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RATIONAL
NUMBERS
CHAPTER
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70 | CHAPTER 2 | RATIONAL NUMBERS
Section 2.1
Recognizing, Reading, Writing and Simplifying
Fractions
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What is a fraction?
You have a circle. Cut it into two equal parts. Each part is called a half of a circle. Each part
is a fraction of the circle.
We can write a half as 1
2
We now cut a circle into 4 equal parts (fractions). Each part is called one fourth (or ‘one
quarter’) of a circle.
We write this as
1
4
If we take one part away, there are now three quarters left.
We can write this as 3
4
MATHEMATICS FOUNDATION 1
RECOGNIZING, READING AND WRITING FRACTIONS | 71
A fraction is made of two parts:
The numerator tells you how many parts you have.
The denominator tells you how many equal parts in total.
Note that all of the parts in the fraction must be of equal size.
There is 1 part shaded, so the numerator is 1.
There are 3 total parts in the shape, so the denominator is 3.
So the fraction shaded is
1
.
3
There are 5 parts shaded, so the numerator is 5.
There are 12 total parts in the shape, so the denominator is 12.
So the fraction shaded is
5
.
12
Don’t forget that the parts must be of equal size!
72 | CHAPTER 2 | RATIONAL NUMBERS
Practice 1
What fraction is shaded in the shapes below ?
a)
________________________
b)
________________________
c)
________________________
d)
________________________
e)
________________________
f)
________________________
To write a fraction in words, we use numbers for the numerator, and ordinal numbers for
the denominator.
MATHEMATICS FOUNDATION 1
RECOGNIZING, READING AND WRITING FRACTION
EXAMPLE
Write the fractions in words.
a)
3
5
b)
1
3
one-third
c)
5
6
d)
2
4
two-fourths
two-quarters
Practice 2
Write the fractions in words.
2
5
________________________
c)
1
10
________________________
e)
4
9
________________________
a)
3
8
________________________
d)
2
3
________________________
f)
1
4
________________________
b)
or
________________________
Exception – when the denominator is “2”, we do NOT say “second”. Instead, we say “half” (or
the plural, “halves”, if there is more than one).
EXAMPLE
1
2
one-half
3
2
three-halves
| 73
74 | CHAPTER 2 | RATIONAL NUMBERS
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Write the fractions in words.
a)
b)
c)
two-thirds
3
5
5
8
2
3
Practice 3
Write the words as fractions
a)
six-sevenths
_______
b)
seven-tenths
c)
_______
d)
one-half
e)
_______
f)
three quarters
_______
_______
_______
Write the fractions found in the sentences.
a) Seven out of ten people enjoy going swimming.
7
10
b) Two of the seven Emirates begin with the letter A.
2
7
MATHEMATICS FOUNDATION 1
PROPER FRACTIONS, IMPROPER FRACTIONS AND MIXED NUMBERS | 75
Practice 4
Write the fractions found in the sentences.
a)
_____________
b)
Six of the seven Emirates joined at the same time in 1971.
_____________
c)
Two of my three brothers like ice cream.
_____________
d)
_____________
e)
Six of my nine notebooks are blue.
_____________
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EXAMPLE
2
2
=
9
9
=
15
15
=
105 = 1 whole
105
Practice 5
Write the numerator to make each of the fractions below, equal to 1 whole.
__
__
__
__
__
3
5
12
7
22
There are three different kinds of fractions; proper fractions, improper fractions and mixed
numbers.
3
4
The numerator is less than the denominator.
This is called a proper fraction.
A proper fraction is less than one whole one.
76 | CHAPTER 2
| RATIONAL NUMBERS
7
The numerator is greater than the denominator.
4
This is called an improper fraction.
An improper fraction is greater than one whole one.
A proper fraction is less than one whole one.
We can also write
7
3
as 1
4
4
1 3 is called a mixed number.
4
We have added a whole number to a fraction:
1+
whole
3
3
=1
4
4
fraction
mixed numbers
We say this as, one and three-quarters.
MATHEMATICS FOUNDATION 1
PROPER FRACTIONS, IMPROPER FRACTIONS AND MIXED NUMBERS | 77
EXAMPLE
State whether each of these is a proper fraction, an improper fraction, or as a mixed
number. Then write the fraction in words.
9
7
5
10
Improper fraction
Nine-sevenths
Proper fraction
Five-tenths
23
4
Mixed number
Two and three-quarters
a)
b)
c)
Practice 6
State whether each of these is a proper fraction, an improper fraction, or as a mixed
number. Then write the fraction in words.
a)
b)
7
6
4
1
5
c)
2
3
d)
9
4
e)
8
3
6
78 | CHAPTER 2
| RATIONAL NUMBERS
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Percent means out of 100.
90% =
90
100
What percent of each diagram is shaded?
a)
Since there are a total of 100 squares, the
denominator is 100. There are 3 shaded squares, so
the fraction is 3 .
100
This means that 3% of the diagram is shaded.
b)
There are 34 shaded squares,
34
.
so the fraction is
100
This means that 34% of the diagram is shaded.
MATHEMATICS FOUNDATION 1
PERCENT FRACTIONS | 79
Practice 7
What percent of each diagram is shaded?
a) _______ %
b) _______ %
There are also “special” percentages and their related fractions and decimals that you should
be able to remember:
25 = 1 = 0.25 = 25%
100
4
75 = 3 = 0.75 = 75%
100
4
50 = 1 = 0.5 = 50%
100
2
100 = 1 = 100% (one whole)
100
80 | CHAPTER 2
| RATIONAL NUMBERS
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Compare the diagrams below:
2
4
1
2
1
2
,
2
4
We can say that
,
4
8
3
6
3
and
6
4
8
are all the same part of a whole.
They are called equivalent fractions because they have the same value. They are equal.
We use the sign = for (equal to) or (equivalent to):
1
2
=
2
4
or
2
4
=
4
8
or
3
6
=
4
8
Finding a fraction that is equal but with smaller numbers is called simplifying a fraction.
This is done very easily with a calculator.
On a calculator the fraction button looks like this:
MATHEMATICS FOUNDATION 1
ab
c
SIMPLIFYING FRACTIONS WITH A CALCULATOR | 81
EXAMPLE
Using your calculator, simplify the fractions.
a) 3 Enter
9
3 a
b
c
9 =
b)
12
Enter
20
12 a
b
c
20 =
3
5
1
3
So with our calculator, we found that 3 1
9
3
3
12 3
, or
=
.
5
20 5
, or 3
9
= 1
3
and that
12
20
Practice 8
Using your calculator, simplify the fractions.
a)
e)
5
10
___________
18
30
___________
b)
f)
4
10
___________
8
9
___________
c)
g)
d)
35
50
___________
50
___________ h)
100
11
20
___________
12
15
___________
!simplest form? That’s okay! Not
all fractions will simplify.
82 | CHAPTER 2 | RATIONAL NUMBERS
Section 2.1 Exercises
1. Write the fractions in words.
b)
1
3
________________________
________________________
d)
1
2
________________________
________________________
f)
3
4
________________________
a)
4
5
________________________
c)
1
10
e)
5
12
or
________________________
2. Write the words as fractions
a) one-third
________________________
b) two quarters ________________________
#
________________________
d) one-half
e) three-thirtieths________________________
________________________
f) seven-eighths ________________________
3. Write the fractions found in the sentences.
a) Six out of ten people go to college.
________
b) Two boxes of chocolate are shared by six people.
________
#
$$$$$$$$
d) My mother had four of her seven brothers and sisters over for lunch.
