Fractions in standard form

Fractions in standard form DEFAULT

Writing Equations in Standard Form

We know that equations can be written in slope intercept form or standard form.

Let's quickly revisit standard form. Remember standard form is written:

Ax +By= C

We can pretty easily translate an equation from slope intercept form into standard form. Let's look at an example.


Example 1: Rewriting Equations in Standard Form


Rewrite y = 2x - 6 in standard form.


Standard Form:  Ax + By = C

This means  that we want the variables (x & y) to be on the left-hand side and the constant (6) to be on the right-hand side.

When we move terms around, we do so exactly as we do when we solve equations!  So, remember... Whatever you do to one side of the equation, you must do to the other side!

Solution

Rewriting equations in standard form

That was a pretty easy example. We just need to remember that our lead coefficient should be POSITIVE!

Let's take a look at another example that involves fractions. There is one other rule that we must abide by when writing equations in standard form.


Equations that are written in standard form:

Ax + By = C

CANNOT contain fractions or decimals! A, B, and C MUST be integers!

Let's take a look at an example.


Example 2: Standard Form Equations


Rewrite y = 1/2x + 4 in standard form.


We now know that standard form equations should not contain fractions. Therefore, let's first eliminate the fractions.

Since the only fraction is is 1/2, we can multiply all terms by the denominator (2) to eliminate the fraction.

Solution

Writing Standard Form Equations

Now, let's look at an example that contains more than one fraction with different denominators.

Need More Help?

If you find that you need more examples or more practice problems, check out the Algebra Class E-course. You'll find additional examples on video, lots of practice problems with detailed solutions and little "tips" to help you through!


Example 3: Eliminating Fractions


Rewrite y = 3/4x - 1/8 in standard form.

Our first step is to eliminate the fractions, but this becomes a little more difficult when the fractions have different denominators!

We need to find the least common multiple (LCM) for the two fractions and then multiply all terms by that number!


Solution

Rewriting equations in standard form

Slope intercept form is the more popular of the two forms for writing equations. However, you must be able to rewrite equations in both forms.

For standard form equations, just remember that the A, B, and C must be integers and A should not be negative.


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Julia asked her kids, "do you know how old the Earth is?"

The Earth is over 4.5 billions years old

They thought and said that the Earth was older than her, them, and their grandparents. Julia told them that the Earth was believed to have been formed over 4.5 billion years ago. She asked them if they could try counting this number on their fingers. When they replied that they couldn't, she told them that a number as huge as this can be expressed mathematically in its standard form only. 

In this chapter, we will explore the world of standard form. The journey will take us through standard form of fractions, standard form of equation, standard form formula, standard form of slope and standard form in algebra, a few tips, such as the thumb rule, and interesting ways to understand standard form in simpler way. 

Lesson Plan

What is Standard Form?

A standard form is a form of writing a given mathematical concept like an equation, number, or an expression in a form that follows certain rules.

4.5 billion years is written as 4,500,000,000 years

As you can see here, writing a large number like 4.5 billion in its number form is not just ambiguous but also time-consuming and there are chances that we may write a few 0’s less or more while writing in the number form.

So, to represent very large or very small numbers concisely, we use the standard form.


Standard Form Rules

The process of writing a given mathematical concept like an equation, number, or an expression in the standard form follows certain rules.

Depending upon which mathematical concept we are dealing with, the standard form formula will vary.

For example, the standard form of \(4,500,000,000 = 4.5 \times 10^9\)

What about the fractions \(\dfrac{14}{22} \text {and} \dfrac{13}{6} \)? Are these standard forms of fractions?

In the case of fractions, we need to ensure that in the standard form of fractions, the numerator and denominator must be co-prime.

That means they have no common factor other than 1

So, the standard form, also called as the simplest form of the fraction \(\dfrac{14}{22} \text {is} \dfrac{7}{11} \)

The fraction \(\dfrac{13}{6}\) is already in its simplest form as 13 and 6 are co-prime.

What we have observed in the above cases are the standard forms of writing a decimal number and fractions.


How do we write in Standard Form?

Standard form has different meanings depending on which country you are in.

In the United Kingdom and countries using UK conventions, the standard form is another name for scientific notation.

