Math 6 Honors ( Periods 1, 2, & 3)

Changing a Fraction to a Decimal 6-5
There are two methods that can be used to change a fraction to a decimal.

1) find an equivalent fraction whose denominator is a power of 10. ( this method does not always work but when it does it becomes really easy to change to a decimal)

13/25 multiply the numerator and the denominator by 4 to get 52/100 and then just close your eyes and see Chapter 3... and 0.52


2) divide the numerator by the denominator. It's a great way to determine your score out of 100 and then figure out your percent.

If you got 67/75 on the last test

divide 67 by 75 carefully 0.8933333... you earned a B+

take 3/8 and divide 3 by 8 8 goes into 3 0.375 times

so 3/8 = 0.375

If the numerator is smaller than the denominator we know our number must be between 0 and 1--> it must be a decimal.


When the remainder is 0 as in the case of dividing 3 by 8, it is called a terminating decimal.


By examining a fraction in lowest terms, we can determine whether the fraction can be expressed as a terminating decimal.

If the denominator has no prime factors other than 2 or 5, the decimal representation will terminate.

7/40

looking at 40 we notice the prime factorization ( oh no, it's Chapter 5)
40 = 2³·5 Since the only prime factors are 2 and 5

7/40 must terminate.

What about 5/12 ?

12 = 2²· 3 Since 3 is a prime factor of the denominator, the fraction cannot be expressed as a terminating decimal.


What about 9/12 ? At first it looks the same as the one above, but look carefully and realize 9/ 12 = 3/4

Since 4 = 2² , 4 has no other prime factors except 2, this can be expressed as a terminating decimal.


Let's look at 15/22
Since 22 has the prime factor of 11 we know that this will not terminate. In fact when you divide 15 by 22 you end up with 0.681818181...


We write this as ( Please check page 196) Notice that the bar is only over the 81 and represents a block numbers that continues to repeat indefinitely and is called a repeating decimal.


EVERY FRACTION CAN BE EXPRESSED AS EITHER A TERMINATING DECIMAL OR A REPEATING DECIMAL.


Let's look at 4/9 = 0.4444444....

5/9 =

7/9 =


31/99 = 0.3131313131...

8/ 11 = 72/99 = .72727272...

Math 6 Honors ( Periods 1, 2, & 3)

Comparing Fractions 6-4
When two fractions have equal denominators it is easy to tell which of the fractions is greater.
We simply compare their numerators.
3/11 < 5/11 since 3< 5 If the fractions have different denominators, there are a variety of methods to consider. We could find a common denominator, which we will need to do when we add or subtract fractions... but when comparing let's try other methods... Take 2/3 and 4/5 Comparing Fractions


Or compare 5/6 and 7/9

again this time you would multiply
5(9) = 45 and 7(6) = 42
so 5/6 > 7/9

Also if the numerator is the same
2/3, 2/7, 2/9, 2/11, 2/21, 2/35

The larger the denominator the smaller the fractions so to list in order from least to greatest start with the largest number in the denominator!!

and if you have fractions with the numerator just one away from the denominator
such as 3/4, 5/6, 7/8, 9/10, 23/24, 45/46
the smallest fraction will be the one with the smallest numbers
3/4 is the smallest fraction and that list is in order from least to greatest!!

What if you need to name a fraction between two fraction 1/6 and 3/8
you could find the LCD
1/6 = 4/24 and
3/8 = 9/24
so you could state
5/24, 6/24 ( but that is really 1/4), 7/24, or 8/24 ( but that is really 1/3.
There are actually an infinite number of fractions... these are only 4 of them

What if you need to find a fraction between 3/7 and 4/7
sometimes you need to change the denominators just to realize that there really are other fractions between
for instance, 3/7 = 6/14 and 4/7 = 8/ 14 so doesn't 7/14 ( or actually 1/2) work!!
... and that's just one of the fractions!!




There were 10 sunny days in February ( 28 total days)
12 sunny days in November (30 total days)
Which month has higher fraction of sunny days?

10/28 or 12/30
First simplify each fraction... then compare

10/28 = 5/14 and 12/30 = 2/5

Now use one oc the methods to compare
Cross products works really well and you discover
NOVEMBER is the month with the higher fraction of sunny days

The Bears won 11 out of 16 games.
The Eagles won 17 out of 24 games.
Which team won a great fraction of their games?

11/16 or 17/24
Again, I would use cross multiplication and we discovered that THE EAGLES won a greater fraction of their games.
If n > 0
Then
if a < b a/n < b/n Think about this one!! Plug in some numbers and see what happens and if a < b, then n/a > n/b
Again, plug in some numbers and see what happens!!

If a/b and c/d are fractions and if ad > bc, which fraction is greater,
a/b or c/d ?
Post your answer below in the comments for extra credit. Make sure to give your reasoning for your answer.


Ordering or comparing fractions:

Different ways:

I. Benchmarks - 0, 1/4, 1/2, 3/4, and 1 (using your gut feeling)
How do you figure out which benchmark to use?

When the numerator is close to the denominator, the fraction is approaching 1

(Ex: 9/11)

When you double the numerator and it's close to the denominator, the fraction is close to 1/2
(Ex: 4/9)

When the numerator is very far from the denominator, the fraction is approaching zero
(Ex: 1/8)

Also, if one number is improper or mixed number and other is a proper fraction,
then obviously the number greater than 1 will be bigger!

II. LCD - give them all the same denominator using the LCM as the LCD


III. Use cross multiplication when comparing two... do it several times when comparing a list of fractions

IV. Change them to decimals ( works well if you are great at decimals-- but I want you to become GREAT at fractions!!)