________
#
$$$$$$$$
MATHEMATICS FOUNDATION 1
EXERCISES | 83
4. What fraction is shaded in the shapes below?
a)
________________________
b)
________________________
c)
________________________
d)
________________________
5. Write the numerator to make each fraction equal to 1.
a)
b)
5
c)
17
d)
8
331
6. State whether each of these is a proper fraction, an improper fraction, or as a mixed
number. Then write the fraction in words.
a)
b)
c)
9
8
3
2
7
2
3
84 | CHAPTER 2 | RATIONAL NUMBERS
7. What percent of each diagram is shaded?
a) _______ %
b) _______ %
8. Using your calculator, simplify the fractions.
a)
2
4
___________
b)
e)
15
18
___________
f)
MATHEMATICS FOUNDATION 1
36
60
___________
c)
4
12
___________
d)
80
___________
100
6
30
___________
g)
7
8
___________
h)
27
45
___________
READING AND WRITING DECIMALS | 85
Section 2.2
Reading, Writing, Comparing and Rounding Decimals
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Do you recall learning about reading and writing whole numbers?
ten thousands
thousands
hundreds
tens
ones
Ones
hundred thousands
Thousands
3
1
6
0
4
5
?
?
?
?
?
Our table shows that each place gets 10 times bigger as you move to the left. For example, 1
hundred is ten times bigger than 1 ten. 1 thousand is 10 times bigger than 1 hundred, and so
on. This is the ‘decimal system’. We also use it for numbers that are smaller than one whole.
4
5
.
thousandths
0
hundredths
ones
6
Decimal
ttenths
tens
.
hundreds
1
Ones
thousands
3
ten thousands
hundred thousands
Thousands
7
2
9
86 |
CHAPTER 2 | RATIONAL NUMBERS
As you move to the right, the tenths’ place comes after the ones’ place.
0.1 = one tenth
We use a decimal point ( . ) to separate the units and tenths place.
Look carefully at the difference between the place names. All place names to the right of the
decimal point end with “th”.
tenth
hundredth
thousandth
thousandths
.
hundredths
0
Decimal
ttenths
.
ones
tens
hundreds
Ones
thousands
ten thousands
hundred thousands
Thousands
7
2
9
“Zero point seven two nine.”
EXAMPLE
For the number 0.729 above, write the place value of the digit.
a) 7
tenths
b) 2
hundredths
c) 0
ones
d) 9
thousandths
MATHEMATICS FOUNDATION 1
READING AND WRITING DECIMALS | 87
Practice 1
For the number 0.483, write the place value of the digit.
a) 0
______________________________
b) 3
______________________________
c) 8
______________________________
d) 4
______________________________
EXAMPLE
Write the numbers in the correct place on the table.
Thousands
2 432
Hundreds Tens
2
4
3
2.432
24.32
2
Ones
tenths
hundredths
2
&
2
&
4
3
4
&
3
2
thousandths
2
Practise 2
Write the numbers in the correct place on the table.
Thousands
Hundreds
Tens
Ones
tenths
746.278
&
4 628
&
4.628
&
46.28
&
346.5
&
hundredths
thousandths
88 | CHAPTER 2 | RATIONAL NUMBERS
Practice 3
Write the place value of the underlined digit.
a) 0.295 ______________________________
b) 17.435 4 ______________________________
c) 1.448 ______________________________
d) 0.624 58 ______________________________
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Each digit after the decimal place is read separately.
0.48 is read as “zero point four eight,” and NOT “zero point forty-eight.”
0.729 is read as “zero point seven two nine,” and NOT “zero point seven hundred twentynine.
EXAMPLE
Write these numbers in words.
a) 3.61
three point six one.
b) 15.236
Practice 4
Write these numbers in words.
a) 0.8
___________________________________________________________________________
b) 0.25
___________________________________________________________________________
c) 1.461
__________________________________________________________________________
d) 57.829
___________________________________________________________________________
There is also another way to read a decimal, by its place value. We use the place value
furthest to the right to read the decimal.
0.48 is read as forty-eight hundredths because the digit furthest to the right (8) is in the
hundredths place.
0.729 is read as seven hundred twenty-nine thousandths because the digit furthest to the right
(9) is in the thousandths place.
MATHEMATICS FOUNDATION 1
WRITING DECIMALS IN WORDS | 89
EXAMPLE
Write these numbers in words reading the place value.
a) 0.4
four tenths
b) 0.61
sixty-one hundredths
c) 0.236
two hundred thirty-six thousandths
d) 0.008
eight thousandths
Practice 5
Write these numbers in words reading the place value.
a) 0.6 ___________________________________________________________________________
b) 0.37 ___________________________________________________________________________
a) 0.55 ___________________________________________________________________________
b) 0.624___________________________________________________________________________
If there is a whole number in front of the decimal point, we read that number and then say
“and” instead of “point” before reading the decimal part of the number.
EXAMPLE
Write these numbers in words reading the place value.
a) 2.4
two and 4 tenths
b) 45.29
Practice 6
Write these numbers in words reading the place value.
a) 3.7
___________________________________________________________________________
b) 20.49
___________________________________________________________________________
c) 9.261
___________________________________________________________________________
d) 8.009
___________________________________________________________________________
90 | CHAPTER 2 | RATIONAL NUMBERS
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EXAMPLE
Write these numbers in digits.
0.62
a) zero point six two
#
81.351
206.206
c) two hundred six point two zero six
Practice 7
Write these numbers in digits.
a) zero point eight eight
______________________________
b) nine point three six two
______________________________
c) seventy-eight point two
______________________________
#
______________________________
If a number has more than 3 decimal places, it can be written in groups of threes, just like
with whole numbers.
Recall: Whole Numbers
Decimal Numbers
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We can read the decimal 0.3 as ‘three tenths’.
We also read the fraction
So, 0.3 =
3
10
MATHEMATICS FOUNDATION 1
3
as ‘three tenths’.
10
!
G
COMPARING DECIMALS | 91
EXAMPLE
Write these fractions and mixed numbers as decimals.
7
= 5.7
10
a)
543
= 0.543
100
b)
5
c)
6
= 0.06
100
d)
65
9
= 65.09
100
Practice 8
Write these fractions and mixed numbers as decimals.
a)
7
100
=
b)
26
=
1000
c)
229
1000
=
d)
7
46
1000
=
e)
12
f)
1
1
1000
=
23
100
=
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You have already studied equivalent fractions. For example:
2
10
20
100
=
=
When you write these as decimals you see:
2
10
20
100
200
1000
=
0
.
2
=
0
.
2
0
=
0
.
2
0
0
200
1000
92 | CHAPTER 2 | RATIONAL NUMBERS
EXAMPLE
Write equivalent decimals for these:
0.90, 0.5, 2.3, 0.680
tenths
hundredths
thousandths
0.9
0.90
0.900
0.5
0.50
0.500
2.3
2.30
2.300
0.68
0.680
Practice 9
Write equivalent decimals for these:
0.7, 33.9, 0.800, 0.30
tenths
hundredths
thousandths
0.7
33.9
0.800
0.30
We can also compare decimals, using the signs < or > or =
EXAMPLE
Write the correct symbol, < or > or =, between these decimal numbers:
a) 0.45 _________ 0.44
Compare the
tenths place.
0.45
The tenths place is the
Since 5 > 4 we
same so we move to the
have
hundredths place.
0.45 > 0.44
0.45 > 0.44
MATHEMATICS FOUNDATION 1
0.45
COMPARING DECIMALS | 93
b)
0.09 ________ 0.107
0.09
0.107
Compare the
Since 0 < 1 we have
tenths place.
0.09 < 0.107
0.09 < 0.107
c)
0.97 ________ 0.972
0.97
0.972
The tenths’ place is the
The hundredths’ place is
same so move to the
the same so look at the
hundredths’ place.
thousandths’ place.
0.97___
0.972
The thousandths’ place has no digit. We
The thousandths’ place has the
know the value
value of 2.
of this place is 0.
Since 0 < 2 we have 0.97 < 0.972
0.97 < 0.972
d)
0.970 _____ 0.97
The tenths’ and the hundredths’ place have
In the second number the
the same value.
thousandths’ place has no
digit. We know the value of
this place is 0.
0.970 = 0.97
These numbers are the same.
94 | CHAPTER 2 | RATIONAL NUMBERS
Practice 10
Write the correct symbol, > or < = between these decimal numbers:
a) 9.42
_____ 9.04
b) 6.3 _____ 0.75
c) 3.871 _____ 3.9
d) 0.04 _____ 0.039
e) 4.0234 _____ 4.0243
f) 9.987 _____ 9.897
-9V\UKPUN+LJPTHSZ
We round decimals in a very similar way to whole numbers.