\(3890 = 3.89  \times 10^3\)

In the United States and countries using US conventions, the standard form is the usual way of writing numbers in decimal notation.

Standard form \(= 3890\)

Expanded form \(= 3000 + 800 + 90\)

Written form \(=\) Three thousand eight hundred and ninety

We now proceed to study the standard form in algebra.

Standard form in algebra includes some major topics like polynomials, linear and quadratic equations.

Let’s begin with polynomials.


Standard Form of a Polynomial

Consider the algebraic expressions:

\(\begin{align} x^3+7x^2&-8x \\ 4x^5+7x^4&-5x^2-10\\y^2+7y^6&-8y-y^9 \\z^4+7z^5&-8z^2-12z+45\end{align}\)

Can you identify the correlation between all these expressions?

All these belong to the family of Polynomials.

In the above examples, the first two are in the standard form whereas the last two are not.

The rules for writing a polynomial in standard form are very simple.

1. Write the terms in the descending order of their powers (also called as exponents).

2. Ensure the polynomial contains no like terms.

The standard form is:

\(a_nx^n + a_{n-1}x^{n-1}.....+a_0\)

Thus, standard form of \( y^2+7y^6-8y-y^9 = -y^9+7y^6+y^2-8y\) 

Standard form of

\[\begin{align}\!z^4\!+\!7z^5\!-\!8z^2\!-\!12z\!+\!45=\\7z^5\!+\!z^4\!-\!8z^2\!-\!12z\!+\!45\end{align}\]

 


tips and tricks

The thumb rule: D-U for writing a polynomial in its standard form. 

D stands for Descending and U stands for unlike terms. 

Rules for writing a polynomial - DU where D stands for Descending and U stands for Unlike

Standard Form of a Parabola

"What goes up must come down" - this is a well-known law of gravity.

How is it applicable to our topic?

Let's observe the shape of a banana and the motion of a dolphin in the water. 

real life example of parabola

real life example of parabola-dolphin

Both of them form a simple curve shaped like an arch.

This is called a parabola in Mathematical terms.

A parabola is the graph of a quadratic function.

Thestandard form of parabola is:

\(f (x) = ax^2 + bx + c\)

Note: It contains only one squared term.

The word parabola originated in the late 16th Century from para meaning "beside" and bolē meaning "a throw" (from the verb ballein).

The following graphs are two typical parabolas; their x and y-intercepts are marked by green dots, and the vertex of each parabola is marked by a blue dot.

parabola equations

For any quadratic polynomial \(ax^2+bx +c = y \text { where } a\neq 0\), the graph will have one of the two shapes as shown above.

When a > 0, it represents the first parabola (figure 1) which opens upward (U-shaped).

When a < 0, we obtain a parabola that opens downward (inverted U-shaped) as shown in figure 2

Now when we put y = 0 in the above equation, we get the x-intercept which is also called as the roots of the equation: \(ax^2+bx +c = 0\)

Finding the roots provides us with the turning point of the parabola.

Axis of symmetry

The vertical line which is drawn from the turning point is called the axis of symmetry.

It is calculated using the formula:

Example: 

For the equation \( x^2 + 3x-4=0\), the equation of axis of symmetry is \(x = \dfrac{- b}{ 2a}= \dfrac{- 3}{ 2}\).


Standard Form of a Linear Equation

Look at the following images.

standard form of line

All these are images of lines; observe the equations of each one of them.

Each of them has degree 1

Hence, they are called linear equations in two variables.

Can we write them in the form: \(Ax+ By = C?\)

Of course, we can!

Let’s see how.

Line 1: \(\begin{align}  x\,+\,y\,&=\,7\\i.e. 1x+1y&=7\\Here, A= 1, B&=1 \,\text{and }\, C=7\end{align}\)

Line 2: \(\begin{align} \,y\,&=\,3x\\i.e. 3x-1y&=0\\Here, A= 3, B&=-1 \,\text{and } \,C=0\end{align}\)

Line 3: \(\begin{align}  2x\,&=\,3-y\\i.e. 2x+1y&=3\\Here, A= 2, B&=1 \,\text{and }\, C=3\end{align}\)

Therefore, what we see here is the standard form of equation which is linear.


