Metadatics 1 6 2 Fraction

A circle is a geometric shape that we have seen in other lessons. The circle to the left can be used to represent one whole. We can divide this circle into equal parts as shown below.

• Metadatics 1 6 2 Fraction Decimal
• Metadatics 1 6 2 Fraction Calculator
• Metadatics 1 6 2 Fraction Whole

Two fractions are equivalent when they are both equal when written in lowest terms. The fraction 2 12 is equal to 1 6 when reduced to lowest terms.

1. Decimals are a type of fractional number. The decimal 0.5 represents the fraction 5/10. The decimal 0.25 represents the fraction 25/100. Decimal fractions always have a denominator based on a power of 10. We know that 5/10 is equivalent to 1/2 since 1/2 times 5/5 is 5/10. Therefore, the decimal 0.5 is equivalent to 1/2 or 2/4, etc.
2. Calculator for simplifying proper and improper fractions and simplify fractions. Please enter one fraction, decimal or mixed number (mixed fraction). Reduction of fraction is done by dividing numerator and denominator by gcf (greatest common factor). This calculator also simplifies proper fractions by reducing to lowest terms.

This circle has been divided into 2 equal parts.

This circle has been divided into 3 equal parts.

This circle has been divided into 4 equal parts.

We can shade a portion of a circle to name a specific part of the whole as shown below.

Definition: A fraction names part of a region or part of a group. The top number of a fraction is called its numerator and the bottom part is its denominator.

So a fraction is the number of shaded parts divided by the number of equal parts as shown below:

number of shaded parts numerator

number of equal parts denominator

Looking at the numbers above, we have:

There are two equal parts, giving a denominator of 2. One of the parts is shaded, giving a numerator of 1.

There are three equal parts, giving a denominator of 3. Two of the parts are shaded, giving a numerator of 2.

There are four equal parts, giving a denominator of 4. One of the parts is shaded, giving a numerator of 1.

Note that the fraction bar means to divide the numerator by the denominator. Let's look at some more examples of fractions. Iconjar 1 0 6. In examples 1 through 4 below, we have identified the numerator and the denominator for each shaded circle. We have also written each fraction as a number and using words.

Example 1:

One-half

Example 2:

One-third two-thirds

Example 3:

One-fourth Two-fourths Three-fourths

Example 4:

One-fifth Two-fifths Three-fifths Four-fifths

Why is the number 3/4ths written as three-fourths? We use a hyphen to distinguish a fraction from a ratio. For example, 'The ratio of girls to boys in a class is 3 to 4.' This ratio is written a 3 to 4, or 3:4. We do not know how many students are in the whole class. However, the fraction 3/4 is written as three-fourths (with a hyphen) because 3 is 3/4 of one whole. Thus a ratio names a relationship, whereas, a fraction names a number that represents the part of a whole. When writing a fraction, a hyphen is always used.

It is important to note that other shapes besides a circle can be divided in equal parts. For example, we can let a rectangle represent one whole, and then divide it into equal parts as shown below.

two equal parts

three equal parts

four equal parts

five equal parts

Remember that a fraction is the number of shaded parts divided by the number of equal parts. In the example below, rectangles have been shaded to represent different fractions.

Example 5: http://qvidod.xtgem.com/Blog/__xtblog_entry/19462182-macbook-for-developers-2017#xt_blog.

One-half

One-third

One-fourth

One-fifth

Metadatics 1 6 2 Fraction Decimal

The fractions above all have the same numerator. Each of these fractions is called a unit fraction.

Definition: A unit fraction is a fraction whose numerator is one. Each unit fraction is part of one whole (the number 1). The denominator names that part. Every fraction is a multiple of a unit fraction.

In examples 6 through 8, we will identify the fraction represented by the shaded portion of each shape.

Example 6:

In example 6, there are four equal parts in each rectangle. Three sections have been shaded in each rectangle, but not the same three. This was done intentionally to demonstrate that any 3 of the 4 equal parts can be shaded to represent the fraction three-fourths.

