Comparing Fractions

In this lesson, you will learn about comparing fractions. Before you start this lesson, I recommend that you study or review my lesson about fractions. There are many ways to compare fraction.

Comparing Fractions

Equivalent Fractions
Equivalent fractions are fractions that represent the same quantity.

example:
1/2 = 2/4 = 4/8

Fundamental Law of Fractions: the value of a fraction does not change when its numerator and denominator are both multiplied by the same number (not zero).
[Remembering that multiplication by ½ can also be considered division by 2, this law also extends to "division".]


Simplifying Fractions
Reducing Fractions

When working with fractions, it is a common practice to list fractional answers in their "simplest form". The simplest form usually gives the reader a better idea of what the fraction represents.

To simplify (or reduce) a fraction, look for a number that will divide exactly into BOTH the numerator and the denominator. Try to find the "largest" such value (the greatest common divisor).

Example:
3/6 = 1/2

If you don't choose the LARGEST common factor (the GCD), you can still reduce (or simplify) the fraction, but you will have to do the division process more than one time.

Less Than, Greater Than, Equal To
Comparing fractions as to their size.
A. Simple Cut-up Visual Comparisons
These diagrams show a pictorial representation of what is occurring in relation to the size of the fractions. Each of the whole bars are divided into 5, 3, or 6 equal pieces and compared. Unfortunately, it can be difficult to compare fractions in this simple cut-up visual manner. If we did not know that 2/3 = 4/6 in the last comparison, it would be difficult to tell from the diagram whether the fractions were equal or just really close to being equal.

B. Compare using a Common Denominator
1. Comparing fractions that already have the same denominator is easy. The fraction with the larger numerator will be the larger fraction. Since the denominators are the same, you are comparing "pieces" (parts) of the same size.

2. Comparing fractions that have different denominators is harder. The solution requires that we create equivalent fractions that have the same denominators and then compare them. When looking for this same denominator (called the common denominator), look for the smallest possible number (called the least common multiple or the least common denominator) to keep the calculations as simple as possible.

If we re-examine our visual models from above, we can see that when the whole bar is divided equally for BOTH fractions, using the common denominator, we have a more reliable and accurate portrayal of the fraction sizes.

3. Compare using the Cross Multipy Rule
We saw in Ratios and Proportions that the Cross Multiplication Algorithm (Rule) was the result of rewriting fractions to have the same denominator and then comparing the numerators.
Here's how this process works:
To compare two fractions,

• multiply the numerator from the fraction on the left by the denominator of the fraction on the right.
• Place this number above the fraction on the left.
• Now, multiply the denominator from the fraction on the left by the numerator of the fraction on the right. 
• Place this number above the fraction on the right.
• Place the appropriate inequalitiy symbol (< , >, or even =)

As with the Cross Multiplication Algorithm (Rule), this cross multiply application to comparing fractions is actually the result of rewriting fractions to have the same denominators (which we do not write down) and then comparing the numerators.

4. Ordering Fractions
Place in ascending order (from smallest on the left to largest on the right)




This article was originally published on The mathbits notebook. Read the original article.

Source:
https://mathbitsnotebook.com/JuniorMath/FractionsDecimals/FDcomparing.html

Operation of Algebraic Expression

Before we see how to add and subtract integers, we define terms and factors.

Terms and Factors

A term in an algebraic expression is an expression involving letters and/or numbers (called factors), multiplied together.

Example
The algebraic expression

5x

is an example of one single term. It has factors 5 and x.

The 5 is called the coefficient of the term and the x is a variable.

Like Terms

"Like terms" are terms that contain the same variables raised to the same power.

Example
3x and 7x are like terms.

Adding and Subtracting Terms

Important: We can only add or subtract like terms.

Why? Think of it like this. On a table we have 4 pencils and 2 books. We cannot add the 4 pencils to the 2 books - they are not the same kind of object.

We go get another 3 pencils and 6 books. Altogether we now have 7 pencils and 8 books. We can't combine these quantities, since they are different types of objects.

