## Lesson 2: Fractions IV

October 26, 2010

### 7. Word problems on fractions

Fraction Problems are word problems that deal with fractions i.e. parts of a whole.

Remember to read the question carefully. I couldn’t find a web on English with interactive word problems on fractions. Don’t worry we will practise in class.

Here you are a video on a fraction problem.

# Time to relax.

## Lessson 2: Fractions III

October 18, 2010

### 4.2. Multiplying fractions

We do not need a common denominator when multiplying fractions.

To multiply two (or more) fractions,

 a · c b d

you have to:

• Multiply out the numerators and the result will be the numerator
• Multiply out the denominators and the result will be the denominator.
• “Tidy up” the result – simplify it if possible.

 a · c = a · c = Multiply the numerators b d b · d Multiply the denominators

Examples:

 1. Find 2 · 2 3 5

 2 · 2 = 2 · 2 = Multiply the numerators 3 5 3 · 5 Multiply the denominators = 4 15

You can learn why we multiply in this way on Descartes page. If you click on the rightarrow at the bottom of the screen you can also practise multiplication of fractions.

### 4.4. Dividing fractions

The reciprocal of a fraction (fracción inversa in Spanish) is another fraction such as the product of by its reciprocal is 1. It is obtained by interchanging the numerator and the denominator, i.e. by inverting the fraction.

Examples

When you turn the fraction upside down you have a RECIPROCAL of the original fraction.

Turning fraction upside down is also called inverting.

 A reciprocal of 3 is 4 4 3

To divide two fractions multiply the dividend by the reciprocal of the divisor.

You can do this by following few simple steps:

• Turn the second fraction (divisor) upside down.
• Change the divide sign to multiply sign and multiply the fractions.

Example

 Find 1 : 2 2 5

 1 : 2 = 1 · 5 = Change the sign to ·. 2 5 2 2 Invert (turn upside down) the divisor.

 = 1 · 5 = 5 Multiply the numerators. 2 · 2 4 Multiply the denominators.

Another way of dividing two fractions is by multiplying the terms of both fractions “crossing them”:

 a : c = a·d b d b·c

This method is very useful and we will use it most of the times. Practise on the IXL web.

### 5. Combined operations with fractions

Combined operations is an expression formed by fractions in different operations and grouped in different ways – by means of brackets, square brackets and braces {curly brackets}.

The aim of using brackets is to join or to group everything that is between them. You must respect an order to solve combined operations, the hierarchy:

1. What is inside the brackets and square brackets.
2. Powers and roots.
3. Multiplication and division from left to right.
4. Addition and subtraction from left to right

Be careful because mixed numbers appear frequently. Recall they mean an integer added to a proper fraction.

Finally, here you are a couple of videos on multiplying and dividing. Try to understand the explanation. If you have any difficulty, write to me.

### 6.1. Power of a fraction

To raise a fraction to a power you have to raise numerator and denominator to this power:

Example:

• (2/5)3 = 23 / 53 = 8 / 125

Practise on this web. You will find bases which are decimal numbers (fractions can express decimal numbers). Don’ t do those examples.

### 6.2. Root of a fraction

Square root of a fraction is the root of numerator divided by the root of denominator.

Perfect square root or exact square root of a fraction is a number whose square is equal to the fraction.

## Lesson 2: Fractions II

October 17, 2010

2.1. Finding the Lowest Common Denominator

Sometimes, it is interesting to find a common denominator as low as possible. Some applications of it will be adding and subtracting fractions and ordering fractions.

To find the lowest possible common denominator of 2 or more fractions:

1. Simplify the fractions if possible
2. Find the Lowest Common Denominator, which is the Lowest Common Multiple of all denominators.
3. To work out the new numerator of each fraction, divide the Lowest Common Multiple  by each denominator and multiply the quotient by the corresponding numerator.

You can undertand the technique on this link by clicking on the picture

You can practise on

### 3. Comparing fractions

To compare fractions we can work out their values (dividing numerator by denominator) and compare them or we can apply the following criteria:

• If the two fractions have the same denominator, the larger fraction is the fraction with larger numerator.
• If the two fractions have equal numerator and different denominators, the larger fraction is the fraction with smaller denominator.
• If the two fractions have different numerators and different denominators we reduce them to their Lowest Common Denominator and we apply the criterion before.

