400 words in speech

convert words to time .

How long will it take to read a speech or presentation?

Enter the word count into the tool below (or paste in text) to see how many minutes it will take you to read. Estimates number of minutes based on a slow, average, or fast paced reading speed.

Number of words

Reading speed

Check out our other writing tool

Common conversions (average speed)

Want to sponsor this site.

Want some 👀 on your business?

We're looking for sponsors! Get in touch if you want your business to be featured on our site.

Other Tools

Diff Text - Dead simple text comparison 🔎

Prepared.FYI - Find deals on emergency preparedness, camping, and survival products and equipment 🏕️

Grammarly - Grammar, plagiarism, and spell checker.

Hemingway - Editor to make your writing bold and clear.

Power Thesaurus - Simple crowdsourced thesaurus.

Wake Up Time - Fall asleep and actually wake up refreshed.

Copy Arrow - An organized set for easy copying.

Micron Pens - Amazing pens for writing and illustration.

Block Rocker - Portable event speaker with microphone 🎤

TED Talks - The official TED guide to public speaking.

These tools are awesome, and the affiliate income helps keep the site online.

If you've found this tool helpful, please consider paying it forward to keep it ad free!

As a bonus you'll go down in history listed on one of the coolest sites around :)

Want to learn how to make websites like this?

There's some really cool stuff in the works... Enter your email to be the first to know when it's ready!

Enter your email to be the first to know when some really cool stuff is ready...

Enter your email to be notified of some really cool stuff that's in the works...

My name is Spencer, and I actually work here ↑

Looking for the easiest way to grow your business in 2020?

Get more traffic, more sales, and more reviews effortlessly.

Boost your online reviews and become the obvious choice for new customers.

Introducing Speech Time Calculate

Estimate how many minutes your speeches, presentations, and voice-over scripts will take based on your words per minute rate!

How To Speech Time Calculate Using This Tool?

If you have a certain number of words or a piece of text you want to time, you can either type in the word count or paste the text into the provided area. This tool will then calculate how long it would take to read that text out loud.

The talk time estimate is calculated using the average speaking speed of adults, which is determined to be 183 words per minute based on scientific studies. If you’re interested in how long it would take to read silently, it’s estimated at 238 words per minute ( This data is also backed by research )

You can adjust the slider to change the words per minute value, which will affect the talk time estimate. However, the silent reading time estimate remains fixed at 238 words per minute.

For ease of use, we’ve also provided reference points for slow, average, and fast reading rates below the slider.

To begin anew, simply click the ‘clear text’ button to erase the content and restore the slider back to its original setting of 183.

Who is This Words to Minutes Converter Tool For?

If you are a student wondering how long is my essay or you’ve been tasked with writing a speech and need to know how many words to aim for and how many minutes will it take to deliver or perhaps you are a podcaster, just starting out, who wants the ability to easily synchronize music and spoken word without having to painstakingly calculate seconds between them, then this Speech Time Calculate is precisely for you!

From now on, instead of spending long hours in front of the computer trying to figure out how many seconds it takes for one phrase or section of dialogue to end and another to begin, you can let our innovative tool do all the work and convert your text to time quickly and accurately. With this powerful tool at your disposal, whether you’re giving a TED talk or just need to nail a business presentation, your life will become a little bit easier.

So keep reading to learn more about what this fantastic words to minutes converter has in store for public speakers, aspiring students, and professional radio producers alike!

Whether you want to read the text silently or speak aloud, you can use this tool as both:

• Reading time calculator
• Talk time calculator

Explanation of the Reading Time

Reading time refers to the duration it takes for an average person to read a written text silently while still comprehending its content. Based on an extensive analysis of 190 studies that involved 18,573 participants , research conducted by Marc Brysbaert in 2019 suggests that the typical silent reading speed for an adult individual is approximately 238 words per minute .