The only difference when rounding decimals, instead of replacing digits to the right of the
given place value with zeros, we remove those digits.
Therefore the steps to rounding decimals are as follows:
Rules for Rounding Decimals:
Step 1
Underline the digit of the given place value.
Step 2
Step 3
Step 4
MATHS FOUNDATION 1
Circle the digit to its right.
a)
If that circled digit is from 0 to 4, the digit in the given place stays
the same.
b)
If that circled digit is from 5 to 9, add 1 to the digit in the given place.
Remove all digits to the right of the given place.
ROUNDING DECIMALS | 95
Do you remember your decimal place values?
hundredths
thousandths
.
Decimal
tenths
.
7
2
9
= 0.729
Let’s try an example rounding with decimals.
EXAMPLE
Round 0.34 to the nearest tenth
Step 1
Underline the digit of the given place value
(tenths).
0 . 3 4
Step 2
Circle the digit to its right (4).
0 . 3 4
Step 3
Step 4
a)
If that circled digit is from 0 to 4, the digit in the
given place stays the same.
b)
If that circled digit is from 5 to 9, add 1 to the
digit in the given place.
Remove all digits to the right of the given place
value.
0 . 3
96 | CHAPTER 2 | RATIONAL NUMBERS
EXAMPLE
Round 0.761 to the nearest tenth.
Step 1
Underline the digit of the given place value
(tenths).
0 . 7 6 1
Step 2
Circle the digit to its right (4).
0 . 7 6 1
Step 3
a)
If that circled digit is from 0 to 4, the digit in the
given place stays the same.
b)
If that circled digit is from 5 to 9, add 1 to the
digit in the given place. (it is 6, so you add 1 to
the underlined digit 7 making it 8)
Remove all digits to the right of the given place
value.
Step 4
EXAMPLE
Round to the nearest tenth.
a) 0 . 6 2
0.6
b) 0 . 1 7 3
0.2
Practice 11
Round to the nearest tenth.
a) 0.51 ______________
b) 0.88 ______________
c) 0.75 ______________
d) 0.292______________ e) 0.54 ______________
f) 0.388______________
EXAMPLE
Round to the nearest hundredth.
a) 0 . 6 2 4
0.62
b) 0 . 1 7 5
0.18
Practice 12
Round to the nearest hundredth.
a) 0.512______________
b) 0.476______________ c) 0.191______________
d) 0.924______________ e) 0.577______________
MATHEMATICS FOUNDATION 1
f) 0.996______________
0 . 8
ROUNDING DECIMALS | 97
EXAMPLE
Round to the nearest thousandth.
a) 0 . 6 2 4 3 0.624
b) 0 . 1 7 5 5 1
0.176
Practice 13
Round to the nearest thousandth.
a) 0.512 4
d) 0.924 99
________
________
b) 0.476 7
________
c) 0.191 81
________
e) 0.5993
________
f) 1.9995
________
We can also round decimals to a given decimal place.
EXAMPLE
Round 0.38 to one decimal place (1 d.p.).
“3” is the 1st decimal place, so it is the same as rounding to the tenth.
0.38
0.4
EXAMPLE
Round 0.487 to two decimal place (2 d.p.).
“8” is the 2nd decimal place, so it is the same as rounding to the hundredth.
0.487
0.49
EXAMPLE
Round 0.194 7 to three decimal place (3 d.p.).
“4” is the 3rd decimal place, so it is the same as rounding to the thousandth.
0.1947
0.195
98 | CHAPTER 2 | RATIONAL NUMBERS
Practice 14
Round each number as indicated.
a) 0.83 to 1 d.p.
_______
b) 0.67 to one decimal place _______
c) 2.465 to one decimal place
_______
d) 7.809 to 1 d.p.
_______
e) 0.571 to two decimal places
_______
f) 1.345 to 2 d.p.
_______
g) 0.097 2 to 2 d.p.
_______
h) 2.998 to 2 d.p.
_______
j) 5.080 734 to 3 d.p.
_______
i) 0.372 6 to three decimal places _______
.:JPLU[PÄJ5V[H[PVUHUK+LJPTHS-VYT
is a way of writing very large or very small numbers.
KQ
a) A number between 1 and 10. For example, 1.23
b) A power with a base of 10, written as
x 10exponent.
3HYNL5\TILYZ
Look at the table below that uses powers with a base of 10. Can you see the pattern?
We say
Meaning
10 1
10 to the exponent 1
10
2
10 to the exponent 2
10×10
10 3
10 to the exponent 3
10×10×10
10 4
10 to the exponent 4
10×10×10×10
10
5
10 to the exponent 5
10×10×10×10×10
10
6
10 to the exponent 6
10×10×10×10×10×10
10
MATHEMATICS FOUNDATION 1
Decimal Number
10
100
1 000
10 000
100 000
1 000 000
LARGE NUMBERS | 99
EXAMPLE
V
a)
428 000 000
Move the decimal point to the left until you have
a number between 1 and 10.
4.28 000 000
4.28 is between 1 and 10.
8
b)
4.28 x 108
The number of places you moved the decimal point (8) is
the exponent of the power of 10.
56 000
Move the decimal point to the left until you have
a number between 1 and 10.
5.6000
5.6 is between 1 and 10.
4 places
5.6 x 104
The number of places you moved the decimal point (4) is
the exponent of the power of 10.
Practice 15
Write the numbers in a) 750 000
7.5 x 105
b) 574 000 000
____________________
c) 8 200
____________________
d) 406 000
____________________
e) 820
____________________
f) 45 600 000
____________________
100 | CHAPTER 2 | RATIONAL NUMBERS
EXAMPLE
Write the numbers in decimal form.
a) 2.3 x 105
The exponent (5) tells you that you moved the decimal point 5 places
to the left, so you must move it back to the right.
230 000
230 000
b) 8.06 x 107
“7” places to the right.
80600000.
80 600 000.
Practice 16
Write the numbers in decimal form
a) 5.1 x 104
51 000
b) 7.7 x 102
____________________
c) 4.6 x 105
____________________
d) 4.32 x 103
____________________
e) 1.234 x 105 ____________________
f) 2.6 x 107
____________________
MATHEMATICS FOUNDATION 1
LARGE NUMBERS | 101
:THSS5\TILYZ
Now look at the table below that uses powers with a base of 10. Can you see the pattern?
We say
Meaning
Fraction
Decimal
Number
10 -1
B/XB
1
10
1
10
0.1
10 -2
B/X>
1
1
X
10 10
1
100
0.01
10 -3
B/X[
1
1
1
X
X
10 10 10
1
1000
0.001
10 -4
B/X'
1
1
1
1
X
X
X
10 10 10 10
1
10000
0.0001
10 -5
B/X
1
1
1
1
1
X
X
X
X
10 10 10 10 10
1
100000
0.00001
EXAMPLE
Write the numbers in .
a) 0.000 000 052
0 000 000 05.2
Move the decimal point to the right until you have a
number between 1 and 10.
5.2 is between 1 and 10.
8
5.2 x 10
b) 0.007
0 007
The number of places you moved the decimal point (8)
is the exponent of the power of 10. Since you moved in
the opposite direction, the exponent is NEGATIVE.
Move the decimal point to the right until you have a
number between 1 and 10.
7 is between 1 and 10.
3 places
7 x 10
The number of places you moved the decimal point (3)
is the exponent of the power of 10, but NEGATIVE.
102 | CHAPTER 2 | RATIONAL NUMBERS
Practice 17
Write the numbers in a) 0.002 4
2.4 x 10X[
b) 0.000 056
____________________
c) 0.000 000 6
____________________
d) 0.002 04
____________________
e) 0.0345
____________________
f) 0.49
____________________
b) EXAMPLE
Write the numbers in decimal form.
a) 4.3 x 10X
The tells you that you moved the decimal point 5 places to
the right, so you must move it back to the left.
0.000043
.00004
0.000 043
b) 8.06 x 10X*
“7” places to the left.
