Introduction for practice ratio problems

Introduction for practice ratio problems:

              In mathematics, a ratio expresses the magnitude of quantities relative to each other. Specifically, the ratio of two quantities indicates how many times the first quantity is contained in the second and may be expressed algebraically as their quotient. Example: For every Spoon of sugar, you need 2 spoons of flour (1:2)
             In our daily life, by learning ratio and proportion many a times we compare two quantities of the same type. Thus, in convinced situations, comparison by division makes better sense than comparison by taking the difference. The comparison by division is the Ratio. We denote ratio-using symbol ‘:’ . If two ratios are equal, we state that they are in proportion and use the symbol ‘:’ or ‘=’ to equate the two ratios.

Practice Examples for Ratio Problems:

Problem 1:

             In a bag of red and green sweets, the ratio of red sweets to green sweets is 3:4. suppose  the bag contains  totally 120 green sweets then  how many red sweets are there?

Solution:

Step 1: Assign variables:

               Let x = red sweets

            Write the items in the ratio as a fraction.

                Red / green = 4 / 5 = x / 120

Step 2: Solve the equation 

                   Cross Multiply

                      4 × 120 = 5 × x
                             480 = 5x

                  Isolate variable x
                          x = 480 / 5 = 96

Answer: There are 96 red sweets.

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Problem 2:

There are 280 students in a certain school. If the number of boys in a school are 150. Find the ratio between

            i) Number of girls to number of boys

            ii) Number of boys to total number of students

Solution:

           Number of students = 280

           Number of boys = 150

Number of girls = Total number of students – Total number of boys

                             = 280 – 150

                             = 130

i) Number of girls to number of boys

             Number of girls: number of boys

                             = 130: 150

             Dividing each term by 10

                             = 13: 15
             So, ratio between girls to the boys is 13: 12

ii) Number of boys to total number of students

                Number of boys: number of students

                                   = 150: 280

                Dividing each term by 10

                                   = 15: 28
                So, ratio between boys to the students is 15 : 28

Problem 3:

Length and breadth of a rectangular field are 80 m and 20 m respectively. Find the ratio of the length to the breadth of the field.

Solution:

  Length of the rectangular field = 80 m

              Breadth of the rectangular field = 20 m

              The ratio of the length to the breadth is 80: 20

              The ratio can be written as

              = 80 / 20 = 4:1

Thus, the required ratio is 4:1

Practice Ratio Problems:

Practice problem 1:

              Find the ratio of 24 yards to 36 yards.

              Answer: 3: 4

Practice problem 2:

              Find the ratio of 39 days to 30 days.

              Answer: 13: 10

Practice problem 3:

              Mrs. Alice earns $ 160 per month. She spends $ 8 and saves the rest. Find the ratio of her salary to savings?

               Answer: 20: 1

Introduction to factorization in algebra

Introduction to factorization in algebra

                      In Algebra, factorization is the process of simplifying an algebric equation by factoring out the common factors from all terms. Thus, the expression can be expressed in terms of the product of two or more simple expressions. In some cases, the factorization of algebric equations is done by splitting some terms into two or more terms. For example the factorization of a quadratic equation in algebra is done by splitting the middle term in to two terms.

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                      For example, factorization of the polynomial, x² − 16 factors as (x − 4) (x + 4). In all cases, a product of simpler Expressions is obtained. Here x² – 16 is a second degree polynomial.Those factors  (x + 4) and (x – 4) are first degree polynomials. This factorization is useful in simplifying expressions.

Methods of Factorization:

1) Factorization in algebra by removing common factors:

When the terms of an algebraic expression X  have a common factor Y, we divide each term of X by Y and get an expression Z. Now, X is factored as Y × Z.

Ex:Factorize: 6x⁴y³ – 4x²y² + 10xy³.

Solution : We observe that 2xy² is a common factor .

6x⁴y³ – 4x²y² + 10xy³ =2xy²[ (6x⁴y³)/2xy² )-( 4x²y²/2xy² )+( 10xy³/2xy² ) ]=2xy²(3x³y – 2x + 5y).

2)Factorization in algebra  by grouping method:

Solution:Grouping method is used for the given expression contains three or more terms.

Steps:

In the first step, if the terms having common factors, then those terms to be grouped.

In the second step, the greatest common factor (GCF) is taken out.

Finally, in the third step, the distributive law is used to find the factors.

The distributive law is,

a (b + c)=a b + a c

Factoring in algebra by grouping: x³+3x²−6x−18

Solution:In the first step, if the terms have common factors, then those terms to be grouped.

(x³+3x²) + (−6x−18)

Factor x² out of the first two terms, and factor −6 out of the second two terms.

x²(x+3)-6(x+3)

Note that there is a common factor, x+3. So, take (x+3) as common.