To convert word count to read time for a specific text, you can do so by dividing the total word count of the text by this established value of 238. Here is the mathematical equation for determining the duration of reading time in minutes:

Reading Time = Total Word Count / 238

Explanation of the Speech Time

Speech time refers to the duration it takes for an average person to read a text out loud. Based on data from 77 studies involving 5,965 people , it’s been found that most adults read aloud at a speed of approximately 183 words per minute ( research conducted by Marc Brysbaert in 2019 ). To figure out how long it will take to read a specific piece of text aloud, you can divide the total number of words in the text by this average rate of 183 words per minute.

Of course, it’s important to note that talk time can vary depending on factors such as clarity of speech, pauses for emphasis, and use of visual aids. However, using this tool for converting the number of words to minutes can still provide a helpful guideline for planning and practicing your presentation. By having a better understanding of speech rates, you can ensure that your message is delivered effectively and efficiently.

Benefits of Using a Speech Time Calculate

Time management in presentations.

Effective time management during presentations is crucial to ensure the audience remains engaged and the information is accurately conveyed. This is where our words to speaking time converter comes in handy. By using this tool, presenters can easily determine how many words they need to include in their presentation to stay within the allotted time frame.

Not only does it help with time management, but it also ensures that the pacing of the presentation is consistent, making it easier for the audience to follow. With the use of this presentation time calculator, presenters can confidently deliver their presentations without the worry of running over time or rushing through it.

Estimated speech time for public speaking

Public speaking can be nerve-wracking, especially when you have too little or too much information to fill your time slot. You wonder only if there were an accurate public speaking time calculator available so that you could be able to allocate the appropriate amount of time to each section of your presentation, ensuring that you cover all the necessary points without rushing or going over time.

Effective pacing is key in ensuring your message is delivered with clarity and impact.

Most public speakers target an average of 130-150 words per minute for their spoken content, meaning you should aim to limit your speaking time to roughly one minute per 130-150 words. While this may take some practice to achieve, the end result is a confident, well-timed delivery that keeps your audience engaged from start to finish.

Remember, in public speaking, less is often more—take your time to breathe and emphasize key points. Your audience will appreciate your thoughtful and measured approach. For that, you can use this tool and adjust your words to speech time.

Accurate estimations for audiobooks and podcasts

As more and more people turn to audiobooks and podcasts for their entertainment and information needs, accurate estimations of listening time have become more important than ever. After all, there’s nothing worse than settling in for a quick listen only to find yourself trapped in a story that goes on for hours longer than you anticipated.

That’s why it’s great to see publishers and podcast producers taking estimated reading time seriously, providing listeners with the information they need to choose the right content for their schedule. Whether you’re looking for a quick listen on your daily commute or a lengthy distraction for a lazy Sunday afternoon, accurate estimations using this words to speak time calculator make it easier than ever to find the perfect content.

Some Popular Speech Times

how many words in a 2 minute speech

Almost 300 words

how many words in a 3 minute speech

Almost 450 words

how many words in a 4 minute speech

Almost 600 words

how many words in a 15 minute speech

Almost 2250 words

The speech time is calculated taking 150 words per minute as reference value

Common conversions (average speed)

How long does it take to read 500 words?

3.8 minutes

How long does it take to read 750 words?

5.8 minutes

How long does it take to read 1000 words?

7.7 minutes

How long does it take to read 1200 words?

9.2 minutes

How long does it take to read 1500 words?

11.5 minutes

How long does it take to read 1800 words?

13.8 minutes

How long does it take to read 2000 words?

15.4 minutes

How long does it take to read 3000 words?

23.1 minutes

As the world becomes more fast-paced, time is a precious commodity. Determining how long your script will take to read, whether for a presentation or a video, can make a significant difference in engaging and retaining your audience’s attention.

That’s where our Words to Time Converter comes in handy. It’s a valuable tool for anyone working in various professions, from broadcast journalists to teachers to executives. No matter the industry, time is of the essence, and knowing how long your speech or presentation will take is crucial for effective communication.

