Download UNIT 1: REAL NUMBERS Equivalent fractions Two fractions are

UNIT 1: REAL NUMBERS
1. 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)
c)
2. Arrange these fractions in order of size, smallest first:
2. 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.
libredisposicion.es
3. Types of decimal numbers
There are three different types of decimal number: exact, recurring and other
decimals.
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' 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)
 Eventually recurring decimal: When the period is not settled just after de
decimal point. There is a not repeating number placed between the
decimal point and the period.
Other decimals are those which go on forever and don't have digits which
repeat. For example pi = 3.141592653589793238462643.. They are called
irrational numbers.
4. Number sets
All the numbers in the Number System are classified into different sets and those sets are
called as Number Sets.
The set of real numbers is divided into natural numbers, whole numbers, integers,
rational numbers, and irrational numbers.
libredisposicion.es
5. Estimating and rounding decimal numbers
Estimating is an important part of mathematics, since it supposes a very handy tool for
everyday life. Get in the habit of estimating amounts of money, lengths of time or
distances.
There are different ways to estimate a number:
a) Rounding up or down.
To round up, just cut off the number at the place you want, and sum one to the
last digit.
To round down, just cut off the number at the place you want.
See the table below.
b) Rounding is the most usual kind of estimating.
To round a decimal number, find the place value you want and look at the digit just to
the right. If it is 5 or more, then round up. If it is 4 or less, round down.
Then, sometimes we round up, and sometimes we round down. It depends on the digit
to the right.
15,2753
Rounding down
15,2
15,27
15,275
8,7451
Rounding up
15, 3
15,28
15,276
Rounding
15,3
15,28
15,275
Examples of Number Sets



N= {1, 2, 3…….} is the set of natural numbers.
(0, 1, 2, 3……} is the set of whole numbers.
I= {……-3, -2, -1, 0, 1, 2, 3….} is the set of integers.

The numbers

10.3, 102.25, etc. are rational numbers but not integers.
,
, etc. are irrational numbers.
More exercises:
3. Do the exercises 81, 82, 84, 87 on page 20.
4. Express the following numbers in a fraction format. Write down which type of decimal
number is everyone.
a. 2.25
b. 0.2…
c. 0.18…
d. 0.2544444….
libredisposicion.es
5. 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?
6. Three – eighths of a cake that weighed 1. 2 kg has been eaten. How much does the
remaining cake weigh?
7. 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?
8. 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?
9. Three-quarters of a kilo of cheese costs €9.75. How much does one kilo of cheese cost?
10. 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?
11. 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?
12. Alberto moves forward 5/6 of a metre with each step. How many steps must he take
to complete 9 kilometer walk?
13. 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?
14. 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?
libredisposicion.es