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Adults.
Algebra for Adults:
Part 4, Review, Fractions, Decimals, and Percents:
December 16, 2002:
Here are the basic facts about fractions.
A fraction can be part of a whole thing or a part of a _Group of things.
A fraction can be used to indicate Division.
A fraction can mean a ratio of two numbers.
A common fraction is a fraction whose numerator is a whole number and whose denominator is a nonzero whole number.
An improper fraction is a fraction whose numerator is equal to or larger than it's denominator.
A proper fraction is a fraction whose numerator is less than its denominator.
2/3, 10/15, and 2/5 are examples of proper fractions.
8/5, 3/2, and 13/2 are examples of improper fractions.
We also defined a mixed number as a number that can be expressed
as two components, these being a whole number and a fractional
component.
For example:
6/4 = 1 and 1/2
5/3 = 1 and 2/3
16/5 = 3 and 1/5
18/7 = 2 and 4/7
We find we cannot change the value of a fraction by multiplying
its numerator and denominator by the same number, excepting zero.
For example;
2/5 x 5/5 = 10/25
We cannot change the value of a fraction by dividing its numerator and denominator by the same number.
We can reduce fractions to their lowest terms.
For example:
1,008/1,890 = 8/15
1,440/23,328 = 5/81
Reciprocals and division of fractions.
Reciprocals are two numbers whose product is equal to 1.
For example:
The reciprocal of 3/4 is 4/3
The reciprocal of 5/2 is 2/5
To divide a nonzero number by a fraction, we can just invert
and multiply.
For example:
3/2 divided by 2/3 = 3/2 x 3/2 = 9/4
7/8 divided by 2/5 = 7/8 x 5/2 = 35/16
10 divided by 3/16 = 10 x 16/3 = 160/3
4/5 divided by 2/9 = 4/5 x 9/2 = 36/10 = 18/5
An equation is:
A statement asserting the equality of two expressions, usually
written as a linear array of symbols that are separated into left
and right sides and joined by an equal sign.
We can move any term across an = sign without changing the value of the equation by also moving the term across the division line.
We can express fractions in higher terms.
For example:
3/10 = ?/30
? = 3 x 30/10
? = 9
3/10 = 9/30
We can express fractions in terms of common denominators.
For example
Express 1/2 and 3/4 as 8ths.
1/2 = 4/8
3/4 = 6/8
We can add and subtract like fractions. We can find the least
common denominator (LCD) of a set of fractions. Then we can combine
them.
For example:
For the fractions 3/5, 7/10, 15/16.
LCD = 80. This is the smallest multiple of 16 that is divisible
by 5 and 10.
3/5 = 48/80
7/10 = 56/80
15/16 = 75/80
48/80 + 56/80 + 75/80 = 179/80
We can add, subtract, multiply, and divide mixed numbers.
For example:
1 & 1/2 x 2 & 3/8
3/2 x 19/8 = 57/16 = 3 & 9/16
3 & 1/4 divided by 3 & 3/8
13/4 x 8/27 = 104/108 = 26/27
Decimal numbers.
We define a decimal fraction as a fraction whose denominator is 10, 100, 1000, or any other place value.
A decimal is a symbol for an equivalent decimal fraction.
This allowed us to understand parts of dollars as decimal numbers.
The number of decimal places of the decimal is equal to the number of zeros in the denominator.
The generalized procedure to express a common fraction in decimal form is:
First convert the common fraction to an equivalent decimal
fraction.
Then express the decimal fraction as a decimal.
The place values to the right of the decimal follow the same pattern as they do to the left. To the right of the decimal point we have tenths (.1), hundredths (.01), thousandths (.001), and so on.
The value of a decimal does not change when zeros are annexed to the right.
We can round decimals. For example we can round decimal numbers to the nearest hundredth.
.0456 = .05
.0444 = .04
We can add decimal numbers.
For example:
We can subtract decimal numbers.
For example:
The number of decimal places in the product of two or more factors is the Sum of the decimal places of the factors.
To multiply a decimal number by a place value, we can shift
the decimal point to the right as many places as there are zeros
in the place value.
For example:
.25 x 1000 = 250
17.01 x 100= 1701
In division of decimals the quotient will have the same number of decimal places as the dividend.
We can shift the position of the decimal place an equal distance in the divisor and the dividend without changing the outcome of the problem.
We can convert common fractions to equivalent decimals.
For example:
5/16 = .3125
Percents.
Percent means hundredths!
The symbol for percent is %.
A given number of percent is equivalent to a fraction whose numerator is the given number of percent and whose denominator is 100.
This means that, one buck ($1.00) is 100% of a dollar.
To convert a percent to a decimal, we need only eliminate the
percent sign and move the decimal two places to the left.
for example:
8.5% .085
The rules for comparing numbers are:
Any positive number is greater than 0.
Any negative number is less than 0.
Any positive number is greater than any negative number.
The larger of two positive numbers has the greater absolute value.
The larger of two negative number has the smaller absolute value.
To subtract a larger number from a smaller number, subtract the smaller number from the larger number and give the result a Negative sign.
To subtract a signed number, change its sign and add.
To multiply two signed numbers with like signs, multiply their absolute values. The product will be a positive number.
To multiply two signed numbers with unlike signs, multiply their absolute values. The product will be negative number.
Zero times any signed number is zero.
This completes our review of basic math. In our next exercise
we will begin the transition from math to algebra.
Back to Algebra for Adults.
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