0.000000806
.00000080
0.000 000 806
MATHEMATICS FOUNDATION 1
LARGE NUMBERS | 103
Practice 18
Write the numbers in decimal form
a) 4.6 x 10X
0.000 046
b) 2.1 x 10X[
____________________
c) 3.25 x 10X@ ____________________
d) 1.06 x 10X> ____________________
e) 7.8 x 10X'
____________________
f) 8.204 x 10X* ____________________
/*HSJ\SH[VY5V[H[PVU
\!]
calculator.
EXAMPLE
Multiply. 34 000 x 1 000 000
3.410
The answer is too large for the calculator, so you see 3.410 on your
display.
3.4 x 1010
This display actually means 34 000 000 000
You now know that this can also be written in decimal form.
104 | CHAPTER 2 | RATIONAL NUMBERS
EXAMPLE
If a calculator displays 5.2XBB!
^
5.2
The answer is too large for the calculator, so you see 3.410 on your
display.
5.2 x 10
This display actually means .
0.000 000 000 052
You now know that this can also be written in decimal form.
Practice 19
Write the calculator displays in and decimal form.
a) 5.513
5.5 x 1013
55 000 000 000 000
b) 7.3XB'
7.3 x 10"
0.000 000 000 000 073
c) 1.04XBB
____________________
____________________
d) 6.3510
____________________
____________________
e) 9.3XB>
____________________
____________________
f) 9.917
____________________
____________________
MATHEMATICS FOUNDATION 1
EXERCISES | 105
Section 2.2 Exercises
1. For the number 0.167, write the place value of the digit.
a) 0
____________________
b) 1
____________________
c) 6
____________________
d) 7
____________________
2. State the place value of the underlined digit.
a) 0.365
____________________
b) 2.376 ____________________
c) 1.299
____________________
d) 0.452 ____________________
3. Write these numbers in words.
a) 0.8
__________________________________________________
b) 0.25
__________________________________________________
c) 12.1
__________________________________________________
d) 6.259
__________________________________________________
e) 350.12
__________________________________________________
f) 0.990
__________________________________________________
4. Write these numbers in words using place value.
a) 0.8
__________________________________________________
b) 0.25
__________________________________________________
c) 12.1
__________________________________________________
d) 6.259
__________________________________________________
e) 350.12
__________________________________________________
f) 0.990
__________________________________________________
106 | CHAPTER 2 | RATIONAL NUMBERS
5. Write these numbers in digits.
a) zero point one seven
______________
#G
c) seventy and twenty-six hundredths
$$$$$$$$$$$$$$
______________
#
$$$$$$$$$$$$$$
e) six and two thousandths
______________
f) two hundred twenty-one and three hundred six thousandths
______________
6. Write these fractions and mixed numbers as decimals.
a)
75
=
100
b)
32
=
1000
c) 14
54
100
d) 2
4
1000
=
=
7. Write equivalent decimals for these:
0.60. 0.8, 9.4, 0.200
tenths
hundredths
thousandths
0.60
0.8
9.4
0.200
8. Write the correct symbol, > or < = between these decimal numbers:
a) 8.32
_____ 8.04
b) 5.37 _____ 0.64
c) 7.843 _____ 12.9
d) 0.06 _____ 0.029
e) 3.0234 _____ 3.0423
f) 4.217 _____ 4.017
9. Round the number to the nearest tenth.
a) 0.49 _____
b) 0.81 _____
c) 0.66 _____
d) 0.354_____
e) 0.19 _____
f) 0.958_____
MATHEMATICS FOUNDATION 1
EXERCISES | 107
10. Round to the nearest hundredth.
a) 0.471
_____
b) 0.778
_____
c) 0.660
_____
d) 0.485
_____
e) 0.495
_____
f) 0.997
_____
11. Round to the nearest thousandth.
a) 0.881 6
_____
b) 2.292 2
_____
c) 11.448 51
_____
d) 0.338 85
_____
e) 0.573 49
_____
f) 1.9995
_____
12. Round each number as indicated.
a) 0.835 to two decimal places
_______
b) 0.44 to 1 d.p.
_______
c) 1.119 5 to three decimal places_______
d) 2.498 to 2 d.p.
_______
e) 6.145 299 to 3 d.p.
_______
f) 3.299 to one decimal place
_______
g) 0.998 to one decimal place
_______
h) 0.998 to 2 d.p.
_______
i) 23.77 to 1 d.p.
_______
j) 6.199 720 to 3 d.p.
_______
B[V
a) 7.5XBB
____________________
____________________
b) 4.1X`
____________________
____________________
c) 1.04X[
____________________
____________________
d) 6.354
____________________
____________________
108 | CHAPTER 2 | RATIONAL NUMBERS
Section 2.3
Converting, Comparing and Ordering Decimals,
Fractions and Percents
(*VU]LY[PUNIL[^LLU-YHJ[PVUZHUK+LJPTHSZ
How do we compare fractions, decimals and percents to each other? We need to put them
all in the same format.
-YHJ[PVUZ[V+LJPTHSZ
You can use your calculator to easily change fractions into decimals.
This is done by dividing the numerator by the denominator.
EXAMPLE
Convert the fractions and mixed numbers to decimals.
keystrokes
on the calculator
a)
3
8
3'8
b)
5
2
5'2
c)
2
3
2'3
d) 2
3
4
e) 10
1
5
3'4
1'5
answer
on the calculator
0.375
2.5
0.666666…….
2.75
10.2
If your answer repeats, like in c), we can write this in an easier way.
2
= 0.666666…… = 0.6
3
The bar over top of a digit or digits means that they repeat forever.
MATHEMATICS FOUNDATION 1
CONVERTING BETWEEN FRACTIONS AND DECIMALS | 109
Practice 1
Convert the fractions to decimals.
a)
7
________
10
d)
8
________
5
b)
3
5
e) 3
________
c)
5
________
6
48
________
50
f)
5
7
________
9
+LJPTHSZ[V-YHJ[PVUZ
Recall your decimal place values:
thousandths
.
hundredths
Decimal
tenths
.
7
2
9
The place value of the digit furthest to the right tells you the denominator of the fraction.
EXAMPLE
Write each decimal number as a fraction.
a) 0.2 2 is in the tenths place, so…
2
10
b) 0.31 1 is in the hundredths place, so…
31
100
110 | CHAPTER 2 | RATIONAL NUMBERS
c) 0.075
5 is in the thousandths place, so…
75
1000
If there is a number (other than 0) in front of the decimal point, it simply becomes the
whole number part of the mixed number.
d) 2.8
8 is in the tenths place, so…
2
8
10
e) 6.25
5 is in the hundredths place, so…
6
25
100
f) 15.207
7 is in the thousandths place, so…
15
207
1000
Practice 2
Write each decimal number as a fraction.
a) 0.3
= _____
b) 0.27
= _____
c) 2.13
= _____
d) 30.88
= _____
e) 1.22
= _____
f) 0.9
= _____
g) 4.35
= _____
h) 0.167
= _____
Note: Anytime your answer is a fraction,
(remember this?) Again this is easy to do
with our calculators.
MATHEMATICS FOUNDATION 1
CONVERTING BETWEEN FRACTIONS AND DECIMALS | 111
EXAMPLE
Write the fractions from the last example in simplest form.
a) 0.2
=
2
10
=
b) 0.31
=
31
100
(simplest form)
c) 0.075
=
75
1000
=
d) 2.8
= 2
8
10
= 2
e) 6.25
= 6
25
100
=6
f) 15.207
= 15
207
1000
1
5
3
40
4
5
1
4
(simplest form)
Practice 3
Write each of your answers from Practice 2 in simplest form (remember, you can use your
calculator).
a)
= _____
b)
= _____
c)
= _____
d) = _____
e)
= _____
f)
= _____
g)
= _____
h) = _____
112 | CHAPTER 2 | RATIONAL NUMBERS
)*VU]LY[PUN)L[^LLU7LYJLU[ZHUK+LJPTHSZ
7LYJLU[Z[V+LJPTHSZ
To convert a percent to a decimal, you divide by 100.