A.

B.

Example 7:

In example 7, each circle is shaded in different sections. However, both circles represent the fraction two-thirds. The value of a fraction is not changed by which sections are shaded.

Call of duty advanced warfare support line. Example 8:

In example 8, each rectangle is shaded in different sections. However, both rectangles represent the fraction two-fifths. Once again, the value of a fraction is not changed by which sections are shaded.

A.

B.

In the examples above, we demonstrated that the value of a fraction is not changed by which sections are shaded. This is because a fraction is the number of shaded parts divided by the number of equal parts.

Let's look at some more examples.

Example 9:

In example 9, the circle has been shaded horizontally; whereas, in example 10, the circle was shaded vertically. The circles in both examples represent the same fraction, one-half. The positioning of the shaded region does not change the value of a fraction.

Example 10:

Example 11:

In example 11, the rectangle is positioned horizontally; whereas in example 12, the rectangle is positioned vertically. Both rectangles represent the fraction four-fifths. The positioning of a shape does not change the value of the fraction it represents.

Example 12:

Remember that a fraction is the number of shaded parts divided by the number of equal parts.

In example 13, we will write each fraction using words. Place your mouse over the red text to see if you got it right.

Summary: A fraction names part of a region or part of a group. A fraction is the number of shaded parts divided by the number of equal parts. The numerator is the number above the fraction bar, and the denominator is the number below the fraction bar.

Exercises

In Exercises 1 through 5, click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR. Note: To write the fraction two-thirds, enter 2/3 into the form.

Composition of Functions:
Composing with Sets of Points (page 1 of 6)

Sections: Composing functions that are sets of point, Composing functions at points, Composing functions with other functions, Word problems using composition, Inverse functions and composition

Until now, given a function f(x), you would plug a number or another variable in for x. You could even get fancy and plug in an entire expression for x. For example, given f(x) = 2x + 3, you could find f(y² – 1) by plugging y² – 1 in for x to get f(y² – 1) = 2(y² – 1) + 3 = 2y² – 2 + 3 = 2y² + 1.

In function composition, you're plugging entire functions in for the x. In other words, you're always getting 'fancy'. But let's start simple. Instead of dealing with functions as formulas, let's deal with functions as sets of (x, y) points:

• Let f = {(–2, 3), (–1, 1), (0, 0), (1, –1), (2, –3)} and
  let g = {(–3, 1), (–1, –2), (0, 2), (2, 2), (3, 1)}.
  Find (i)f (1), (ii) g(–1), and (iii) (gof )(1).

(i) This type of exercise is meant to emphasize that the (x, y) points are really (x, f (x)) points. To find f (1), I need to find the (x, y) point in the set of (x, f (x)) points that has a first coordinate of x = 1. Then f (1) is the y-value of that point. In this case, the point with x = 1 is (1, –1), so:

Advertisement

f (1) = –1

(ii) The point in the g(x) set of point with x = –1 is the point (–1, –2), so:

g(–1) = –2

(iii) What is '(gof )(1)'? This is read as 'g-compose-f of 1', and means 'plug 1 into f, evaluate, and then plug the result into g'. The computation can feel a lot easier if I use the following, more intuitive, formatting:

(gof )(1) = g( f(1))

Now I'll work in steps, keeping in mind that, while I may be used to doing things from the left to the right (because that's how we read), composition works from the right to the left (or, if you prefer, from the inside out). So I'll start with the x = 1. I am plugging this into f(x), so I look in the set of f(x) points for a point with x = 1. The point is (1, –1). This tells me that f(1) = –1, so now I have: Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved


(gof )(1) = g( f(1)) = g(–1)

Working from the right back toward the left, I am now plugging x = –1 (from 'f(1) = –1') into g(x), so I look in the set of g(x) points for a point with x = –1. That point is (–1, –2). This tells me that g(–1) = –2, so now I have my answer:

(gof )(1) = g( f(1)) = g(–1) = –2

Note that they never told us what were the formulas, if any, for f(x) or g(x); we were only given a list of points. But this list was sufficient for answering the question, as long as we keep track of our x- and y-values.