Standard Form of a Line

As discussed in the above section, a linear equation geometrically represents a line.

Hence, the standard form of a line is same as that of a linear equation.

i.e. \(Ax+By=C\)

Slope:

We can also determine the slope of a given line.

If we come across an equation in standard form that we are required to present graphically, we must convert it to slope-intercept form.

To do this, we must solve the equation for y.

For slope, we write the equation in the form: \(y = mx+c\)

Here \(m\) represents the slope of the line.

Example 1:

Consider \( 5x+4y=10\)

Let us solve this equation.

\(\begin{align} 5x+4y&=10\\4y&=10-5x\\y&=\dfrac{-5x+10}{4}\\&=\dfrac{-5x}{4}-\dfrac{10}{4}\\&=\dfrac{-5}{4}x-\dfrac{5}{2}\end{align}\)

\(\therefore Slope= \dfrac{-5}{4}\)

Given two points in a plane, slope can be defined as the ratio of change in the value of y to the change in the value of \(x.\)

Standard form of slope is:

\( m=\dfrac {y_2 – y_1}{x_2 – x_1}\)

slope

Example 2:

Let's apply (4, 2) and (5, 6) into the slope formula.

We get slope \(=  \dfrac {y_2 – y_1}{x_2 – x_1}=\dfrac {6 – 2}{5 – 4}= 4\)

Types of Slope:

Observe the figure below to understand the nature of slope.

types of slope

You may calculate slope using the formula \(Slope = \dfrac{\text {Change in y}}{\text {Change in x}}\).

In case 1, the value of slope is positive; in case 2, it is negative.

In case 3, \(Slope = \dfrac {0}{4}=0\) and in case 4 \(Slope = \dfrac {4}{0}\) which is not defined.

Take a look at the slope being formed by different lines in the simulation below.

 

important notes to remember
  1. Degree means the highest power of the variable present in the equation. 
  2. The word “Linear in two variables" stands for a line on a plane.

Solved Examples

 

 

Liza is trying to find out which of the following equations represent the given graph.

As there are no values of the coordinates given, she is not able to decide.

\(  1. \,x^4+7x^2-5x =4\hspace{1cm} 2.\,4x+5y=0\)

\( 3.\,y^2+7y^6-9y=y^3 \hspace{1cm} 4. \,3z^4+7z^5=-12z\)

equation of line

How can we use standard form concept to solve her problem?

Solution

The graph given here represents a line.

Now, a line represents a linear equation in two variables which has a degree 1

Out of the four equations given above, only option (b) is linear.

\(\therefore\)The graph represents the equation \( 4x+5y =0 \)

 

 

Anna showed her class notes on polynomials to her teacher.

The teacher returned her notes with a remark "Write the polynomial  \(x^2 -10x+16-x^2+x^5-3x^4 + 3x^2\) in standard form."

What is the correct format Anna should have used?

Solution

To write a polynomial in standard form, two rules must be taken care of.

Rule : 1. Write the terms in descending order of their powers.

2. All terms must be unlike.

Let us first arrange in descending order.

\(x^2 - 10x +16 - x^2 +x^5 -3x^4 +3x^2 = x^5 - 3x^4 + x^2 -x^2 + 3x^2 - 10x +16\)

After adding like terms, we get

\(x^5- 3x^4+3x^2-10x+16\)

\(\therefore \text{ Standard form is }x^5- 3x^4+3x^2-10x+16\)

 

 

James is looking at the parabola and trying to find the equation of axis of symmetry.

upside down parabola

How will he get the equation?

Solution

The x-coordinate of the vertex gives the equation of the axis of symmetry.

The vertex has coordinate \((-2,4)\)

Its x-coordinate is \(-2\)

Thus, the equation of the axis of symmetry is \(x = -2\)

\(\therefore \; x=-2 \) is the equation of axis of symmetry

Interactive Questions

Here are a few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result.

 

 

 

 

 

 


Let's Summarize

We hope you enjoyed learning about standard form with the simulations and practice questions. Now you will be able to easily solve problems on standard form of fractions, standard form of equation, standard form formula, standard form of slope and standard form in algebra.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


Frequently Asked Questions (FAQs)

1. How do you write in standard form?

We write numbers in standard form using the rules for respective mathematical concepts. 