EXAMPLE
Convert the percents into decimals.
a) 75%
75 ÷ 100
0.75
b) 22%
22 ÷ 100
0.22
c) 3%
3 ÷ 100
0.03
d) 52.5%
52.5 ÷ 100
0.525
e) 258%
258 ÷ 100
2.58
Did you notice something? All the answers look similar to the questions. The only difference
is that the decimal point has moved 2 places to the left.
Practice 4
Convert the percents into decimals.
a) 89%
___________
b) 37%
___________
c) 47%
___________
d) 99%
___________
e) 9%
___________
f) 2.5%
___________
g) 8.29%
___________
h) 100%
___________
i) 385%
___________
j) 3500%
___________
MATHEMATICS FOUNDATION 1
CONVERTING BETWEEN PERCENTS AND DECIMALS | 113
+LJPTHSZ[V7LYJLU[Z
To convert a decimal to a percent, you multiply by 100.
EXAMPLE
Convert the decimals into percents.
a) 0.35
0.35 x 100
35%
b) 0.82
0.82 x 100
82%
c) 1.7
1.7 x 100
170%
d) 0.003
0.003 x 100
0.3%
e) 10.5
10.5 x 100
1050%
Practice 5
Convert the decimals into percents.
a) 0.48
_________ %
b) 0.17
_________ %
c) 0.2
_________ %
d) 0.487
_________ %
e) 0.8216
_________ %
f) 1
_________ %
g) 2.38
_________ %
h) 0.002
_________ %
i) 0.00575
_________ %
j) 23.8
_________ %
**VU]LY[PUN)L[^LLU7LYJLU[ZHUK-YHJ[PVUZ
7LYJLU[Z[V-YHJ[PVUZ
Remember, percent (%) means out of 100.
EXAMPLE
Write each percent as a fraction.
75% =
75
100
20% =
20
100
54% =
54
100
114 | CHAPTER 2 | RATIONAL NUMBERS
Practice 6
Write each percent as a fraction.
a) 30% = ______
b) 8% = ______
c) 6% = ______
When the percentage is greater than 100, the result is a mixed number.
EXAMPLE
Write each percent as a fraction or mixed number.
250% =
250
100
=2
50
100
120% =
120
100
=1
20
100
305% =
305
100
=3
5
100
Practice 7
Write each percent as a fraction or mixed number.
a) 250% = ______ = ______
b) 305% = ______ = ______
c) 225% = ______ = ______
d) 197% = ______ = ______
e) 552% = ______ = ______
f) 101% = ______ = ______
-YHJ[PVUZ[V7LYJLU[Z
EXAMPLE
Write the fractions as percents.
a)
3
5
Step 1: Convert the fraction into a decimal using
your calculator.
Step 2: Convert the decimal into percent ( x100)
b)
5
8
3
= 0.6
5
0.6 x 100 = 60%
= 0.625
0.625 x 100 = 62.5%
MATHEMATICS FOUNDATION 1
CONVERTING BETWEEN PERCENTS AND DECIMALS | 115
Practice 8
V
a) 2
6
20
2
6
= 2.3
20
2.3 x 100 =
230%
0.41 6 x 100 =
41. 6 %
b)
5
12
c)
3
8
__________
d)
5
9
__________
e) 4
5
= 0.416
12
1
20
__________
f)
11
16
__________
g)
7
2
__________
h)
7
1000
__________
+*VU]LY[PUNIL[^LLU-YHJ[PVUZ+LJPTHSZHUK7LYJLU[Z
With the skills learned in this section, we can now convert between all three number forms - fractions, decimals and percents.
EXAMPLE
Fill in the chart below using the skills learned in this section.
Fraction
Decimal
Percent
3
4
3 ÷ 4 = 0.75
0.75 x 100 = 75%
1.5
1.5 x 100 = 150%
a)
b)
c)
1
5
1
=1
10
2
0.792 =
792
99
=
1000 125
79.2 ÷ 100 = 0.792
79.2%
116 | CHAPTER 2 | RATIONAL NUMBERS
Reminder:
Don’t forget to simply your answers with fractions!
Practice 9
Complete the chart below using the skills learned in this section.
Fraction or Mixed
Number (simplest form)
a)
Decimal
23
50
b)
0.45
80%
c)
2
3
d)
e)
1.2
0.3%
f)
g)
Percent
2
h)
i)
MATHEMATICS FOUNDATION 1
4
5
0.525
99.5%
CONVERTING BETWEEN PERCENTS AND DECIMALSE | 117
,*VTWHYPUNHUK6YKLYPUN-YHJ[PVUZ+LJPTHSZHUK7LYJLU[Z
The easiest way to compare and order fractions, decimals and percents is to convert everything
to decimals. You can then use the rules for ordering decimals that you learned in Module 1.
EXAMPLE
Which number is greater
a)
7
3
or
8
4
3
= 0.75
4
7
is greater
8
7 = 0.875
8
0.875 > 0.75
b)
3
or 0.72
4
3
= 0.75
4
3
is greater
4
c)
3
or 80%
4
3
= 0.75
4
80% is greater
80% = 0.80
Practice 10
Which number is greater?
a)
2
1
or
5
3
__________________
b)
2
or 0.42
5
__________________
c)
2
or 35%
5
__________________
d) 1.81 or 235%
__________________
118 | CHAPTER 2 | RATIONAL NUMBERS
EXAMPLE
Order the numbers in ascending order:
2
1
1
, 35%, 0.42,
, 50%,
5
3
4
First, convert each number to decimal form:
2
5
35%
0.42
1
3
50%
1
4
0.40
0.35
0.42
0.33…
0.50
0.25
0.42
0.50
Now, order the numbers using the decimal equivalents.
0.25
0.33…
0.35
0.40
Finally, write the original numbers that match these decimal equivalents.
1
4
1
3
35%
2
5
0.42
Practice 11
a) Order the numbers in ascending order: 0.88, 0.91,
b) Order the numbers in ascending order:
42
, 90%, 94%
50
3
5
, 0.7, 72%, 0.66,
4
8
c) Order the numbers in descending order: 1.99, 150%,
MATHEMATICS FOUNDATION 1
9
8
,
, 2.1
5 10
50%
EXERCISES | 119
Section 2.3 Exercises
Remember to simplify all fraction answers in this section and from now on!
1. Convert the fractions to decimals.
7
8
a)
3
4
________
b)
d)
12
13
________
e) 7
4
5
11
________
20
________
c)
________
f) 5
7
________
12
2. Write each decimal number as a fraction
a) 0.8 = _____
b) 0.94 = _____
c) 4.446 = _____
d) 8.12 = _____
e) 0.07 = _____
f) 0.054= _____
g) 1.005 = _____
h) 0.250 = _____
3. Convert the percents into decimals.
a) 35%
___________
b) 89%
___________
c) 52%
___________
d) 5.8%
___________
e) 6%
___________
f) 2.5%
___________
g) 0.41%
___________
h) 200%
___________
4. Convert the decimals into percents.
a) 0.31
_________ %
b) 0.49
_________ %
c) 0.9
_________ %
d) 0.458
_________ %
e) 0.0425
_________ %
f) 3
_________ %
120 | CHAPTER 2 | RATIONAL NUMBERS
5. Write each percent as a fraction or mixed number.
a) 25% = ______
b) 74% = ______
c) 5% = ______
d) 120% = ______
e) 50% = ______
f) 12% = ______
g) 515% = ______
h) 105% = ______
i) 30% = ______
j) 88% = ______
k) 62% = ______
l) 100% = ______
6. Write the fractions and mixed numbers as percents.
a) 2
1
4
3
b)
15
1
c) 1
5
11
d)
12
__________
__________
__________
__________
7. Complete the chart below using the skills learned in this section.
Fraction or Mixed Number
(simplest form)
a)
Decimal
5
8
b)
0.02
c)
d)
e)
f)
MATHEMATICS FOUNDATION 1
Percent
125%
2
3
4
0.95
5.4%
EXERCISES | 121
8. Which number is greater?
a)
3
2
or
5
3
_______________________________
b)
3
or 0.47
5
_______________________________
c)
3
or 35%
5
_______________________________
d) 1.61 or 235%
9. Put the numbers in ascending order.
3
3
, 0.31,
, 35%, 34%
8
10
10. Put the numbers in descending order.
122%,
9
10
, 1.31, 140%, 1.19,
4
8
_______________________________
122 | CHAPTER 2 | RATIONAL NUMBERS
Section 2.4
Adding, Subtracting, Multiplying and
Dividing Fractions
Performing the four arithmetic operations with fractions is very easy when using your
V
module.