Next, our sister comes in and grabs 5 pencils. We are left with 2 pencils and we still have the 8 books.

Similarly with algebra, we can only add (or subtract) similar "objects", or those with the same letter raised to the same power.

Example
Simplify 13x + 7y − 2x + 6a

13x + 7y − 2x + 6a

The only like terms in this expression are 13x and −2x. We cannot do anything with the 7y or 6a.

So we group together the terms we can subtract, and just leave the rest:

(13x − 2x) + 6a + 7y

= 6a + 11x + 7y

We usually present our variables in alphabetical order, but it is not essential.

Multiplication of Algebraic Expressions

When we multiply algebraic expressions, we need to remember the Index Laws from the Numbers chapter.

Let's see how algebra multiplication works with a series of examples.

Example
Multiply (x + 5)(a − 6)

We multiply this out as follows. We take each term of the first bracket and multiply them by the second bracket. Then we expand out the result.

(x + 5)(a − 6)

= x(a − 6) + 5(a − 6)

= ax − 6x + 5a − 30

We cannot do any more with this answer. There are no like terms, so we cannot simplify it in any way.

Dividing by a Fraction

Recall the following when dividing algebraic expressions.

The reciprocal of a number x, is 1/x

For example, the reciprocal of 5 is 1/5 and the reciprocal of 5/3 is 3/5

To divide by a fraction, you multiply by the reciprocal of the fraction.




This article was originally published on The Interactive Mathematics. Read the original article.

Source:
https://www.intmath.com/basic-algebra/1-addition-subtraction-algebra.php

https://www.intmath.com/basic-algebra/2-multiplication-algebra.php

https://www.intmath.com/basic-algebra/3-division-algebra.php

Ratio

Proportions are built from ratios. A "ratio" is just a comparison between, or a relating of, two different things. For instance, someone can look at a group of people, count noses, and refer to the "ratio of men to women" in the group. Suppose there are thirty-five people, fifteen of whom are men. Then the rest are women, so:

35 – 15 = 20

...there are twenty women in the group. The language "the ratio of (this) to (that)" means that (this) comes before (that) in the comparison. So, if one were to express "the ratio of men to women", then the ratio, in English words, would be "15 men to 20 women" (or just "15 to 20").


The order of the items in a ratio is very important, and must be respected; whichever word came first in the ratio (when expressed in words), its number must come first in the ratio. If the expression had been "the ratio of women to men", then the in-words expression would have been "20 women to 15 men" (or just "20 to 15").

source: twinkl.co.uk

Expressing the ratio of men to women as "15 to 20" is expressing the ratio in words. There are two other notations for this "15 to 20" ratio:

odds notation: 15 : 20

fractional notation: 15/20

 You should be able to recognize all three notations; you will probably be expected to know them, and how to convert between them, on the next test. For example:

There are 16 ducks and 9 geese in a certain park. Express the ratio of ducks to geese as a ratio with a colon, as a fraction (do not reduce), and in words.
They want "the ratio of ducks to geese", so the number for the ducks comes first (or, for the fractional form, on top). So my answer is:

16:9, 9/16, 16 to 9

The ratio from a representative group can also be used to provide percentage information.

In the class above, what percentage of students passed the class? (Round your answer to one decimal place.)
I already know that the representative group contains 12 students, of which 7 passed the class. Converting this to a percentage (by dividing, and then moving the decimal point, as explained here), I get:

7/12 = 0.583333... = 58.3333...%

They want the answer rounded to one decimal place, so my answer is:

58.3% passed



This article was originally published on The purplemath. Read the original article.

Source:
https://www.purplemath.com/modules/ratio.htm

Aplication of the Pythagorean Theorem

The Pythagorean Theorem is a statement in geometry that shows the relationship between the lengths of the sides of a right triangle – a triangle with one 90-degree angle. Being able to find the length of a side, given the lengths of the two other sides makes the Pythagorean Theorem a useful technique for construction and navigation.

