Friday, December 10, 2010

Ratios and Proportions

A ratio is a pair of positive numbers that is used to compare two sets. 
A ratio gives the relative sizes of two sets but not the actual numbers of objects in those sets.
Ex)
There are 21 students in a classroom that consists of 3 males and 18 females. What is the ratio of males to females? 
Answer: 3/18 = 1/6, (can be reduce to 1/6 in lowest terms), 1 to 6, or the most common way to write a ratio, 1:6
A proportion is a part considered in relation to the whole.It's an equality of ratios. Each ratio gives rise to many pairs of equal ratios. 

For any two ratios a/b and c/d, a/b = c/d - proportion
Ex)
If the ratio of teachers to students in a school is 1 to 18 and there are 360 students, how many teachers are there? 

 1 teacher  =  x teachers 
18 students   360 students       = 20 Teachers

We know that the numerator and denominator of 1/18 must be multiplied by the same number to result in an equal fraction. Since the denominator of 1/18 must be multiplied by 20 to get 360 (360/18=20),
So the number of teachers is 20.

Another example of how you could solve a proportion-
Ratios and Proportions - This website helped me understand the definitions of ratios and proportions.

Writing Decimals as Percents, vice versa

To convert from decimal to percentage, just multiply the decimal by 100, but remember to put the "%" sign so people know it is per 100.


1). 0.426 = 42.6%
    Because, 0.426 x 100 = 42.6


2.) 0.003 = 0.3%
    Because, 0.003 x 100 = 0.3


To convert from percentage to decimal: divide by 100. And then remove the "%" sign.
Moving the decimal 2 places to the right is an easy way to do this.
Ex)
1). 42.6% = 0.426                                                           2.) 0.3% = 0.003
     Because, 42.6/100 = 0.426                                            Because, 0.3/100 = 0.003



Monday, December 6, 2010

Denominator of 9's


We learned a pattern for fractions that have denominators with 9, 99, 999,..and so on, that the numerator is repeated in its decimal form.
For Example:
Here are Single-digit Repeating Decimal Numbers-
0.111...
0.222...
0.333...
0.444...
0.555...


Any single-digit repeating decimal  number can be written as a fraction with 9 as the denominator and the repeating single-digit as the numerator. 

1/9 = 1 1203.h17.jpg (931 bytes) 9 = 0.111...

2/9 = 2 1203.h17.jpg (931 bytes) 9 = 0.222...

3/9 = 3 1203.h17.jpg (931 bytes) 9 = 0.333...
4/9 = 4 1203.h17.jpg (931 bytes) 9 = 0.444...
5/9 = 5 1203.h17.jpg (931 bytes) 9 = 0.555...
6/9 = 6 1203.h17.jpg (931 bytes) 9 = 0.666...
7/9 = 7 1203.h17.jpg (931 bytes) 9 = 0.777...
8/9 = 8 1203.h17.jpg (931 bytes) 9 = 0.888...
So....

0.111... = 1/9
0.222... = 2/9
0.333... = 3/9
0.444... = 4/9
0.555... = 5/9
0.666... = 6/9
0.777... = 7/9
0.888... = 8/9


Equal Decimals & Rational Numbers

Two decimal numbers are equal if, they have the same numbers in the same place values not including leading zeros. Leading zeros of a decimal number are not significant.
Ex) 0.7=0.7000    or     0.85000=0.85

A rational number is any number other than zero that can be rewritten as a fraction
OR cab be any fraction a/b that can be written as a decimal.
Ex of Rational Numbers:
1/10 can be written as 0.1
1/3 can be written as 0.3333...
π(3.141592654...) is not a rational number because it's decimal cannot be written into a fraction.

A fraction or rational number can be written as a terminating decimal if prime factorization of the denominator contains either 2's and/or 5's with the fraction being in lowest terms.
Ex) 11/40
      11/ 2x2x2x5
      11(5x5) / 2x2x2x5 (5x5)
      275/ 1,000 = 0.275
Rational Numbers - This website helped me understand what Rational Numbers are through explanation and examples.

Models For Decimals

To visually show how decimals are represented, we can use Decimal Squares or Base Ten Blocks.
Decimal Squares-
Since the whole is a flat, of 10 columns where 3 are only shaded.
This represents the decimal, 0.3 or the fraction, 3/10, (three tenths)
Now the whole is shown to be a flat of 100 squares, with 73 squares shaded. This represents the decimal, 0.73, seventy-three hundredths, since there are 73 out of the 100 squares shaded.
Base Ten Blocks-


The whole here is the 1,000 block. So there are 2 flats of 100 squares, 5 longs of 10s, and 3 cubes. But when comparing the flats, longs, and cubes to the whole, which is the block, is how they receive their value.
10 flats make up a block-2/10
100 longs make up a block- 5/100
1000 cubes makes up a block- 3/1,000
So 1.253 can be shown by the image above, when using base 10 blocks.

Place Value, Terminating & Converting Decimals

The week we learned about decimals, we first learned about the place value.
Place Value: The value of a digit as determined by its position in a number. These are the place values-
This is how you would write out and/or say a number that has a decimal in it:
1.)  23.52 = twenty-three and fifty-two hundredths
2.)  6.4193 = six and four thousand, one hundred, ninety-three thousandths

We then learned what terminating decimals are.
A terminating decimal is a decimal that doesn't keep going or repeat.
A terminating decimal is 3.2
A non- terminating decimal is 3.2394847395883...
Since the fraction, 9/80 is written like this, another way to look at it is saying, 9 divided by 80. When you do this expression..9 divided by 80, you end up with a decimal answer of 0.1125.