((KKPUN-YHJ[PVUZ
Your calculator makes adding and subtracting fractions very easy.
EXAMPLE
Add the fractions and mixed numbers.
on your calculator
3 2
+
=
3 8 + 2 5
a)
8 5
b) 1
3 5
+
8 7
=
MATHEMATICS FOUNDATION 1
1 3 8+5 7
=
5
6
= 2
5
56
ADDING, SUBTRACTING, MULTIPLYING AND DIVIDING FRACTIONS | 123
Practice 1
Add the fractions and mixed numbers.
a)
1
3
+
=
3
4
c) 1
3
15
+3
8
24
=
________
b) 2
________
d)
5
5
+6
9
12
18 5
+
9
7
=
=
________
________
The fraction button can be used the same way for all four operations (add, subtract, multiply,
divide).
)(KKPUN:\I[YHJ[PUN4\S[PWS`PUNHUK+P]PKPUN-YHJ[PVUZ
The way you used your calculator in Section A is the same for all math operations. Just be
sure which operation is being asked for in the question!
EXAMPLE
Perform the given operations on the fractions and mixed numbers.
a)
2
5
X
5
8
b)
1
4
3
x
7
8
c)
2
1
3
÷
10
10
d)
1
8
11
+2
15
12
on your calculator
5 8 - 2 5
=
1 =
=
=
3 8 x 4 7
2 1 10 ÷ 3 10
1 11 12 + 2 8 15
=
9
40
=
11
14
= 7
= 4
27
60
124 | CHAPTER 2 | RATIONAL NUMBERS
Practice 2
Perform the given operations on the fractions and mixed numbers.
a)
3
1
x
4
3
c)
1
e)
18
g)
3
1
3
÷
8
2
=
=
2
5
X14
3
6
=
1
7
÷1
=
2
10
________
b) 2
________
d)
________
f) 2
________
h)
5
X 2 5
12
9
=
5
18
+
=
9
7
5
12
+
6
5
8
1
x2
5
4
________
________
=
=
________
________
*-PUKPUNH-YHJ[PVUVMH5\TILY
The word “of” tells you that you must multiply.
EXAMPLE
a) How much is 1
2
of 10? Show with a diagram.
Of course, this can be done on a calculator, but it is also important that you can see and
understand what is being asked for.
1
means that there are 2 total parts and 1
2
is shaded. We need to separate 10 into two equal parts, and
choose one of those parts.
Remember that
We have now separated the 10 shapes into 2 parts, and
chosen 1 of those parts (the shaded circles). Counting the
shaded circles, we can see that
1
of 10 = 5
2
Check on your calculator:
b) How much is
1
x 10 = 5
2
3
of 16? Show with a diagram.
4
3
means that there are 4 total parts and 3 are shaded.
4
We need to separate 16 into 4 equal parts, and choose 3 of
those parts. Counting the shaded circles, we can see that
3
of 16 = 12
4
Check on your calculator:
MATHEMATICS FOUNDATION 1
3
x 16 = 12
4
FINDING A FRACTION OF A NUMBER | 125
Practice 3
a) How much is
Diagram:
b) How much is
Diagram:
c) How much is
Diagram:
1
of 14? Show with a diagram then check with your calculator.
2
Check on your calculator:
1
of 15? Show with a diagram then check with your calculator.
3
Check on your calculator:
2
of 25? Show with a diagram then check with your calculator.
5
Check on your calculator:
126 | CHAPTER 2 | RATIONAL NUMBERS
+-PUKPUNH7LYJLU[VMH5\TILY
Remember from the last section that ‘of’ means to multiply.
1
of 20
2
=
1
x 20
2
= 10
!percent of a
number.
You need to remember how to change a percent to a decimal.
25% =
25
= 0.25 (remember to simply ÷ 100)
100
EXAMPLE
Find the percent of the numbers below.
a)
Change the percent to a decimal
25% of 50
0.25 x 50
= 12.5
0.082 x 500
= AED41
and multiply.
b)
8.2% of AED500
Change the percent to a decimal
and multiply.
In the last example, notice that a unit was used (AED), so we put the units in the answer.
Practice 4
Find the percent of the numbers below. Show how you change the percent to a decimal
a) 15% of 50
=
= ______
b) 18% of AED1200 =
= ______
c) 8% of 250kg
=
= ______
d) 5.5% of AED150 =
= ______
MATHEMATICS FOUNDATION 1
FINDING A PERCENT OF A NUMBER | 127
Section 2.4 Exercises
1. Perform the given operations on the fractions and mixed numbers.
=
________
b)
1
5
÷
6
8
=
________
1
2
+5
=
2
3
________
d)
7
3
x
8
4
=
________
=
________
f)
12
2
X
5
3
=
________
=
________
h)
7
2
÷1
18
12
=
________
a)
9
3
X
10
4
c)
1
e)
g)
2
9
1
2
x
4
7
5
3
x
6
4
2. Draw a diagram and check with your calculator to solve the multiplication questions.
a)
1
of 12
3
Diagram:
b)
Check on your calculator:
7
of 25
8
Diagram:
Check on your calculator:
3. Find the percent of the numbers below. Show how you change the percent to a decimal
a) 30% of 50
=
= ______
b) 15% of AED650
=
= ______
c) 5% of AED1250
=
= ______
d) 2.75% of 50
=
= ______
128 | CHAPTER 2 | RATIONAL NUMBERS
Section 2.5
Ratios and Proportions
(+LÄUPUN9H[PVZ
Ratio – the comparison of two numbers.
For example, from the diagram below, the ratio of shaded squares to non-shaded squares is
2 to 3. This means that there are 2 shaded squares and 3 non-shaded squares.
There are three ways that we write a ratio:
2 to 3
2:3
2
3
However, all three ways are said the same way, “two to three.”
EXAMPLE
Use the shapes above to write the ratios. Write the ratios in three different ways.
Description
Ratio
a) The ratio of shaded circles to non-shaded circles.
5:2, 5 to 2, 5
2
b) The ratio of non-shaded squares to non-shaded circles.
4:2, 4 to 2,
4
2
c) The ratio of shaded circles to all circles.
5:7, 5 to 7,
5
7
MATHEMATICS FOUNDATION 1
EQUIVALENT RATIOS AND PROPORTIONS | 129
In example c), the ratio included all circles, which means you must add the shaded circles and
non-shaded circles together.
Practice 1
Use the shapes above to write the ratios. Write the ratios in three different ways.
Description
Ratio
a) The ratio of shaded squares to non-shaded squares.
b) The ratio of non-shaded squares to all squares.
c) The ratio of all squares to all circles.
d) The ratio of shaded circles to shaded squares.
e) The ratio of all squares to non-shaded circles.
),X\P]HSLU[9H[PVZHUK7YVWVY[PVUZ
The ratio of non- shaded squares to shaded squares is 1 to 2. When we write this ratio in
another form, it is
1
.
2
This means that there is 1 non-shaded square for every 2 shaded squares.
The picture above still shows a ratio of 1 non-shaded square for every 2 shaded squares.
Looking at all the squares together, the ratio of non-shaded squares to shaded squares is 2 to
4. Since both ratios describe the same thing, they are called equivalent ratios.
2
1
=
4
2
130 | CHAPTER 2 | RATIONAL NUMBERS
EXAMPLE
Write equivalent ratios for the shapes in the diagram above.
1
2
non-shaded squares to
shaded squares
=
2
4
=
3
6
=
Practice 2
Write equivalent ratios for the shapes in the diagram above.
moons
to
stars
1
5
=
=
The order of a ratio is very important! For example, 2 is not the same as 3 .
2
3
Let’s look at the relationship between equivalent ratios.
1
2
=
2
4
=
3
6
=
4
8
Ratios are equivalent if they have their cross products are equal.