Source: passnownow.com

Architecture and Construction

Given two straight lines, the Pythagorean Theorem allows you to calculate the length of the diagonal connecting them. This application is frequently used in architecture, woodworking, or other physical construction projects. For instance, say you are building a sloped roof. If you know the height of the roof and the length for it to cover, you can use the Pythagorean Theorem to find the diagonal length of the roof's slope. You can use this information to cut properly sized beams to support the roof, or calculate the area of the roof that you would need to shingle.

Laying Out Square Angles

The Pythagorean Theorem is also used in construction to make sure buildings are square. A triangle whose side lengths correspond with the Pythagorean Theorem – such as a 3 foot by 4 foot by 5 foot triangle – will always be a right triangle. When laying out a foundation, or constructing a square corner between two walls, construction workers will set out a triangle from three strings that correspond with these lengths. If the string lengths were measured correctly, the corner opposite the triangle's hypotenuse will be a right angle, so the builders will know they are constructing their walls or foundations on the right lines.

Navigation

The Pythagorean Theorem is useful for two-dimensional navigation. You can use it and two lengths to find the shortest distance. For instance, if you are at sea and navigating to a point that is 300 miles north and 400 miles west, you can use the theorem to find the distance from your ship to that point and calculate how many degrees to the west of north you would need to follow to reach that point. The distances north and west will be the two legs of the triangle, and the shortest line connecting them will be the diagonal. The same principles can be used for air navigation. For instance, a plane can use its height above the ground and its distance from the destination airport to find the correct place to begin a descent to that airport.

Surveying

Surveying is the process by which cartographers calculate the numerical distances and heights between different points before creating a map. Because terrain is often uneven, surveyors must find ways to take measurements of distance in a systematic way. The Pythagorean Theorem is used to calculate the steepness of slopes of hills or mountains. A surveyor looks through a telescope toward a measuring stick a fixed distance away, so that the telescope's line of sight and the measuring stick form a right angle. Since the surveyor knows both the height of the measuring stick and the horizontal distance of the stick from the telescope, he can then use the theorem to find the length of the slope that covers that distance, and from that length, determine how steep it is.





This article was originally published on The sciencing. Read the original article.

Source:

https://sciencing.com/real-life-uses-pythagorean-theorem-8247514.html

Patterns

The Pattern is defined as the series or sequences that are replicates. The sequence of objects that are arranged based on particular rule is known as number pattern.

Algebraic pattern:

The sequence of number pattern based on the addition or subtraction is known as algebraic pattern.

algebraic pattern

Example:

Consider the sequence 1, 5, 9, 13, 17, 21, 25,...

Here, the numbers are arranged by following certain rule, which is explained below:

(i) First number in the given number pattern is 1.

(ii) Add 4 to the first number of the sequence. That is, 1 + 4 = 5, is the second number of the sequence.

(iii) Add 4 to the second number of the sequence. That is, 5+ 4 = 9, is the third number of the sequence.

(iv) Add 4 to the third number of the sequence. That is, 9 + 4 = 13, is the fourth number of the sequence.

(v) Add 4 to the fourth number of the sequence. That is, 13 + 4 = 17, is the fifth number of the sequence. Similarly, the remaining numbers will be obtained.

Thus, the given sequence is ordered by using the addition rule. Hence, the pattern followed by the sequence is an algebraic pattern.



Geometric pattern:

The sequence of number pattern based on the multiplication or division is known as geometric pattern.

Geometric pattern

Example:

Consider the sequence 90, 45, 22.5, 11.25,...

Here, the numbers are arranged by following certain rule, which is explained below:

(i) First number in the given number pattern is 90.

(ii) One-half of the first number gives the second number of the pattern. That is, , is the second number of the sequence.

(ii) One-half of the second number gives the third number. That is, , is the third number of the sequence. Similarly, the remaining numbers will be obtained.

Thus, the given sequence is ordered by using the division rule. Hence, the pattern followed by the sequence is a geometric pattern



This article was originally published on The cheeg. Read the original article.

Source:
https://www.chegg.com/homework-help/definitions/algebraic-and-geometric-patterns-67