For example, a linear equation has a standard form \(Ax+By = C \). 

2. How does the standard form look like?

The standard form of a line is: \(Ax+By = C \)

For polynomials, the standard form is: \(a_nx^n + a_{n-1}x^{n-1}.....+a_0\)

3. What is the meaning of standard form?

A standard form is a form of writing a given mathematical concept like an equation, number or expression in a form that follows certain rules.

Sours: https://www.cuemath.com/algebra/standard-form/
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How do you write fractions in standard form?

How do you write fractions in standard form?

Standard Fraction Standard form of Fraction: When numerator and denominator are co-prime, then the fraction is said to be in standard form. Two numbers are co-prime if they have no common factor other than 1. Following are some examples of fractions in standard form.

What is 25.413 in word form?

How to Write Out Number 25.413 in Words: twenty-five and four hundred thirteen thousandths in (US) American English, Number Converted (Spelled Out) to Words.

What is a standard form in math?

Standard form is a way of writing down very large or very small numbers easily. 103 = 1000, so 4 × 103 = 4000 . So 4000 can be written as 4 × 10³ . This idea can be used to write even larger numbers down easily in standard form.

How do I get rid of standard form?

There are two ways to add and subtract with numbers in standard form. The first is to write them both as the same power of ten and add or subtract the decimals. To add the numbers in standard form both numbers need to written using the same power of ten.

What does x10 mean on a calculator?

â stands for “x10” and the calculator interprets the entire number as (123.45 x 10-2) as if it was entered with parentheses. Click to see full answer. Keeping this in view, how do you do scientific notation on a calculator? Using Scientific Notation on a Scientific Calculator. Enter the number 10.

How do I get my Casio calculator out of standard form?

Casio models: Press [SHIFT][MODE][6:Fix]. You are then prompted to enter a number between 0 and 9. The number you enter fixes the number of decimal digits your results will have.

How do you get rid of zeros on a calculator?

Your calculator should shut off on its own. Use a button combination to force it off. This combination of buttons will turn off most Citizen brand calculators: ON ÷ × % Check Correct Correct.

What is fix mode on calculator?

The Fix command puts the calculator in fixed-point display mode: all numbers will be displayed with a fixed number of digits (0-9) after the decimal, depending on the argument of Fix. This could be useful if you’re trying to display potentially fractional numbers in a limited amount of space.

Why is my calculator rounding automatically?

there are 10 digits before the decimal point. For example: any natural logarithm (ln) of a whole number you can input, except 1, will have an infinite number of digits, and the screen of your calculator, or any calculator has only a limited number of digits, so the result has almost always a rounding error.

Why does LCD screen go black?

The most common reason for a cellphone’s screen to go black is a simple hardware failure. This can be caused by the actual LCD going bad, by the cable that runs between the LCD and control board going bad, or even just from the cable’s connectors coming loose.

Why does my calculator have a black screen?

Troubleshooting: Calculators Screen Is Black ➢ Remove all of the batteries including the round lithium battery for five minutes. After five minutes, reinsert all of the batteries and turn the calculator on. You may need to adjust the contrast again after doing this.

Why is my TI 86 screen dark?

To make the display darker, press 2nd, release that key, and then press and hold the up arrow until the display is dark enough. To make the display lighter, press 2nd, release that key, and then press and hold the down arrow until the display is light enough.

How do I make my TI 83 screen brighter?

Press the “On” key (row 10, column 1 on the TI-83 keypad) to turn your calculator on. Press and release the “2nd” key(row 2, column 1, often a yellow-orange color) and then press and release the “down arrow” key (row 3, between the fourth and fifth column, with a white arrow pointing down) to lighten the screen.

How do you lighten the screen on a TI 86?

To lighten the screen contrast, press and hold #. After about four minutes of inactivity, the TI-86 turns off automatically. If you release $ or # while adjusting the contrast, you must press – again to continue the adjustment.

How do you brighten the screen on a TI 85?