How to convert decimal numbers
Steps in order to convert 42/100 into a decimal. Using Division
7/1,000= 0.007
82/10,00= 0.0082

Another trick I used was, moving the decimal. How many zeros there are, is how many times I would move the decimal from the original spot to the new spot, to convert the fraction.
Ex) 64193/10,000. The decimal is at the end of the number 64193.0, so I move the decimal over to the right 4 decimal places..
So 64193/10,00 = 6.4193

Converting Fractions to Decimals - On WebMath.com, you can put in any fraction you want to be converted into a decimal. By providing the number of decimal places you want, it shows you step-by-step on how to get the answer using the division technique.


  • First, interpret the fraction bar to mean "divided by." This means that 42/100 is the same as 42 divided by 100.
  • Second, since there are two zeros..you will put your answer into two decimal places, so we'll write the 42 as 42.00 instead (a 42 with 2 zeros after the decimal point).
  • Now, just do what the fraction bar says: divide 42.00 by 100:


42/100 written as a decimal to 2 decimal places is 0.42.
31/100= 0.31

Methods for Converting Fractions to Decimals
ALSO, we can convert a fraction to an equivalent fraction with a denominator that is a power of ten before converting to a decimal. Like this-
1/2 = 5/10 = 0.5
3/20 = 15/100 = 0.15
1/8 = 125/1000 = 0.125
The decimal equivalent of a proper or improper fraction can be calculated by dividing the numerator by the denominator.  The result will be a terminating or repeating decimal.
3/4 = 0.75 terminating decimal
3/8 = 0.375 terminating decimal
4/15 = 0.0666... repeating decimal
Method for Converting Terminating Decimals to Fractions
Use the place value to convert the terminating decimal to a fraction that is a power of ten.  Then, reduce to lowest terms.
The denominator of a fraction converted from a terminating decimal will be a multiple of 2 and/or 5.
0.5 = 5/10 = 1/2
0.2 = 2/10 = 1/5
0.15 = 15/100 = 3/20
0.125 = 125/1000 = 1/8

Monday, November 22, 2010

Fractions

Today we learned about fractions. The definition of a fraction is an ordered pair of integers a and b, with b not equaling 0. Written a/b. A fraction is a number that can represent part of a whole. The parts of a whole must be equal
The top part of the fraction is called the numerator and the bottom part is called the denominator. The numerator says how many is considered. The denominator is how many parts it's breaking into. A fraction is the same as saying, a number divided by another number. 
A fraction can be represented in many ways
For Example:
1/2 = 2/4, 3/6, 4/8, 5/10, 6/12..and so on.
All of these fractions are considered one-half (1/2) because 2 is half of 4, 3 is half of 6, 4 is half of 8...
The process of taking 6/12 and making it into 1/2, is called putting the fraction in lowest terms. [1/2 is 6/12  at its lowest terms]
A fraction is said to be at its lowest terms when the numerator and the denominator cannot be divided by the same number to be at a lower valued numerator and denominator
Ex) 3/4 is in lowest terms because because the numerator (3) and the denominator (4) cannot be divided by the same number to lower their values.
Types of Fractions:
Proper Fraction- A proper fraction is a fraction that has a numerator smaller than the denominator. Examples: 1/2, 3/5, 7/8, 15/25
Improper Fraction- An improper fraction is a fraction that has a numerator larger than the denominator. Examples: 9/2, 4/3, 5/3, 21/8
Addition and Subtraction of Fractions
When adding and subtracting fractions, I always like to find the common denominator so that it's easier to do the operation.
Example: 
1.)  1/2 + 2/8, the common denominator would be 8, because I can multiply the 2 in 1/2 by 4, and then I would have to multiply 4 by the numerator, 1, to make the equation, 4/8 + 2/8 = 6/8. 6 and 8 can both be divided by 2, so then 6/8 is 3/4 at its lowest terms.
1/2 + 2/8 = 3/4
2.) 3/5 - 4/10, the common denominator is 10, so then we would have to multiply 5 by 2 because 5x2=10. Then multiply 3 by 2 as well, making 3/5=6/10. Then 6-4 is 2 and 10 is the denominator. This equation equals 2/10, but 2/10 can be reduced to 1/5 because 2 and 10 can be divided by 2.
3/5 - 4/10 = 1/5
Multiplying and Dividing Fractions
- 1/3 x 4/5 = 4/15 
because 1x4 = 4 and 3x5 = 15
- 1/2 x 1/6 = 1/12
because 1x1 =1 and 6x2 =12
Dividing Fractions - This showed me how to divide fractions by "multiplying the reciprocal" of a fraction.
Properties of Fractions
Closure: fraction x or / fraction = fraction
Commutative: a/b x c/d = c/d x a/b
Associative: (a/b x c/d) x e/f = a/b x (c/d x e/f)
Distributive: a/b(c/d + e/f) = a/b x c/d + a/b x e/f
Identity: a/b x 1 = a/b x 1/1 = a/b
Inverse: For every fraction a/b, a, b not equaling 0, There exists an inverse b/a so that a/b x b/a = 1
(aka- the reciprocal) 

* These three websites helped me learn the basic terms of fractions and the operations fractions can perform. 
(addition, subtractions, multiplication, and division)