To master division with decimals, you have to become very familiar with exponents and place values. The equation of a decimal being divided is the decimal being divided by the power of ten. Depending on the exponent is how many times you move the decimal place to the left to get your answer.
Example: 5.06 divided by 10 ^ 4 = .000506
No matter if the exponent is negative of positive the decimal still moves to the left.
9's TRICK:
If any number is over 9, 99, 999, 9999, etc. that number will be repeated in the decimal form. It just depends on the place value of the 9.
1/9 = 0.11111111....
8/9 = 0.8888888...
* The pattern changes when the place value of 9 changes:
7/99 = 0.0707070707...
- the 0 represents the missing tens place value in the number 7
37/99 = 0.3737373737.....
- Notice how the 0 was replaced but a 3, because 37 has a number in the tens place.
* The pattern changes once again when the place value of 9 gets larger:
49/999 = 0.049049049049...
-Once again the 0 is representing how the number 49 does not have a number in the hundreds place
749/999 = 0.749749749749...
- The 0 was replaced with 7 because that us the number in the hundreds place.
***The reason why this numbers constantly repeat is because the number .999999 is actually equal to 1.
Converting Decimals to Fractions:
Method #1:
0.27777....
= 0.2 + 0.7777... ( separate the lonely number from the repeating number)
= 2/10 + 0.777...
= 2/10 + 0.777... / 10
= 2/10 + 7/9 * 1/10 (just found out any repeating number is over a 9 number)
= 2/10 + 7/90
= 18/90 + 7/90 (common denominator multiply 2/10 by 9)
= 25/90
Method #2:
0.4191919...
10n = 4.191919...
100n = 41.91919...
1000n = 419.191919...
* you want to get two answers that have the same repeating decimal. (in this case 10n and 1000n do)
1000 = 419.191919...
- 100 = 4.191919...
990 415
The answer then comes to 415/990
Showing posts with label Fractions. Show all posts
Showing posts with label Fractions. Show all posts
Thursday, December 9, 2010
Multiplication with Fractions
Today's lesson was pretty easy! It was basically overview of simple multiplication only in a fraction form. Basically when multiplying two fractions together the product is made by just multiplying the two numerators together and the two denominators together.
For Example: a/b * c/d = a * c/b * d
Another way of showing this besides just doing it in your head is by making a grid and shading each fraction to get your answer. For instance, if you were using the equation 3/4 * 1/3 you would make a grid with 3 across and 4 down creating a grid of 12 boxes. This is because you base the grid off of the two denominators. (the first denominator is always the amount going down, the second is always the amount going across) Next you should shade in 3/4 of the grid (9 boxes), then you should shade in 1/3 of the boxes (4 boxes). This leaves exactly 3 out of the 12 boxes shaded by both fractions. That makes your answer 3/12.
Through multiplication familiar property rules apply:
For Example: a/b * c/d = a * c/b * d
- 3/4 * 1/3 = 3/12 <-- this is also known as 1/4
- 2/5 * 1/3 = 2/15
- 3/2 * 1/4 = 3/8
Another way of showing this besides just doing it in your head is by making a grid and shading each fraction to get your answer. For instance, if you were using the equation 3/4 * 1/3 you would make a grid with 3 across and 4 down creating a grid of 12 boxes. This is because you base the grid off of the two denominators. (the first denominator is always the amount going down, the second is always the amount going across) Next you should shade in 3/4 of the grid (9 boxes), then you should shade in 1/3 of the boxes (4 boxes). This leaves exactly 3 out of the 12 boxes shaded by both fractions. That makes your answer 3/12.
Through multiplication familiar property rules apply:
- closure: fraction * fraction = fraction
- commutative: a/b * c/d = c/d * a/b
- associative: (a/b * c/d) * e/f = a/b * (c/d * e/f)
- distributive: a/b (c/d + or - e/f) = a/b * c/d + or - a/b * e/f
- inverse: for every fractions a/b, a & b do NOT = 0; there exists an b/a so that a/b * b/a = 1 (reciprocal) ---> practice the inverse operation
- identity: a/b * 1 = a/b * 1/1 = a/b
Fractions
When learning how to use fractions there is one thing you must know from the beginning. That is that the top number (a) is called the numerator, and the bottom number (b) is called the denominator. The denominator can NOT be zero. There are a few different ways to look at a fraction:
1. Part-to-Whole, which is when the numerator equally goes into the denominator and the fraction is just representing a part to the whole unit.
2. Division Concept, which is just converting the fraction into a division problem. For example, 1/4 now becomes 1 divided by 4.
3. Ratio Concept, which is when you compare one amount to another.
I had also learned the Fundamental Law of Fractions: let a/b be a fraction. Then for any number k can NOT equal 0. (a/b = ka/kb) Also, Equivilent Fractions can be formed by multiplying or dividing the numerator and denominator by a number.
You can find the answer to the addition of two fractions by adding the top two numbers across and the two bottom numbers across leaving you with a fraction answer still.
1. Part-to-Whole, which is when the numerator equally goes into the denominator and the fraction is just representing a part to the whole unit.
2. Division Concept, which is just converting the fraction into a division problem. For example, 1/4 now becomes 1 divided by 4.
3. Ratio Concept, which is when you compare one amount to another.
I had also learned the Fundamental Law of Fractions: let a/b be a fraction. Then for any number k can NOT equal 0. (a/b = ka/kb) Also, Equivilent Fractions can be formed by multiplying or dividing the numerator and denominator by a number.
You can find the answer to the addition of two fractions by adding the top two numbers across and the two bottom numbers across leaving you with a fraction answer still.
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