Remember:

Descending order means from the largest to the smallest (or going down).
Ascending order means from the  smallest to the largest (or going up).

Do you remember signs “>” and “<“?

Sign  >  means  greater ( 2 > 1 means “2 is greater than 1”)

Sign  <  means  smaller ( 2 < 99 means “2 is smaller than 99”)

﻿Practise and play a funny game on

You will find harder exercises on

### 4. Operations

To add or subtract fractions with the same denominator you have to do is to add or subtract the numerators.

And it is always a good idea to make your result “nice” by simplifying if possible.

Example:

 Find 1 + 2 5 5

Both fractions have the same denominator of 5, so we can simply add the numerators:

 1 + 2 = 3 5 5 5

To add or subtract fractions with different denominators we must first make the denominators the same (by finding the Lowest Common Denominator and using equivalent fractions), and finally add or subtract the new numerators and write the new denominator.

Remember, simplify if possible!

Here you can see examples step by step and progress  by practising. In order to advance you have to click on the right arrow placed at the bottom of the screen.

Here you are a video on adding fractions very easy to understand

## Lesson 2: Fractions. First definitions (I)

October 16, 2010

Vocabulary on Fracions

Lesson 2: Fractions (Notes)

Word Problems on Fractions
I Have to Know by the End of this Lesson

### 1. Fractions

Definition: A fraction is an expression

 a b

where   and are integers and  can never be zero. We call  numerator and denominator

NOTE: Fraction: (from the Latin fractus, broken)

Fractions have three different meanings.

#### 1.1.  Fraction as a part of a whole

The denominator represents the parts the whole is divided into and the numerator shows the number of parts are taken.

### 1.2. Fraction as a quotient

A fraction

 a b

expresses the quotient .

To work out the value of the fraction we divide numerator by denominator.

Example:

 4 4:8 = 0,5 8

1.3. Fraction as operator

A fraction  can act as an operator over a number.

To work out the fraction of a number multiply the numerator by the number and divide by the denominator.

Example:

In a class ¾  of 24 pupils wear glasses:

¾ of 24 = (3 · 24) : 4 = 18 pupils wear glasses

### 1.4. Types of fractions

There are 4 types of fractions:

• Proper fractions: Numerator < denominator. Proper fractions nominator part smaller than the denominator part. They have values less than 1, for example , or
• Improper fractions: Numerator> denominator. Improper fractions have the nominator par grater than the denominator part. They have values greater than 1, for example .
• Unit fractions: Numerator = denominator. Unit fractions have value 1, for  example
• Mixed fractions: Mixed fractions have a whole number plus a proper fraction 2

If you want to practise this concepts click on the picture

### 2. Equivalent fractions

Definition:

Two fractions

 a b

and

 c d

areequivalentand we write it if  a · d = b · c  and  they are named crossing products (productos cruzados)

What equivalent means?

The wordEQUIVALENTmeans the same as EQUAL or, more precisely, of equal value.

We can observe the following facts:

1. Graphically, two equivalent fractions represent the same part of the whole.

For example, you can see that the colored part of each of the circles below is exactlythe same.

2. Two equivalent fractions have the same value

Since equivalent fractions represent the same part of the whole, therefore have the same value, the quotient is the same when we divide

3. If we multiply both numerator and denominator of a fraction by the same number (not zero) we obtain an equivalent fraction.

4. If we divide both numerator and denominator of a fraction by the same number (not zero) we obtain an equivalent fraction.

How can we obtain equivalent fractions?

There are to ways:

−  Amplifying:Equivalent fractions are obtained by multiplying both the numerator and the denominator by the same number (not equal to zero)

−  Simplifying:Equivalent fractions are obtained by dividing both the numerator and the denominator by the same number (not equal to zero)

Definition: We saya fraction is in the Simplest form or it is a reduced fractionif  we can’t simplify it any more.

Example:

Simplify

SOLUTION:

Both sides are divided by 8, so the fraction is canceled by 8

But the result is not in the Simplest form. It still can be simplified:

=

Both sides are divided by 2, so the fraction is reduced by 2

Now the result is in the Simplest Form.

You could get this result by reducing the fraction once only by 16. (The Highest Common Factor).

You can practise the concept of equivalent fractions and simplifying fractions through several exercises if you click on the picture.