Push the “2nd” key on the TI-85. Push and hold the down triangle key on the directional keypad to decrease the contrast. Release the down triangle key when you have lightened the screen enough. The longer you hold down the key, the less contrast the screen will have.

Why is my TI 84 screen dark?

High Contrast If the whole screen is black it is likely that the contrast may have been turned up on the calculator. Press the (2ND) button and then the down arrow button to lower the contrast. Repeat this until the contrast is at the level you like.

How do you darken the screen on a TI 82?

Press 2nd and release. 2. Hold the DOWN ARROW key to make the screen lighter. Hold the UP ARROW key to make the screen darker.

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Algebra/Standard Form and Solving Slope

Algebra

Algebra
 ← SlopeStandard Form and Solving Slope[[Algebra/|]] → 

Standard Form[edit | edit source]

Standard form is another way to write slope-intercept form (as opposed to y=mx+b). It is written as Ax+By=C. You can also change slope-intercept form to standard form like this: Y=-3/2x+3. Next, you isolate the y-intercept(in this case it is 3) like this: Add 3/2x to each side of the equation to get this: 3/2x+y=3. You can not have a fraction in standard form so you solve this. 2(3/2x+y)=3(2). To get: 3x+2y= 6. Now you have a standard form equation! However, there are some rules for standard form. A, B, C are integers (positive or negative whole numbers) No fractions nor decimals in standard form. "Ax" term is positive. If these are not followed, it is not standard form. i.e. -1/3x+1/4y=4 is NOT standard form.

Solving Slope[edit | edit source]

Solving Standard Form[edit | edit source]

Slope intercept equations (y=mx+b) are the easiest to graph. So if you encounter an equation in standard form that you are required to graph, you must convert it to slope intercept form. To do this you must take the equation and solve for Y.

Example:

{\displaystyle 9x+7y=-3}

{\displaystyle 9x-9x+7y=-3-9x}

{\displaystyle 7y=-3-9x}

{\displaystyle {\frac {7y}{7}}={\frac {-3-9x}{7}}}

{\displaystyle y={\frac {-3}{7}}-{\frac {9}{7}}x}

That is technically slope intercept form, but if you want to make it true (y=mx+b) simply follow the rule of negatives (a - b = a + -b):

{\displaystyle y={\frac {-9}{7}}x+{\frac {-3}{7}}}

Solving an equation in slope-intercept form- how?

If you come upon an equation in slope-intercept form and require it to be in standard form, simply solve for m (c).

Example:

{\displaystyle y=10x+9}

{\displaystyle y-10x=9+10x-10x}

{\displaystyle y-10x=9}

Standard form cannot have fractions. If they are in a fraction, you must multiply each side to get rid of it.

Example:

{\displaystyle {\frac {9}{10}}x+9y=5}

{\displaystyle 10[{\frac {9}{10}}x+9y]=10[5]}

{\displaystyle 9x+90y=50}

Sours: https://en.wikibooks.org/wiki/Algebra/Standard_Form_and_Solving_Slope

Standard fractions form in

Standard Fraction

Standard form of Fraction: When numerator and denominator are co-prime, then the fraction is said to be in standard form.

Two numbers are co-prime if they have no common factor other than 1.Following are some examples of fractions in standard form.

`1/2, 3/4, 4/5, (11)/(12)`

In these fractions there are no common factors other than 1 in the numerator or denominator. Sometimes, standard form of fraciton is also called simple fraction.


Sample Problems

Question 1: Reduce the following fractions in their standard form.

`2/4, 6/9, (16)/(20), (9)/(18)`, `(12)/(30), (18)/(27), (19)/(57)`, `(20)/(50), (27)/(108)`, `(55)/(275), (57)/(108), (102)/(625)`, `(625)/(1275), (212)/(1000), (500)/(1500)`

(a) Solution: To solve it write the prime factorization of numerator and denominator.

`2/4=(2)/(2xx2)=1/2`

Now, cancel the equal numbers, if they are present in numerator and denominator both.

Or, `2/4=1/2`

Hence, `1/2` is the standard form of the fraction `2/4`.