• Let f = {(–2, 3), (–1, 1), (0, 0), (1, –1), (2, –3)} and
  let g = {(–3, 1), (–1, –2), (0, 2), (2, 2), (3, 1)}.
  
  Find (i) ( fog)(0), (ii)( fog)(–1), and (iii)(gof )(–1).

(i) To find ( fog)(0), ('f-compose-g of zero'), I'll rewrite the expression as:

( fog)(0) = f(g(0))

This tells me that I'm going to plug zero into g(x), simplify, and then plug the result into f(x). Looking at the list of g(x) points, I find (0, 2), so g(0) = 2, and I need now to find f(2). Looking at the list of f(x) points, I find (2, –3), so f(2) = –3. Then:

( fog)(0) = f(g(0)) = f(2) = –3

(ii) The second part works the same way:

( fog)(–1) = f(g(–1)) = f(–2) = 3

(iii) I can rewrite the composition as (gof )(–1) = g( f(–1)) = g(1).

Uh-oh; there is no g(x) point with x = 1, so it is nonsense to try to find the value of g(1). In math-speak, g(1) is 'not defined'; that is, it is nonsense.Then (gof )(–1) is also nonsense, so the answer is:

(gof )(–1) is undefined.

Part (iii) of the above example points out an important consideration regarding domains and ranges. It may be that your composed function (the result you get after composing two other functions) will have a restricted domain, or at least a domain that is more restricted than you might otherwise have expected. This will be more important when we deal with composing functions symbolically later.

Another exercise of this type gives you two graphs, rather than two sets of points, and has you read the points (the function values) from these graphs.

• Given f(x) and g(x) as shown below, find ( fog)(–1).
  

In this case, I will read the points from the graph. I've been asked to find ( fog)(–1) = f(g(–1)). This means that I first need to find g(–1). So I look on the graph of g(x), and find x = –1. Tracing up from x = –1 to the graph of g(x), I arrive at y = 3. Then the point (–1, 3) is on the graph of g(x), and g(–1) = 3.

Now I plug this value, x = 3, into f(x). To do this, I look at the graph of f(x) and find x = 3. Tracing up from x = 3 to the graph of f(x), I arrive at y = 3. Then the point (3, 3) is on the graph of f(x), and f(3) = 3.

Then( fog)(–1) = f(g(–1)) = f(3) = 3.

• Given f(x) and g(x) as shown in the graphs below, find (gof )(x) for integral values of x on the interval –3 <x< 3.
  

This is asking me for all the values of (gof )(x) = g( f(x)) for x = –3, –2, –1, 0, 1, 2, and 3. So I'll just follow the points on the graphs and compute all the values:

(gof )(–3) = g( f(–3)) = g(1) = –1 Dr web antivirus 11 0 1.

I got this answer by looking at x = –3 on the f(x) graph, finding the corresponding y-value of 1 on the f(x) graph, and using this answer as my new x-value on the g(x) graph. That is, I looked at x = –3 on the f(x) graph, found that this led to y = 1, went to x = 1 on the g(x) graph, and found that this led to y = –1. Similarly:

(gof )(–2) = g( f(–2)) = g(–1) = 3
(gof )(–1) = g( f(–1)) = g(–3) = –2
(gof )(0) = g( f(0)) = g(–2) = 0
(gof )(1) = g( f(1)) = g(0) = 2
(gof )(2) = g( f(2)) = g(2) = –3
(gof )(3) = g( f(3)) = g(3) = 1

You aren't generally given functions as sets of points or as graphs, however. Generally, you have formulas for your functions. So let's see what composition looks like in that case.

Metadatics 1 6 2 Fraction Calculator

Top | 1 | 2 | 3 | 4 | 5|6| Return to IndexNext >>

Metadatics 1 6 2 Fraction Whole