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Lesson 1B: Decimals, Fractions, Proportions and Percent
1. Common Conversions
4. Scientific Notation
By this point, you should have a basic grasp of how to work with numbers. You should be comfortable with basic number operations, like arithmetic (adding, subtracting, multiplying, and dividing). You should be able to perform these actions with whole numbers, integers, fractions and decimals.
Decimal = Fraction = Proportion
1 1/1 = 2/2 =3/3=100/100
0.50 = 50/100 = 5/10 = ½
0.25 = 25/100 = ¼
0.10 = 10/100 = 1/10
0.01 = 1/100
.33 = 1/3
.66 = 2/3
Percent (%) or percentage means per one hundred
45%, for example, is 45/100 or 0.45
A pie chart is often used to represent the breakdown of percentages.
If you are doing a standard grid question and end up with a percent as your answer, convert it to a decimal because there is no percent symbol (%) on the standard grid.
Problem - Proportions:
At the market, a man buys six avocados for $13.35. All produce is 30% off today only. How much will two avocados cost you tomorrow at full price?
You want to find the regular price per avocado because the question is asking you to calculate the cost of two of them.
You can find this by figuring out the sale price. The man paid $13.35 for 6 avocados. So divide the total cost by the number of avocados to get the price for one avocado.
13.35/6 = 2.225 this is the discounted price per avocado.
The discount is 30% less than the regular price.
If you take away 30%, you have 70% (since 100% - 30% = 70%)
So you know that $2.225 is 70% of the full price.
Full $ x 0.70 = 2.225
(Remember that if A x B = C, then A = C /B)
So full $ = 2.225 / 0.70 which, when we work it out either with the calculator or by hand, gives us = $ 2.50
At $2.50 per avocado, buying two would cost us $5.00, which is Answer C.
When you multiply 10s together, the product is called a power of ten. We use exponential notation to show a power of ten. The exponent is the number that shows the number of times that ten is a factor and is written in superscript above and to the right of the number. For example, 3 to the fifth equals 3 times 3 times 3 times 3 times 3 equals 243. Written numerically: 35 = 3 x 3 x 3 x 3 x 3 = 243
When number X is taken to power Y
It means that X is multiplied against itself Y times.
32 is the same as 3 x 3 = 9
42 is 4 x 4 = 16
43 is 4 x 4 x 4 = 4(16) = 64
A negative exponent simply means that number is a denominator or bottom number of a fraction.
4-2 = 1/4 2 = 1/16
Rules for When Exponents Interact:
The way to remember the last operation, when an exponent is taken to a power, is to think of what, for example,
(Y3)2 really means. It is
(Y * Y * Y) x 2
(Y * Y * Y) * (Y * Y * Y) = Y * Y* Y* Y* Y* Y = Y 6
And that should help you remember to multiply the power (in this case, 2 and 3)
1. Exponentials can be multiplied or divided only if they have the same “base” (the number or variable being taken to a power.)
Thus: x2 * x3 = x5
But: x2 * y3 is simply x2y3
2. Exponent operations do not apply to addition/subtraction.
X 2 cannot be added to X 3
x 2 + x 3 is simply x 2 + x 3
In numerical terms, think of it this way:
3 2 + 3 3
3 * 3 + 3 * 3 *3
9 + 27 =36
36 is not a power of 3.
Now you are ready for something that sounds complicated, but really isn’t. The skills involved with this concept apply to other types of questions you will find on the test.
Scientific Notation is a way of writing any decimal number as the product of
1) a decimal number between 0 and 10
2) a power of ten
For example, the number 100 would be expressed as:
1.0 x 10 2
2000 would be
2.0 x 10 3
2500 would be
2.5 x 10 5
3.75 would be
3.75 x 10 0
Remember that any number to the power of zero equals 1.
Numbers less than one are represented similarly, but with ten to a negative power.
0.48 would be 4.8 x 10 -1
And 0.0515 would be 5.15 x10 -2
0.007 would be 7.0 x10 -3
And 0.00008214 would be 8.214 x 10 -5
For numbers less than 1, a simple way to figure out the power of 10 is to count the zeros, starting with the one's place on towards the right and ending at the first non-zero number.
0.00000009081 would be 9.081 x 10 -8 because we count eight zeroes before we get to the '9'.
The way to figure out numbers greater than zero is to start at the one's place and count the places toward the left until just before the last (greatest) digit.
2560.0 would be 2.56 x 10^3 because we count three places before we get to the '2'.
Scientific Notation is used as a standard or uniform way to express a wide variety of rational numbers. You will likely see some form of it on the test.
√ is the root symbol (or radical).
x2 = 9, what does x equal?
Another way of asking this question is
√9 = ?
3 √ means cubed root.
4 √ means fourth root and so on.
3 √ 64 = 4 because 64 = 4 x 4 x 4.
Many of the concepts covered will also be useful in the next lesson on Algebra.
Back: Math Lesson 1A | Next: Math Lesson 2A
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