• Games, topic printables & more
• The 4 main speech types
• Example speeches
• Commemorative
• Declamation
• Demonstration
• Informative
• Introduction
• Student Council
• Speech topics
• Poems to read aloud
• How to write a speech
• Using props/visual aids
• Acute anxiety help
• Breathing exercises
• Letting go - free e-course
• Using self-hypnosis
• Delivery overview
• 4 modes of delivery
• How to make cue cards
• How to read a speech
• 9 vocal aspects
• Vocal variety
• Diction/articulation
• Pronunciation
• Speaking rate
• How to use pauses
• Eye contact
• Body language
• Voice image
• Voice health
• Public speaking activities and games
• Blogging Aloud
• About me/contact
• Speech delivery
• How many words per minute in a speech

How many words per minute in a speech?

A quick guide for slow, average and fast rates of speech.

By:  Susan Dugdale  

If you're preparing a speech or presentation with a strict time limit it’s useful to have an estimate of the number of words you’ll need to fit the number of minutes you've been given.

And it's even more useful to have those guidelines before you rush into writing, only to discover when you’re done, you’ve written far too much. Something I used to do on a regular basis! (And then, I had the tedious job of pruning to fit.)

What's on this page:

How many words are there in a 1-minute speech, how many words are there in a 2-minute speech, how many words are there in a 3-minute speech, how many words are there in a 4-minute speech, how many words are there in a 5-minute speech, how many words are there in a 6-minute speech, how many words are there in a 7-minute speech, how many words are there in an 8-minute speech, how many words are there in a 9-minute speech, how many words are there in a 10-minute speech, how many words are there in a 15-minute speech, how many words are there in a 20-minute speech, how many words are there in a 25-minute speech, how many words are there in a 30-minute speech.

• How to calculate your own speech rate: 2 easy methods
• How many pages is a 4, 7 or 20-minute speech?     

About this quick reference guide

The following table (below) shows the average number of words spoken per minute for speeches from 1 to 30 minutes long. It's a quick reference guide: one only intended to provide general information.

Who, and what, is truly average?

An average word count for any timed speech (one minute, two minutes, three, four, five, six...or more, minutes) is impossible to give. As there really is no such thing as an average person, who uses an average rate of words per minute when they speak.

We are individuals: each of us different. Speech patterns and speech (speaking) rates vary considerably between person to person for a great many reasons. The best you’ll ever get is an educated guess.

Use as an estimate: test yourself to be sure

That aside, the following estimates will give you a handy indication of the number of minutes it takes to say x number of words depending on whether you talk at a slow speed, a medium speed or you're one of the world's fast talkers.

(And, no, it's never a good idea to gabble at a mile a minute to fit everything you've prepared into the length of time you've been given!)

To be absolutely sure what you've done will fit the time allocation you've been given it's a good idea to test yourself to establish your own speaking rate or speed of speech .

How many pages is a 'x' minute speech?

Multiple variations on the question how many pages are needed for a speech are frequently asked. For instance:

• How many pages is a 4-minute speech?
• How many pages is a 7-minute speech?
• How many pages is a 20-minute presentation?

Unfortunately, the answer is not simple. Calculating the number of A4 pages of text you need for a speech of any number of minutes long is tricky for two reasons.

How many words are on an A4 page?

The first reason is that the number of words on an A4 page depends on how the page has been formatted. The answers to the questions below make a very big difference to the total number of pages needed to print a speech.

• What is the font family being used? (Different fonts take up differing amounts of space because of the shape/design of their characters.)  
• What is the size of the font? (The font size you're reading right now is 20px.)
• What is the spacing between each of the letters in a word?
• What is the spacing between each line of text? Is it single spacing, 1.5 or double line spacing?
• What spacing has been set for paragraphs?
• What size have the margins of the page been set at?
• Are there headings? Sub-headings or lists?

On average if the font is plain, (sans-serif, without embellishment), like Arial or Verdana, its size is set for 12px, and the line spacing is set for 1.5, then a page may have between 400-500 words on it.