Answer: `6/9=(2xx3)/(3xx3)=2/3`

Answer: `(16)/(20)=(2xx2xx2xx2)/(2xx2xx5)=4/5`

Answer: `(9)/(18)=(3xx3)/(2xx3xx3)=1/2`

Answer: `(12)/(30)=(2xx2xx3)/(2xx3xx5)=2/5`


Answer: `(18)/(27)=(2xx3xx3)/(3xx3xx3)=2/3`

Answer: `(19)/(57)=(19)/(19xx3)=1/3`

Answer: `(20)/(50)=(2xx2xx5)/(2xx5xx5)=2/5`

Answer: `(27)/(108)=(3xx3xx3)/(2xx2xx3xx3xx3)=1/4`

Answer: `(55)/(275)=(5xx11)/(5xx5xx11)=1/5`

Answer: `(57)/(108)=(3xx19)/(2xx2xx3xx3xx3)=(19)/(36)`

Answer: `(102)/(625)=(2xx3xx17)/(5xx5xx5xx5)=(102)/(625)`

Answer: `(625)/(1275)=(5xx5xx5xx5)/(3xx5xx5xx17)=(25)/(51)`

Answer: `(212)/(1000)=(2xx2xx53)/(2xx2xx2xx5xx5xx5)=(53)/(250)`

Answer: `(500)/(1500)=(2xx2xx5xx5xx5)/(2xx2xx3xx5xx5xx5)=1/3`

Question 2: Reduce the following fractions in their standard form:

`7/5, (16)/(15), (25)/(20)`, `(27)/(12), (12)/(27), (120)/(150)`, `(150)/(120), (45)/(50), (50)/(45)`, `(14)/(20), (20)/(14), (76)/(95)`, `(95)/(76), (12)/(100), (100)/(12)`

(a) Solution: In this fraction, numerator is greater than the denominator hence it is an improper fraction. And there is no common factor in the numerator and denominator. Therefore change this fraction into mixed fraction.

Follow these steps to change a fraction into mixed fraction:
  • Step 1: Divide the numerator with denominator.
  • Step 2: Write the quotient as the whole number.
  • Step 3: Write the remainder as numerator.
  • Step 4: Write the divisor as denominator.

When we divide 7 (numerator of fraction) by 5 (denominator of fraction), we get Quotient = 1, Remainder = 2.

`text(Mixed Fraction) = text(Quotient)\text(Remainder)/text(Denominator)`

`7/5=1\2/5`

Note: If the numerator is greater than the denominator and they are not co-prime then change the fraction into standard fraction. After that change it into a mixed fraction.


Answer: `(16)/(15)=1\(1)/(15)`

Answer: `(25)/(20)=(5xx5)/(2xx2xx5)=5/4=1\1/4`

Since, in this fraction the numerator and denominator are not co-prime. Therefore, make them co-prime first and then change it into a mixed fraction.

Answer: `(27)/(12)=(3xx3xx3)/(2xx2xx3)=9/4=2\1/4`

Answer: `(12)/(27)=(2xx2xx3)/(3xx3xx3)=4/9`

Answer: `(120)/(150)=(12)/(15)=(2xx2xx3)/(3xx5)=4/5`

Answer: `(150)/(120)=(15)/(12)=(3xx5)/(2xx2xx5)=5/4=1\1/5`

Answer: `(45)/(50)=(3xx3xx5)/(2xx5xx5)=(9)/(10)`

Answer: `(50)/(45)=(2xx5xx5)/(3xx3xx5)=(10)/(9)=1\1/9`

Answer: `(14)/(20)=(2xx7)/(2xx2xx5)=(7)/(10)`

Answer: `(20)/(14)=(2xx2xx5)/(2xx7)=(10)/(7)=1\3/7`

Answer: `(76)/(95)=(2xx2xx19)/(5xx19)=4/5`

Answer: `(95)/(76)=(5xx19)/(2xx2xx19)=5/4=1\1/4`

Answer: `(12)/(100)=(2xx2xx3)/(2xx2xx5xx5)=(3)/(25)`

Answer: `(100)/(12)=(2xx2xx5xx5)/(2xx2xx3)=(25)/(3)=8\1/3`



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Ex 2: Write General Equation of a Circle in Standard Form (Fractions)

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