3
6
1
2
2 x 3 = 6 and 1 x 6 = 6, so these ratios are equivalent (equal).
4
8
2
3
2 x 8 = 16, and 3 x 4 = 12, so these ratios are not equivalent.
When two or more ratios are equivalent, they are said to be in proportion.
1
2
=
3
6
The above ratios are equivalent, so they are in proportion.
MATHEMATICS FOUNDATION 1
=
4
8
EQUIVALENT RATIOS AND PROPORTIONS | 131
EXAMPLE
Are the ratios equivalent (in proportion)? Show your work.
a)
1
2
and
5
10
b)
2
3
and
3
4
1 x 10 = 10
5 x 2 = 10
Yes, the ratios are in proportion (equivalent).
2x4=8
3x3=9
No, the ratios are not in proportion (not equivalent).
Practice 3
Are the ratios equivalent (in proportion)? Show your work.
a)
6
2
and
9
3
b)
3
12
and
5
20
c)
1
1
and
4
6
d)
9
3
and
12
4
e)
1
3
and
6
12
132 | CHAPTER 2 | RATIONAL NUMBERS
*:PTWSPM`PUN9H[PVZ
Very similar to simplifying fractions, we can simplify ratios.
Recall simplifying fractions with your calculator.
4
2
=
10
5
4
2
and
10
5
are equivalent (have the same value), but
2
5
is in simplest form.
The same is true for ratios. However be careful, because a ratio must always have 2 parts, or
terms!
EXAMPLE
Simplifying the ratios:
a)
3
1
=
6
2
Simplifying the ratio
is exactly the same as
simplifying a fraction.
b)
3
6
=
2
4
If this were a fraction, it
would simplify to 1 1 .
2
However we need to keep
2 parts or terms, so it must
remain as 3 .
2
Practice 4
Simplify the ratios using your calculator. Be sure your answers have 2 terms!
a)
6
8
=
b)
9
12
=
c)
15
=
45
d)
10
4
=
e)
15
10
=
f)
21
7
=
g)
16
20
=
h)
18
24
=
i)
6
2
=
MATHEMATICS FOUNDATION 1
SOLVING PROPORTIONS
+:VS]PUN7YVWVY[PVUZ
We learned that a proportion is when 2 ratios are equal to each other. For example, the ratios
6
2
and
make a proportion because they are equal (2 x 9 = 18; 3 x 6 = 18).
9
3
We write a proportion like this:
6
2
=
9
3
K!
6
2
=
3
Since we know the cross products must be equal, we know that
2 x ___ = 18
3 x 6 = 18
Therefore, the missing value must be 9 (2 x 9 = 18)
To complete the proportion, it looks like this:
6
2
=
9
3
EXAMPLE
Complete the proportions. Show your work.
a)
1
=
2
8
1x 8 =8
2 x __ = 8
The missing number must be
4, because 2 x 4 = 8.
The completed proportion is
4
1
=
8
2
b)
5
=
15
6
5 x 6 = 30
__ x 15 = 30
The missing number must be
2, because 2 x 15 = 30.
The completed proportion is
15
5
=
6
2
| 133
134 | CHAPTER 2 | RATIONAL NUMBERS
Practice 5
Complete the proportions. Show your work.
a)
6
1
=
2
b)
5
20
=
10
c)
d)
e)
6
3
=
12
8
=
18
9
5
=
2
6
MATHEMATICS FOUNDATION 1
EXERCISES | 135
Section 2.5 Exercises
1. Use the shapes above to write the ratios. Write the ratios in three different ways.
Description
Ratio
a) The ratio of shaded circles to non-shaded circles.
b) The ratio of all circles to all squares.
c) The ratio of shaded squares to shades circles.
d) The ratio of shaded squares to all squares.
e) The ratio of non-shaded circles to shaded circles.
2. Write equivalent ratios from the picture above.
boys
to
girls
2
3
=
=
136 | CHAPTER 2 | RATIONAL NUMBERS
3. Are the ratios equivalent (in proportion)? Show your work using cross products.
a)
6
2
and
9
3
b)
3
12
and
5
20
c)
1
1
and
4
5
d)
9
3
and
12
4
e)
1
3
and
6
12
f)
4
2
and
10
7
4. Simplify the ratios using your calculator. Be sure your answers have 2 terms!
a)
6
8
=
b)
9
=
12
c)
15
=
45
d)
10
=
4
e)
15
=
10
f)
21
=
7
g)
16
=
20
h)
18
=
24
i)
6
2
5. Complete the proportions. Show your work.
a)
b)
c)
d)
e)
2
=
3
12
5
=
6
15
3
9
=
8
7
=
14
6
4
=
6
24
MATHEMATICS FOUNDATION 1
=
SUBTITLE | 137
SKILL BUILDERS
138 | CHAPTER 2 | RATIONAL NUMBERS
SKILL BUILDERS – Section 2.1
Recognizing, Reading, Writing and Simplifying Fractions
(9LJVNUPaPUN9LHKPUNHUK>YP[PUN-YHJ[PVUZ
1. What fraction is shaded in the shapes below?
a)
_____________
b)
_____________
c)
_____________
d)
_____________
2. Write the fractions in words.
a)
5
8
___________________
b)
1
4
___________________
c)
2
9
___________________
d)
1
2
___________________
e)
11 ___________________
20
f)
3
4
___________________
or
___________________
MATHEMATICS FOUNDATION 1
SKILL BUILDERS | 139
)>YP[PUN>VYKZHZ-YHJ[PVUZ
2. Write the words as fractions
a) one-half
____
#
$$$$
b) three fourths ____
e) nine-twelfths ____
# $$$$
f) one-tenth
____
3. Write the fractions found in the sentences.
a) One out of every three students go to college.
________
b) Five out of eight people went to Dubai to shop.
________
#
$$$$$$$$
d) Three of my four sisters are coming home for dinner tonight.
________
e) I scored nine out of ten on the quiz today.
________
*7YVWLY-YHJ[PVUZ0TWYVWLY-YHJ[PVUZHUK4P_LK5\TILYZ
4. Write the numerator to make each of the fractions below, equal to 1 whole.
5
11
35
47
28
5. State whether each of these is a proper fraction, an improper fraction, or as a mixed
number. Then write the fraction in words.
4
7
a)
3
b)
10
4
c)
1
12
140 | CHAPTER 2 | RATIONAL NUMBERS
+7LYJLU[-YHJ[PVUZ
6. What percent of each diagram is shaded?
a) _______ %
b) _______ %
E. Simplifying Fractions with a Calculator
7. Using your calculator, simplify the fractions.
a)
3
_______
6
b)
10
_______
20
f) 9 _______
45
e) 25 _______
30
c)
8
_______
12
d)
44
_______
50
g)
5
_______
11
h)
