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UNIT 2 VOCABULARY: FRACTIONS AND DECIMALS
0. How to read fractions?
REMEMBER! TERMS OF A FRACTION
•
•
Fractions are written in the form
a
where a and b are whole numbers, and the
b
number b is not 0.
• The number a is called the numerator, and tells us how
many equal parts we have.
The number b is called the denominator, and tells us how many
equal parts are available.
REMEMBER! READING FRACTIONS
We use the cardinals to name the numerator and the ordinals for the denominator.
For example:
2
--> two thirds
3
7
--> seven fifths
5
1
--> one eight
8
Three exceptions:
• When the denominator is 2, it is read "half".
For example:
•
3
--> three halves
2
When the denominator is 4, it can be read as "fourth" or "quarter".
For example:
•
1
--> one half
2
1
--> a fourth or a quarter
4
3
--> three quarters or three fourths
4
For denominators greater than 10, we say “over” and do not use ordinal.
For example:
12
--> twelve over fifteen
15
17
--> seventeen over thirty-two
32
BE CAREFUL WITH PLURALS!
If the numerator is greater than 1, you must use plurals with ordinals.
5
--> five halves
2
3
--> three quarters
4
Exercise. Write in words and read the following fractions:
11
4
17
2
1
12
_______________
________________
_______________
3
_______________
8
4
_______________
9
14
______________
93
7
--> seven tenths
10
9
_______________
6
19
______________
62
17
_____________
100
Exercise. Write and read the fractions that represent the shaded portions.
Exercise. Which fraction of this window is broken?
Which fraction is OK?
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Equivalent fractions are fractions that look different from each other, but are really the same.
We can test if two fractions are equivalent by taking the cross-product.
Exercise.
3
5
and
8
4
2. Write the missing numerators or denominators:
1. Write three equivalent fractions of
x 3
=
28 2
−12
5
=
x
15
72
x
=
30
2
35
7
=
4
x
REMEMBER! SYMBOLS
To compare two numbers, we can use these symbols:
Symbol
Is read
=
Is equal to / equals
Example
1
2
≠
Isn't equal to / doesn't equal
2
3
<
Is less than
2
3
>
Is greater than
5
3
=
≠
<
>
Is read
A half equals two fourths
2
4
Two thirds doesn't equal three halves
3
2
Two thirds is less than three quarters
3
4
Five thirds is greater than five sixths
5
6
Exercise. For each fraction pair, put them boyh over a common denominator to see which is bigger.
Write out the original fraction pair using the "greater than" sign (>).
3
4
and
4
5
1
and
3
2
5
13
and
20
Simplifying (or reducing) fractions means to make the fraction as simple as possible.
Why say four-eighths
()
4
8
when you really mean half
()
1
2
?
7
10
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There are two ways to find the simplest form of a fraction:
• Try dividing both the top and bottom of
the fraction until you can't go any
further (try dividing by 2,3,5,7,... etc).
For example, to reduce
24
to its lowest
108
terms:
•
You can divide both the top and bottom of the fraction by the
Greatest Common Factor, (you have to work it out first!).
For example, to find the simplest form of
▪
▪
8
:
12
The GCF of 8 and 12 is 4.
Divide both top and bottom by 4.
Exercise. Find the simplest form of these fractions:
24
75
=
=
36
55
84
240
50
=
120
=
1.1. Addition and subtraction with fractions
REMEMBER!
+  PLUS
-  MINUS
To add or subtract fractions:
• If the fractions have the same denominator, the numerator of the sum or the
difference is found by simply adding or subtracting the numerators over the denominator.
2 3 2+3 5
2 3 2−3
1
+ =
=
− =
=−
For example,
4 4
4
4
4 4
4
4
REDUCE ALWAYS WHEN POSSIBLE !!!!
•
If the
▪
▪
▪
denominators are different, follow these steps:
Reduce them to a common denominator (see comparing fractions).
Add or subtract the numerators and do not change the denominator.
Reduce (if possible).
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For example, to find out
5
5
6
13
9 10−24+27
−2+ =
=
6
4
12
12
−2+
9
4
. The LCM of 6 and 4 is 12, so
.
Exercise. Work out:
a)
3
1 3
+ +
8 4 16
1 3
b) 2+ −
9 5
4 2
−
7 3
c)
( )
1 1
1 1
+ − +
2 3 4 5
d)
e)
( )
1 1
1 1
− − +
4 8 3 6
1.2. Multiplication with fractions
REMEMBER!
●  TIMES/MULTIPLIED BY
To multiply fractions:
• Multiply the top numbers (numerators).
• Multiply the bottom numbers (denominators).
• Simplify the fraction (if possible).
Exercise.
1. Work out:
a)
2 1 3
⋅ ⋅
3 4 5
( )
3 2
3
− ⋅ ⋅−
7 9
5
b)
2. Work out:
3
a)
of 90
5
b)
3
4
of
c)
2
9
c)
4
7
( )( )
1
3
4
− ⋅− ⋅−
9
11
7
of 490
1.3. Division with fractions
REMEMBER!
•
:  DIVIDED BY
To divide fractions:
• Multiply the numerator of the first fraction by the denominator of the second.
• Multiply the denominator of the first
fraction by the numerator of the second.
Simplify the fraction (if possible).
Exercise. Work out:
a)
4 3
:
5 7
b)
9 7
− :
12 5
c)
3
7
2
8
−
4
11
−
3
16
d)
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1.4. Order of Operations
When you have several operations to do, which one do you
calculate first?
