marbles are green. How many marbles
are green?
To answer this question, you have to calculate 83
x 16.
83 x 16 = 8
3 x 116
= 1x816x3
1
2
= 1123
=16 = 6
322
38
32
32
32
32
32
4
=
=
+++=
EXAMPLE 14
Exp ress 16 a s 1
16 .
u lt ip ly the num erato ran d th e d en o m in a to r.Simplify.
M
Dith
v ide the numerato r ande d enom inato r by 8 .
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c. Multiplying Mixed Numbers
Rini has a big photo album. Eight and one third
pages of the album are still empty. Rini wants tofill one-half of the empty pages with her favorite
artists photographs. How many pages of the
album will be filled with artists photographs?
To answer this question, you need to calculate 21
of 318 or
21 x 3
18 .
Before multiplying a mixed number, you have to
express the mixed number as an improper
fraction.
318 =
325
31)38(
=+
.
Then multiply the fractions.
625
325x2
1 = = 614
Therefore, as many as 614 pages of the photo
album will be filled with the artists photographs.
You may also illustrate the above case by using
the following model.
318 p a g e s
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21 of 8 is 4, and 2
1 of 31 is 6
1 .
Hence, 3182
1 is 4 + 61 or 6
14 .
You may also use the model of an area to multiply
mixed numbers.
Draw a quadrilateral with a length of 211 cm and a
width of 412 cm as shown in the following figure.
4122
11 = 1 + 1 + 41 + 2
1 + 21 + 8
1 = 833 .
The area of the quadrilateral is 833 cm 2.
Find the result of 4133
22 .
21 o f
318 p a g e s
EXAMPLE 15
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Solution:
4133
22 = 413
38
=1
2
413
38
= 13132
= 3283
26 =
Solving Problems Related to the Real World
Suppose you can read one-page full of text in41
hours. At least, how many hours do you need to
read 421
pages?
Solution:
To read 421
pages of the text needs 421
41 hours =
89
hours. So, you need at least 181 hours to read 4
21
pages of the text.
d. Dividing Fractions
To understand the meaning of division of
fractions, follow the situations below.
Direction
You could determinethe result of
Simplify.
Multiply.
Simp lify t hthe G CF of
em b y using8 an d 4
Write d oa s a fra c
w n eac h m ixed n u m b ertion.
2128
by applying thefollowing method.
8 2 = 168 = 416 + 4 = 20
Thus2128 = 20
EXAMPLE 16
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1. You have 6 candies that will be given to your
friends. Each of your friends will get 2 candies.
How many people will get candies?
Answer: 3
Written as 6 : 2 = 3.
2. Suppose the 6 candies are given to your
friends. Each of them will get 1 candy. How
many people will get the candies?
Answer: 6
Written as 6 : 1 = 6.
3. Suppose the 6 candies are divided so that each of
your friends gets 21 candy. How many people will
get the candies?
Answer: 12
Written as 6 : 21 = 12.
Notice that:
6 : 21 = 12 relates to 6 1
2 = 12.
The numbers 21 and 2 have a specific relation, that
is, the product is 1. 12x21 = .
One half or 21 is called the multiplicative inverse of
2.
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Notice that any two numbers whose product is 1
are called the multiplicative inverse to each other.
Find other numbers that are the multiplicative
inverses to each other.
e. Dividing Mixed Numbers
To divide mixed numbers, firstly, express every
mixed number as a fraction and then multiply
them.
Calculate533:
412 .
Solution:
8524 51
185
49
518:
49
533:
412
2
1
=
=
=
=
Evaluate 3:3210 .
Solution:
953
932
3313231332
13:
3323:
3210
=
=
=
=
=
EXAMPLE 17
Chang e the m ixed numbe rs.
Divide the nume rato r and thede nominator by the GCF of 9 and 18,tha t is 9.
Find the p rod uc t.
EXAMPLE 18
Conver t the mixed numbe r.
Multiply b y the multiplic ative inverse o f 3.
Multiply thedenomina
nume rato rs a nd t hetors.
Write d ow n t
Exp ress a s a
h e p ro d u c t .
mixed number.
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1. Approximate the results of the following
operations.
a. 71 +
83
b. 9111
- 397
c. 1321
x31
d. 7 53
: 1 21
2. Estimate the product of51 and 248.
a. Use the result to approximate the product of53
and 248.
b. Explain how to estimate the result of85
x 55.
3. Open-ended . Write two numbers (either
fractions or mixed numbers) and then estimate the
computational result of the two numbers.
4. Which one of the following results is between 6
and 7?
a. 21
x 1421
b. 2 x 31615
c. 5 1211
+ 2524
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a. How many parts of Tinis after-school time are
used to sleep, eat, and study at school?
b. How many parts of Tinis after-school time are
used to do other activities?
10. Find the result of the following additions.
a. 3218 + b. 26
13 + c. 433
518 +
d.1612
8311 + e.
438
1219 +
11. Writing. Describe the method you use to find
the sum of .516
322
543
315 +++
12. Open Question . Write down two mixednumbers so that their sum is a whole number.
13. Explain why the result of addition of two mixed
numbers is not always a mixed number.
14. Cuisine . To make a kind of cake, we need
431 cups of sugar, while another kind of cake
needs 211 cups of sugar. Suppose you have about
3 cups of sugar. Is your sugar enough to make the
two kinds of cake? Explain .
15. Find the result of the following subtractions.
a. 83
343
7
b. 41
1165
2
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c. 534
549 d. 16
1118121 e. 2
111121115
16. Number Pattern . Write down the two
successive numbers in the following pattern.
319 , 6
18 , 7, 655 , 3
24 , ....., ......
17. Research . Determine the fraction representing
the number of female teachers at your school.What is the ratio of the female teachers to the
male teachers that teach at year I? What is the
ratio of the female teachers that teach at year I to
all teachers at your school?
18. Find the product of the following
multiplication.
a. 51 of 2
1 b. 21 of 5
1 c. 81
52
d. 21 of 3
1 e. 43
53
19. Evaluate ab if a =3
1 and b =7
6 .
20. Garden . Suppose two-thirds of a garden will
be planted grass. The remaining will be planted
plants. Three-fourths of the garden that are
planted plants, will be planted flowers. What is
the part of the garden that will be planted
flowers?
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21. Biology . About 107 of a human body consists
of liquid. If the weight of a man is 60 kg, how
many kilograms would the weight of the liquidbe ?
22. Find the results of the following
multiplication.
a.2
172
14 b.10
963
23 c.3
272
16
d. 2182
18
23. Writing . List the objects the area of which
you can find their area by the methods of
multiplication of mixed numbers.
24. Geometry Calculate the area of the
quadrilateral as shown on the left.
25. Application . Tine needs 433 metres of textile
to make a pair of dresses. If Tine wants to make 4
pairs of dresses, at least how many metres oftextile does Tine have to buy?
26. Geometry. Calculate the area of a rectangular
yard with a length of 215 m and a width of 3
24 m.
27. Critical Thinking . Is 2143
22 more or less
than 10? Explain your answer without calculating
the result.
21
4 cm
312 cm
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28. Measurement . How many parts of a rope of41
metre are there on a 2-metre rope?
29. Find the result of the following divisions.
a.83:5 b.
43:15 c.
53:
94 d.
31:12
30. Suppose you cut 3 apples. Each is cut into
eight equal parts. How many parts of apples will
you get?
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