50
_______
75
SKILL BUILDERS – Section 2.2
Reading, Writing, Comparing and Rounding Decimals
(9LHKPUNHUK>YP[PUN+LJPTHSZ
1. For the number 0.946, write the place value of the digit.
a) 4
____________________
b) 9
____________________
c) 6
____________________
d) 0
____________________
2. State the place value of the underlined digit.
a) 0.141
____________________ b) 3.556
____________________
c) 0.496
____________________ d) 0.401
____________________
MATHEMATICS FOUNDATION 1
SKILL BUILDERS | 141
)>YP[PUN+LJPTHSZPU>VYKZ
3. Write these numbers in words.
a) 0.2
__________________________________________________
b) 0.36
__________________________________________________
c) 56.7
__________________________________________________
d) 8.829
__________________________________________________
e) 771.84
__________________________________________________
f) 7.009
__________________________________________________
4. Write these numbers in words using place value.
a) 0.5
__________________________________________________
b) 0.89
__________________________________________________
c) 7.6
__________________________________________________
d) 2.105
__________________________________________________
e) 200.02
__________________________________________________
f) 0.921
__________________________________________________
*>YP[PUN>VYKZHZ+LJPTHSZ
5. Write these numbers in digits.
a) zero point three
______________
b) one point two two six
______________
c) six and fourteen hundredths
______________
d) nine and eight thousandths
______________
#
$$$$$$$$$$$$$$
#
$$$$$$$$$$$$$$
142 | CHAPTER 2 | RATIONAL NUMBERS
++LJPTHSZHUK-YHJ[PVUZ
6. Write these fractions and mixed numbers as decimals.
a)
49
100
c)12
=
b)
26
=
100
72
=
1000
d) 3
15
=
1000
7. Write equivalent decimals for these:
0.40. 0.3, 7.2, 0.600
tenths
hundredths
thousandths
0.40
0.3
7.2
0.600
,*VTWHYPUN+LJPTHS5\TILYZ
8. Write the correct symbol, > or < = between these decimal numbers:
a) 7.32
_____ 8.06
b) 5.41 _____ 0.64
c) 7.843 _____ 11.6
d) 0.04 _____ 0.018
e) 3.0135 _____ 3.0351
f) 6.207 _____ 6.017
-9V\UKPUN+LJPTHSZ
9. Round the number to the nearest tenth.
a) 0.58 _____
b) 0.72 _____
c) 1.39 _____
d) 2.511_____
e) 0.27 _____
f) 0.955_____
MATHEMATICS FOUNDATION 1
SKILL BUILDERS | 143
10. Round to the nearest hundredth.
a) 0.336
_____
b) 0.492
_____
c) 0.293
_____
d) 1.478
_____
e) 0.994
_____
f) 0.998
_____
11. Round to the nearest thousandth.
a) 0.223 6
_____
b) 2.292 2
_____
c) 11.448 51
_____
d) 0.444 76
_____
e) 0.339 5
_____
f) 1.989 7
_____
12. Round each number as indicated.
a) 0.85 to one decimal place
_______
b) 0.32 to 1 d.p.
_______
c) 1.195 to two decimal places _______
d) 3.808 to 2 d.p.
_______
e) 6.145 91 to 3 d.p.
_______
f) 3.29 to one decimal place
_______
g) 0.799 to one decimal place
_______
h) 0.005 to 2 d.p.
_______
i) 15.15 to 1 d.p.
_______
j) 1.055 72 to 3 d.p.
_______
.:JPLU[PÄJ5V[H[PVUHUK+LJPTHS-VYT
B[V
a) 4.5X<
____________________
____________________
b) 3.7X'
____________________
____________________
c) 5.23X
____________________
____________________
d) 8.354
____________________
____________________
144 | CHAPTER 2 | RATIONAL NUMBERS
SKILL BUILDERS – Section 2.3
Converting, Comparing and Ordering Decimals,
Fractions and Percents
(*VU]LY[PUNIL[^LLU-YHJ[PVUZHUK+LJPTHSZ
1. Convert the fractions to decimals.
a) 4 ________
5
d)
6
________
2
b)
1
________
8
e) 3
c)
1
________
2
4
________
20
f) 4
12
________
16
2. Write each decimal number as a fraction
a) 0.2 = _____
b) 0.25 = _____
c) 0.182 = _____
d) 4.72 = _____
e) 0.05 = _____
f) 0.106= _____
g) 2.001 = _____
h) 0.119= _____
)*VU]LY[PUNIL[^LLU7LYJLU[ZHUK+LJPTHSZ
3. Convert the percents into decimals.
a) 25% ___________
b) 14% ___________
c) 8% ___________
d) 7.5%___________
e) 99% ___________
f) 5.75%___________
g) 0.82%___________
h) 150%___________
4. Convert the decimals into percents.
a) 0.20 _________ %
b) 0.75 _________ %
c) 0.7 _________ %
d) 0.425_________ %
e) 0.003_________ %
f) 10
MATHEMATICS FOUNDATION 1
_________ %
SKILL BUILDERS | 145
**VU]LY[PUNIL[^LLU7LYJLU[ZHUK-YHJ[PVUZ
5. Write each percent as a fraction or mixed number.
a) 20% = ______
b) 75% = ______
c) 10% = ______
d) 175% = ______
e) 60% = ______
f) 88% = ______
g) 750% = ______
h) 101% = ______
i) 81% = ______
j) 64% = ______
k) 0.8% = ______
l) 5 000% = ______
6. Write the fractions and mixed numbers as percents.
a) 1
__________
5
20
b)
c) 2
d)
2
5
__________
1
2
__________
7
8
__________
+*VU]LY[PUNIL[^LLU-YHJ[PVUZ+LJPTHSZHUK7LYJLU[Z
7. Complete the chart below using the skills learned in this section.
Fraction or Mixed Number
(simplest form)
a)
Decimal
Percent
3
6
b)
0.15
c)
210%
d)
1
e)
f)
6
24
1.4
2.75%
146 | CHAPTER 2 | RATIONAL NUMBERS
,*VTWHYPUNHUK6YKLYPUN-YHJ[PVUZ+LJPTHSZHUK7LYJLU[Z
8. Which number is greater?
a)
3
2
or
5
3
_________________________
b)
3
or 0.47
5
_________________________
c)
3
or 35%
5
_________________________
d) 1.61 or 235%
9. Put the numbers in ascending order.
4
85
, 0.78,
, 81%, 95%
5
100
10. Put the numbers in descending order.
275%,
10
7
, 2.9, 300%, 2.41,
4
2
MATHEMATICS FOUNDATION 1
_________________________
SKILL BUILDERS | 147
SKILL BUILDERS – Section 2.4
Adding, Subtracting, Multiplying and Dividing Fractions
(HUK)(KKPUN:\I[YHJ[PUN4\S[PWS`PUNHUK+P]PKPUN-YHJ[PVUZ
1. Using your calculator, perform the given operations on the fractions and mixed numbers.
a)
2
9
X
5
10
=
________
b)
1
5
÷
4
8
=
________
1
2
+
8
7
=
________
d)
1
3
x
2
4
=
________
=
________
f)
4
1
X
10
3
=
________
=
________
h)
6
1
÷1
9
2
=
________
c) 2
e)
7
1
x
8
4
g) 2
1
9
x
4
10
*-PUKPUNH-YHJ[PVUVMH5\TILY
2. Draw a diagram and check with your calculator to solve the multiplication questions.
a)
1
4 of 12
Diagram:
b)
Check on your calculator:
2 of 9
3
Diagram:
Check on your calculator:
148 | CHAPTER 2 | RATIONAL NUMBERS
+-PUKPUNH7LYJLU[VMH5\TILY
1. Find the percent of the numbers below. Show how you change the percent to a decimal
a) 30% of 20
=
= ______
b) 15% of AED300
=
= ______
c) 4% of AED 840
=
= ______
d) 5.5% of 8
=
= ______
SKILL BUILDERS – Section 2.5
Ratios and Proportions
(+LÄUPUN9H[PVZ
1. Use the shapes above to write the ratios. Write the ratios in three different ways.
Description
Ratio
a) The ratio of all squares to all circles.
b) The ratio of shaded squares to all circles.
c) The ratio of non-shaded squares to shaded circles.
d) The ratio of shaded squares to all squares.
e) The ratio of all circles to all squares.
),X\P]HSLU[9H[PVZHUK7YVWVY[PVUZ
2. Write equivalent ratios from the picture above.
boys
to
girls
MATHEMATICS FOUNDATION 1
1
4
=
=
SKILL BUILDERS | 149
3. Are the ratios equivalent (in proportion)? Show your work using cross products.
a)
8
2
and
12
3
b)
3
18
and
5
30
c)
1
2
and
4
10
d)
9
4
and
12
5
e)
1
3
and
6
15
f)
1
10
and
10
100
*:PTWSPM`PUN9H[PVZ
4. Simplify the ratios using your calculator. Be sure your answers have 2 terms!
8
=
10
b)
15
=
20
c)
10
=
30
d) 12 =
8
e)
40
=
30
f)
12
=
4
h)
9
=
10
i)
5
=
1000
a)
g)
80
=
100
+:VS]PUN7YVWVY[PVUZ
5. Complete the proportions. Show your work.
a)
b)
c)
d)
e)
5
=
6
12
5
=
1
30
18
9
=
20
2
=
14
21
4
=
6 18