Speech rate changes how we calculate the number of words we need

The second reason why using the number of pages to gauge how long a speech will take to deliver is problematical is because we speak at different rates. 

A 500-word page may take someone speaking at very slow rate 4 minutes to get through.

Another person, who has a much faster speaking rate, may take about 2.5 minutes to deliver exactly the same text.

Using the number of pages as a guide for a speech that has to fit a time limit is completely unreliable.

If you need to use a 'guesstimate' use the speech rate table below. Forget about counting the pages! 

If you are a slow speaker, less than 120 words.

If you speak at an average speed between: 120 - 160 words.

If you are a fast speaker between: 160 - 200 words.

If you are a slow speaker less than 240 words.

If you speak at an average speed between: 240 - 320 words.

If you are a fast speaker between: 320 - 400 words.

If you are a slow speaker less than 360 words.

If you speak at an average speed between: 360 – 480 words.

If you are a fast speaker between: 480 - 600 words.

If you are a slow speaker less than 480 words.

If you speak at an average speed between: 480 – 640 words.

If you are a fast speaker between: 640 - 800 words.

If you are a slow speaker less than 600 words,

If you speak at an average speed between: 600 – 760 words.

If you are a fast speaker between: 760 - 1000 words.

If you are a slow speaker less than 720 words.

If you speak at an average speed between: 720 – 960 words.

If you are a fast speaker between: 960 - 1200 words.

If you are a slow speaker less than 840 words.

If you speak at an average speed between: 840 – 1120 words.

If you are a fast speaker between: 1120 - 1400 words.

If you are a slow speaker less than 960 words.

If you speak at an average speed between: 960 – 1280 words.

If you are a fast speaker between: 1280 - 1600 words.

If you are a slow speaker less than 1080 words.

If you speak at an average speed between: 1080 – 1440 words.

If you are a fast speaker between: 1440 - 1800 words.

If you are a slow speaker a little less than 1200 words.

If you speak at an average speed between: 1200 – 1600 words.

If you are a fast speaker between: 1600 - 2000 words.

If you are a slow speaker, a little less than 1,800 words.

If you speak at an average speed between: 1,800 - 2,400 words.

If you are a fast speaker between: 2,400 - 3,000 words.

If you are a slow speaker, a little less than 2,400 words.

If you speak at an average speed between: 2,400 - 3,200 words.

If you are a fast speaker between: 3,200 - 4,000 words.

If you are a slow speaker, a little less than 3,000 words.

If you speak at an average speed between: 3,000 - 4,000 words.

If you are a fast speaker between: 4,000 - 5,000 words.

If you are a slow speaker, a little less than 3,600 words.

If you speak at an average speed between: 3,600 - 4,800 words.

If you are a fast speaker between: 4,800 - 6,000 words.

Return to Top

Find out more about speech rate

• How to calculate your own words per minute speech rate - two easy methods
• The factors influencing a faster rate or slower rate of speech
• How to develop an ideal rate of speech : one matching content, speech purpose and your audience: six exercises for flexible speaking rate

Words per minute calculator

Lastly here's the link to a useful online words per minutes calculator . Once you've completed your speech enter the total number of words, and select whether you want it to give you an average for a minute of your speech at a slow rate, average or fast rate. Again, it's a 'guesstimate'. ☺

The only really safe way to find out whether your speech fits your time allocation is to say it aloud at an ideal rate or pace: one taking into account the content, and the audience who is going to listen to it, while timing it.

(Use the record function on your phone. It will assist in lots of other ways too! You'll hear where you need more vocal variety, where your pronunciation is blurred ...and, so on. It's super helpful.)

speaking out loud 

Subscribe for  FREE weekly alerts about what's new For more see  speaking out loud  

Top 10 popular pages

• Welcome speech
• Demonstration speech topics
• Impromptu speech topic cards
• Thank you quotes
• Impromptu public speaking topics
• Farewell speeches
• Phrases for welcome speeches
• Student council speeches
• Free sample eulogies

From fear to fun in 28 ways

A complete one stop resource to scuttle fear in the best of all possible ways - with laughter.