Sometimes we have to find the missing number to have two equivalent fractions. We must apply the definition, the cross pruducts must be equal. Practise on this web:

## Lesson 1: Jokes And Games On integers (IV)

October 1, 2010

### Now time to relax. So, I will tell you a joke.

Several students were asked the following problem:

Prove that all odd integers are prime.

Well, the first student to try to do this was a math student. Hey says “hmmm… Well, 1 is prime, 3 is prime, 5 is prime, and by induction, we have that all the odd integers are prime.”

Of course, there are some jeers from some of his friends. The physics student then said, “I’m not sure of the validity of your proof, but I think I’ll try to prove it by experiment.” He continues, “Well, 1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is … uh, 9 is an experimental error, 11 is prime, 13 is prime… Well, it seems that you’re right.”

The third student to try it was the engineering student, who responded, “Well, actually, I’m not sure of your answer either. Let’s see… 1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is …, 9 is …, well if you approximate, 9 is prime, 11 is prime, 13 is prime… Well, it does seem right.”

Not to be outdone, the computer science student comes along and says “Well, you two sort’ve got the right idea, but you’d end up taking too long doing it. I’ve just whipped up a program to REALLY go and prove it…” He goes over to his terminal and runs his program. Reading the output on the screen he says, “1 is prime, 1 is prime, 1 is prime, 1 is prime….”

Do you understand every single word or expression?

### Here you are a couple of games, practise and have fun at the same time!

Ordering numbers

Integers Jeopardy Game – This game has 4 categories: adding integers, subtracting integers, multiplying integers, and dividing integers. You can play it alone or in teams.

## Lesson 1: Integers III

October 1, 2010

I will not treat some concepts you know perfectly: factor, divisor, divisibility criteria, prime and composite number etc.

However, I provide you some links to revise them at the end of the post.

### 7. Prime factor decomposition

Prime factor decomposition of a number means writing it as a product of prime factors using powers.

This expression is unique for each number. For instance:

240 = 24 · 3 · 5

924 = 22 · 3   7 · 11

### 8. Highest common factor (H.C.F) and lowest common multiple (L.C.M)

In this course, we will find H.C.F and L.C.M of big numbers by using their prime factor decomposition.

### 8.1. Highest common factor (H.C.F)

The Highest Common Factor (H.C.F) of two (or more) numbers is the largest number that divides evenly into both numbers. In other words, the H.C.F is the largest of all the common factors.

FINDING THE H.C.F. OF BIG NUMBERS

For larger numbers you can use the following method:

1. Find prime factor decomposition of all numbers.
2. Find which factors are repeating in all the numbers and choose them with the lowest exponent they appear.
3. Multiply them to get  H.C.F.

Example:

Find the Highest Common Factor (H.C.F.) of 240 and -924.

```240 = 24 · 3 · 5
924 = 22 · 3   7 · 11```

Common factors are 2 and 3 but we have to choose the lowest exponent, this is 22 and 3.

Multiply the factors which repeat in both numbers  to get the H.C.F.

The Highest Common Factor is :  22 · 3 = 12

We write:

H.C.F(240,-924)= H.C.F(240,924)=12

### 8.2. Lowest common multiple (L.C.M)

The Least or Lowest Common Multiple of several integer numbers is the smallest positive number that is a multiple of all those numbers at the same time, except for 0.

The simple method of finding the L.C.M of smaller numbers is to write down the multiples of the larger number until one of them is also a multiple of the smaller number.

FINDING L.C.M. OF BIG NUMBERS

1. Find all the prime factors of both numbers.
2. Find which factors there are in all the numbers  and choose them with the highest exponent they appear.
3. Multiply them to get  L.C.M.

Example:

Find the Lowest Common Multiple (L.C.M.) of 240 and -924.

From the example of finding the H.C.F. we know the prime factors of both numbers.

```240 = 24 · 3 · 5
924 = 22 · 3   7 · 11```

Factors are: 2, 3, 5, 7, 11.

We have to choose them with the highest exponent: 24, 3, 5, 7, 11.

`The L.C.M (240, -924) = L.C.M (240, 924)=  24 · 3   5 · 7 · 11 = 18480`

Practise these concepts as well as word problems on them on the following links

Factors

Divisibility rules

Prime or composite

Prime factorization

Greatest common factor

Least common multiple

GCF and LCM: word problems