Download UNIT 2: POWER AND ROOTS

FRACTIONS AND DECIMALS
1. Fractions
When an object is divided into a number of equal parts, then each part is called a fraction.
There are different ways of writing a fraction.
For example, two fifths of an object can be written as



a common fraction  2/5
a decimal  0.4
a percentage  40%
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numerator says how many parts in the fraction
5
denominator says how many equal parts in the whole object
Always remember: denominator can NEVER be 0.
Why? Because you cannot divide by 0.
2. Equivalent fractions

Two fractions are said to be equivalent when simplifying, both of them produces
the same fraction written in its simplest terms.
Equivalent fractions are fractions with identical values.
To create a pair of equivalent fractions, you multiply (or divide, cancelling down)
the top (numerator) and bottom (denominator) of a given fraction by the same
number.



Two fractions are equivalent if the cross-products are the same.
Exercises
1. Cancel down the following fractions into their simplest terms:
a)
b)
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c)
2. Arrange these fractions in order of size, smallest first:
3. Operations involving fractions
a. Adding and subtracting fractions
When adding (or subtracting) fractions with different denominators, they must be
rewritten to have the same denominator before starting the addition.
b. Multiplying and dividing fractions



To multiply: You must simply multiply the two top numbers, and multiply the two
bottom ones.
To divide one fraction by another, turn the second fraction upside down and then
multiply them. (You cross-multiply)
Don’t forget: To multiply or divide by a whole number, just treat it like a fraction
with a denominator of 1.
4. Decimal numbers: decimal places
One decimal place to the left of the decimal point is the ones place. One decimal place to the
right of the decimal place is the tenths place.
Keep your eye on the 9 to see where the decimal places fall.
millions
hundred thousands
ten thousands
thousands
hundreds
tens
ones
tenths
hundredths
thousandths
ten thousandths
hundred thousandths
9,000,000.0
900,000.0
90,000.0
9,000.0
90.0
9.0
0.9
0.09
0.009
0.0009
0.00009
0.000009
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5. Types of decimal numbers
An exact or terminating decimal is one which does not go on forever, so you can write down all
its digits. For example: 0.125
A recurring decimal is a decimal number which do not stop after a finite number of decimal
places, but where some of the digits are repeated over and over again.
For example: 0.1252525252525252525... is a recurring decimal, where '25' (called the period) is
repeated forever.
There exists two types of recurring decimals:
 Pure recurring decimal: It becomes periodic just after the decimal point. Ex. 1.3535…
(35 is called the period). It is usually expressed as
 Eventually recurring decimal: When the period is not settled just after de decimal point.
Ex. 1.457777… or 1.45
6. Scientific notation or standard form
Scientific notation is a way of writing numbers that are too big or too small to be conveniently written
as ordinary numbers. Scientific notation is commonly used in calculators and by scientists,
mathematicians and engineers.
In scientific notation all numbers are written in the form of
(a times ten raised to the power of b), where the exponent b is an integer, and the coefficient a is a
decimal number between 1 and 10. It gives the number’s order of magnitude. If the number is
negative then a minus sign precedes a (as in ordinary decimal notation).
Standard decimal notation Scientific notation
2
2×100
300
3×102
4321.768
4.321768×103
0.2
2×10−1
0.000 000 007 51
7.51×10−9
7. Approximation and rounding.
When a number have a lot of decimal places, it is usual to cut some digits off to make the
number shorter. In this case we are making an approximation.
If we take into consideration the following decimal digit in order to make the best
approximation, we are rounding.
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Let see the following examples:
123,537
Rounding to the nearest ones
(or rounding to the nearest
whole number)
124 (because the following
digit, 5, is ≥ 5)
Rounding to the nearest
tenths (or rounding to two
decimal places)
123,5
(because
the
following number, 3 is <5)
Rounding to the nearest
hundredths (or rounding to
three decimal places)
123,54
(because
the
following number is 7≥5)
4, 092
4
4,1
4,09
34, 736
56, 509
More exercises:
1. Express the following numbers into a fraction format. Write down which type of decimal number is
everyone.
a. 2.25
c. 0.18…
b. 0.2…
d. 0.2544444….
2. Calculate in your head and then answer:
a. How many minutes are there in 1/5 of an hour?
b. How many minutes are there in 5/6 of an hour?
c. What fraction of an hour is 20 minutes?
d. What fraction of an hour is 40 minutes?
3. Three – eighths of a cake that weighed 1. 2 kg has been eaten. How much does the remaining cake
weigh?
4. John had €245.000 in his current account before he invested two-fifths of the money in an insurance
company. How much did he invest? How much is left in the account?
5. A car dealer is selling a new model for €12.000, with one-sixth of the price to be paid upfront and the rest
in forty equal monthly payments. How much is each monthly payment?
6. Three-quarters of a kilo of cheese costs €9.75. How much does one kilo of cheese cost?
7. On a tree plantation, 3 out of every 20 trees have been cut down. If 840 trees have been cut down, how
many are left?
8. A lorry’s tank contains 25 litres of diesel, and the gauge says the tank is ¾ full. How many litres can the
tank hold?
9. Alberto moves forward 5/6 of a metre with each step. How many steps must he take to complete 9
kilometer walk?
10. In a bicycle race, cyclist A has covered 4/5 of the total route and has 21 km left before the finish line.
How many kilometers are left before cyclist B reaches the finish line, if he has covered 2/7 of the route?
11. Vitoria is planning for her holiday. She calculates that if she spends a third of her savings on a plane
ticket and a quarter on a hotel, she will still have €450 left. How much money does she have?
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