Useful pages

• Search this site
• About me & Contact
• Free e-course
• Privacy policy

©Copyright 2006-24 www.write-out-loud.com

Designed and built by Clickstream Designs

Words per Minute Calculator

Table of contents

Have you ever wondered how long it takes to deliver a speech? Or how much time you need to read a book? This words per minute calculator (or WPM calculator for short) will help you convert between the number of words and the time taken to read or say. This way, you will not only learn how many words is a 5-minute speech but also how long it takes you to read a document with 2000 words.

If you want to know how much time you need to finish a book, head straight to our reading speed calculator !

Words per minute (speech)

In order to determine how long your presentation takes, you first need to find your speaking speed — how many words per minute of speech you can deliver. For example, the average speaking speed in English is 130 wpm (words per minute). If, however, you are frequently told that you speak very slowly or very fast, you should adjust this value accordingly.

You could also measure your speaking speed experimentally. Set a timer to one minute and start reading a passage of text aloud. Once the timer stops, count the words you managed to read. This number is your words per minute speech speed — you can input it directly into our calculator after selecting the radio button Other .

Words per minute (reading)

The same principle as your speaking speed applies to your reading speed — the number of words per minute of reading. The average reading speed for the English language is between 170 and 240 wpm , depending on the text difficulty and your English ability.

Naturally, you can measure your reading speed and input it directly into this reading words per minute calculator. All you have to do is set a timer to one minute and start reading a book (or an e-book ), this time silently. Once the timer rings, count the words you read and input the number into the respective field (below the radio button Other in the reading section).

How many words is a 5 minute speech?

Once you know your speaking and reading speed, the calculations are a piece of cake! All you have to do is input the number of words you have to say into this words per minute calculator, which will tell you how long your presentation should take.

Of course, you can also use this calculator to figure out the maximum number of words in a speech with a limited time frame — for example, 5 minutes. Let's look at how to use this tool in more detail.

Choose whether you will speak to an audience or silently read the text. In this case, we are going to choose speech.

Select your speaking speed . We will stay with the average value of 130 wpm.

Determine the duration of the speech — here, we will pick a 5-minute elevator pitch.

Multiply the reading speed by the time to find out how many words are in a 5-minute speech:

130 × 5 = 650 words

Your speech can have a maximum of 650 words.

How many words per minute in a speech?

The average speaking speed in English is 130 words per minute . However, the average speaking rate changes according to the task before a speaker — for presentations, it goes down to 100-120 wpm, while for YouTubers, it's up to 150-160 wpm. When picking the right pace, you should also consider your audience, e.g., the presence of kids or non-native speakers!

How many words per 10 minute speech?

To deliver a successful ten-minute talk, prepare between 1000-1200 words. The absolute maximum you should have is 1300 words.

How many words are in a five minute speech?

A 5-minute speech should have between 500-600 words. Be careful not to exceed 650 words, or you'll talk too fast and, as a result, lose the audience!

How do I calculate the number of words in a talk?

To determine the number of words in a talk that will last m minutes:

Decide on the speaking rate r :

• Average: 130 wpm (words per minute);
• Slow: 100 wpm; and
• Fast: 160 wpm.

Multiply the speaking rate by the time you have in minutes:

The result in 2 is the number of words your talk should contain.