(x+3) (x²-6) is the final factorization.

x³+3x²−6x−18=(x+3) (x²-6).

Few more Methods of Factorization:

3) Factorization of quadratic expression in algebra:

Consider the quadratic expression like 15 – 2x – x².   

Solution: Writing in the standard form,

15 – 2x – x² = –x² – 2x + 15

                      = (–1) (x² + 2x – 15).

Using the splitting method, If we can find two numbers p and q such that

p + q = 2 and p q = -15, then we can get the factors.

So , p + q = 5+(-3)=2 and p*q =5*(-3)= -15

Hence, we get 15 – 2x – x² = (–1) (x² + 2x – 15).

                                                 = (–1) [(x+5) {x + (–3)}]

                                                 = (–1) (x +5)(x – 3)

                                                 = (x + 5) ( 3 – x).

                                                 = (3x + 1) (2x + 5)

4) Factoring Difference of Two Squares:

Factorization of the difference of the two squares in the form x² - y², we should remember the formula

(x²-y²)=(x + y) (x - y).This is the special cases of factorization.

Example: a²-4 = (a + 2) (a - 2)

FRACTIONS

The study of Fractions is an important topic under Math. Fractions are basically a type of numbers. There are different types of numbers:

• Whole Numbers
• Natural Numbers
• Integers
• Fractions
• Rational Numbers
• Irrational Numbers

• Fractions are those numbers which are represented as the 'parts of a whole' . For example: 1/4 is one part out of four total parts. Students can learn how to compare fractions and understand the steps involved.

Various mathematical operations can be conducted on Fractions. Students can learn :

• How to add fractions
• How to subtract fractions
• How to divide fractions
• How to multiply fractions

Division of fractions in turn varies with the different kinds of numbers involved . Students can learn :

• How to divide fractions with whole numbers
• How to divide fractions with mixed numbers

Fractions have 2 parts namely the Numerator and the Denominator. The numerator and the denominator can have different types of values. If they consist of variables, they have different properties. Students can learn :

• How to simplify fractions with variables
• How to divide fractions with variables

Significance of Math

Do you what to know what is arithmetic progression:

The arithmetic series is nothing but the addition of the arithmetic sequences. The series of the arithmetic numbers which are given as the sequences. The sequences are to be added to get the series form. The geometric series is nothing but the addition of numbers of the geometric sequences to form the series. Now we are going to see about the arithmetic and the geometric series.

About the Arithmetic and Geometric Series:

We know that the arithmetic series are given as the sum of sequences of numbers. The arithmetic series formula is used to find the certain number of sequences which can be given as follows,

S_n = n(a₁+ a_n)/2

The infinite geometric series are the sum of the arithmetic sequence formula of numbers. The geometric series formula can be determined by using the formula as follows,

S_n = a₁ (1 - rⁿ)/1 – r

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Familiarization with math

Let us learn steps to divide and simplify:

How to divide a fraction is the technique to perform, it only requires multiplication. keep the first fraction as it is when two fractions are dividing and then multiply the first fraction with the reciprocal of the second fraction (i.e. invert of the second fraction). Example: How to divide fractions 1/6 / 7/8 Keep the first fraction 1/6 as it is = 1/6 And then reciprocal the second fraction = 8/7 Need help with linear programming examples 1/6 x 8/7 = (1 x 8)/ (6 x 7) = 8/42 = 4/21.this is after multiplying both the fractions.
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Incerdible Online Math

Introduction to Solve exponential equations

Online Solving makes the student to interact with the online tutors and let them to clear their doubts in solving exponential equation.

Solving it online makes exponential equation is one of the instant types of learning where student can solve the problem easily with the guidance of online educators.

Equation is which has the variables in the exponent. Same log,logarithmic rules are used to find Exponential equation do you want to learn how to calculate pi.

Solving Exponential Equations Online ...

• To find the exponential equation, take log on either sides and solve the variables.
  
• When the base of the exponential equation is equal on either sides, then the equation can be solved by using the property, if b^x = b^y, then x = y, where b > 0 and b ≠ 0.
• Stay Step forward to math

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Do you work out fractions

* If they add six fraction values, when the fractions have same denominator values means adding six fractions done basically. Denominators are same means, they add the numerator. If there is a necessity, simplify the result fraction values.

* When the fractions have different denominator values means convert the denominator in to common denominator by using the L.C.M and then they perform rename the fraction values and add fractions get help with online graphing calculator with table.

* When the fractions have mixed numbers means they add the mixed numbers in six different ways such as converting mixed numbers in to improper fractions and another way they add the whole numbers and fractional numbers separately.
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