We work out operations in this order:
BRACKETS
EXPONENTS (Powers, roots, etc)
DIVISION and MULTIPLICATION (working from left to right)
ADDITION and SUBTRACTION (working from left to right)
That makes
BEDMAS!
2.1. Reading decimals
REMEMBER! PLACE VALUE WITH DECIMALS!!
In the decimal number system ,the value of a digit depends on where it is placed:
Decimal numbers are read with each figure separate. We use a full stop (called
"point"), not a comma, before the decimal places.
You can also read the full number after the decimal point and then say the word for
the last place value.
Examples:
3.375 ➔ three point three seven five or three point three hundred and seventy-five thousandths.
0.75 ➔ (nought or zero) point seven five or seventy-five hundredths.
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Exercise. Read the following numbers:
120,000.321
34.76
453.897
0.54
700,560
0.054
5,542.678987
8,275.4
3.2. Converting fractions into decimals
Every fraction can be expressed as a decimal number. To convert a fraction into a decimal, you just
have to divide numerator by denominator.
The quotient of a fraction can be:
•
•
•
Integers: no decimal part. For example,
6
=3
2
Exact (or terminating) decimals: decimal numbers that
end (or terminate).
3
=0.3
For example,
10
Recurring decimal numbers: decimal numbers that have
a recurring pattern of a single or multiple digits.
1
2
̂
̂
=0.333333 ...=0. 3
=0. 18...
For example,
3
11
31
=2.58 3̂
12
3.3. Approximations
An approximation of a number is a representation of that number that is not exact, but still close
enough to be useful.
Rounding off a decimal number to a given number of decimal places is the quickest way to
approximate a number.
For example, if you wanted to round off 2,6525272 to three decimal places, you would:
Step 1: Mark off the required number of decimal places. 2,652|5272
Step 2: Check the next digit to see if you must round up or round down. Remember: if the next digit is
5 or more, you must round up, and if it is 4 or less, you must round down. 2,652|5272 must be rounded
up.
Step 3: Write the final answer. 2,653 rounded to 3 decimal places.
Exercises.
1. Round off these distances to the nearest 100 m (to one decimal place):
a) 5.768 km
c) 8.48 km
e) 17.685 km
b) 9.039 km
d) 8.41 km
f) 17.658 km
2. To approximate a number by truncation is to omit the figures at the right of the the decimal
place you want to approximate to. Truncate these numbers to one decimal place:
a) 4.73
b) 19.476
c) 14.006
d) 20.89
e) 0.09871
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4.1. Rational numbers
A rational number is any number which can be written in te form of a fraction. Rational
numbers are represented by the letter ℚ.
So, a rational number can be written as
a
, where a and b are integers and b ≠ 0.
b
But, what numbers can be expressed as a fraction?
Rational numbers include:
RATIONAL NUMBERS
Natural Numbers ℕ
6=
6
1
Integers ℤ
−3=−
ℚ
Exact (or terminating)
decimal numbers
3
1
0.75=
3
4
Recurring decimal
numbers, with a
periodic part.
1.6666...=
5
3
Exercise. Match each number with an appropriate sentence.
4.2. Converting decimals into fractions
Exact and recurring decimals can be expressed as fractions. To convert a decimal into a fraction, you
have to follow these steps:
YOU CAN ONLY WRITE RATIONAL NUMBERS AS A FRACTION, THIS IS, JUST
EXACT AND RECURRING DECIMALS !!!
•
•
If a decimal number is exact:
◦ The numerator is formed by the digits without the decimal point.
◦ The denominator is the number formed by “1” and as many zeros as decimal figures the
number has.
◦ Reduce, if possible.
275 11
=
For example, 2.75=
100 4
If a decimal number is recurring:
◦ The numerator is the difference between the number formed by the figures including the
period (once) and the number formed by the figures BEFORE the period.
◦ The denominator is the number formed by as many 9 as decimal figures the period has and
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as many 0 as decimal figures the non-periodic part has AFTER the decimal point.
Reduce (f possible).
For example,
53−5 48 16
=
=
Here, the period has one figure (3).
9
9
3
2318−23 2295 51
2.3 ̂
18=
=
=
Here, the period has two figures (18) and the
99
990 22
non-periodic part 1 figure (3).
̂
5. 3=
4.1. Irrational Numbers
Irrational numbers are numbers that
denominator as integers.
cannot be written as a fraction with the numerator and
FAMOUS IRRATIONAL NUMBERS
Pi is a famous irrational number. People have calculated Pi to over one million
decimal places and still there is no pattern. The first few digits look like this:
3.1415926535897932384626433832795 (and more ...)
The number e (Euler's Number) is another famous irrational number. People have
also calculated e to lots of decimal places without any pattern showing. The first
few digits look like this:
2.7182818284590452353602874713527 (and more ...)
The Golden Ratio is an irrational number. The first few digits look like this:
1.61803398874989484820... (and more ...)
Many square roots, cube roots, etc are also irrational numbers. Examples:
√ 2 = 1.4142135623715 ... (etc)
√ 3 = 1.732050807568 ... (etc)
BE CAREFUL!
√ 4 = 2 (rational), and
√9
= 3 (rational) ...
so not all roots are irrational.
Real Numbers include:
•
•
Rational numbers, and
Irrational numbers.
In fact a Real Number can be thought of as any point anywhere on the number line:
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