Number of words

.css-m482sy.css-m482sy{color:#2B3148;background-color:transparent;font-family:var(--calculator-ui-font-family),Verdana,sans-serif;font-size:20px;line-height:24px;overflow:visible;padding-top:0px;position:relative;}.css-m482sy.css-m482sy:after{content:'';-webkit-transform:scale(0);-moz-transform:scale(0);-ms-transform:scale(0);transform:scale(0);position:absolute;border:2px solid #EA9430;border-radius:2px;inset:-8px;z-index:1;}.css-m482sy .js-external-link-button.link-like,.css-m482sy .js-external-link-anchor{color:inherit;border-radius:1px;-webkit-text-decoration:underline;text-decoration:underline;}.css-m482sy .js-external-link-button.link-like:hover,.css-m482sy .js-external-link-anchor:hover,.css-m482sy .js-external-link-button.link-like:active,.css-m482sy .js-external-link-anchor:active{text-decoration-thickness:2px;text-shadow:1px 0 0;}.css-m482sy .js-external-link-button.link-like:focus-visible,.css-m482sy .js-external-link-anchor:focus-visible{outline:transparent 2px dotted;box-shadow:0 0 0 2px #6314E6;}.css-m482sy p,.css-m482sy div{margin:0;display:block;}.css-m482sy pre{margin:0;display:block;}.css-m482sy pre code{display:block;width:-webkit-fit-content;width:-moz-fit-content;width:fit-content;}.css-m482sy pre:not(:first-child){padding-top:8px;}.css-m482sy ul,.css-m482sy ol{display:block margin:0;padding-left:20px;}.css-m482sy ul li,.css-m482sy ol li{padding-top:8px;}.css-m482sy ul ul,.css-m482sy ol ul,.css-m482sy ul ol,.css-m482sy ol ol{padding-top:0;}.css-m482sy ul:not(:first-child),.css-m482sy ol:not(:first-child){padding-top:4px;} .css-1h42f0z{margin:auto;overflow:auto;overflow-wrap:break-word;word-break:break-word;}@font-face{font-family:'KaTeX_AMS';src:url(/katex-fonts/KaTeX_AMS-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_AMS-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_AMS-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Caligraphic';src:url(/katex-fonts/KaTeX_Caligraphic-Bold.woff2) format('woff2'),url(/katex-fonts/KaTeX_Caligraphic-Bold.woff) format('woff'),url(/katex-fonts/KaTeX_Caligraphic-Bold.ttf) format('truetype');font-weight:bold;font-style:normal;}@font-face{font-family:'KaTeX_Caligraphic';src:url(/katex-fonts/KaTeX_Caligraphic-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Caligraphic-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Caligraphic-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Fraktur';src:url(/katex-fonts/KaTeX_Fraktur-Bold.woff2) format('woff2'),url(/katex-fonts/KaTeX_Fraktur-Bold.woff) format('woff'),url(/katex-fonts/KaTeX_Fraktur-Bold.ttf) format('truetype');font-weight:bold;font-style:normal;}@font-face{font-family:'KaTeX_Fraktur';src:url(/katex-fonts/KaTeX_Fraktur-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Fraktur-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Fraktur-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Main';src:url(/katex-fonts/KaTeX_Main-Bold.woff2) format('woff2'),url(/katex-fonts/KaTeX_Main-Bold.woff) format('woff'),url(/katex-fonts/KaTeX_Main-Bold.ttf) format('truetype');font-weight:bold;font-style:normal;}@font-face{font-family:'KaTeX_Main';src:url(/katex-fonts/KaTeX_Main-BoldItalic.woff2) format('woff2'),url(/katex-fonts/KaTeX_Main-BoldItalic.woff) format('woff'),url(/katex-fonts/KaTeX_Main-BoldItalic.ttf) format('truetype');font-weight:bold;font-style:italic;}@font-face{font-family:'KaTeX_Main';src:url(/katex-fonts/KaTeX_Main-Italic.woff2) format('woff2'),url(/katex-fonts/KaTeX_Main-Italic.woff) format('woff'),url(/katex-fonts/KaTeX_Main-Italic.ttf) format('truetype');font-weight:normal;font-style:italic;}@font-face{font-family:'KaTeX_Main';src:url(/katex-fonts/KaTeX_Main-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Main-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Main-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Math';src:url(/katex-fonts/KaTeX_Math-BoldItalic.woff2) format('woff2'),url(/katex-fonts/KaTeX_Math-BoldItalic.woff) format('woff'),url(/katex-fonts/KaTeX_Math-BoldItalic.ttf) format('truetype');font-weight:bold;font-style:italic;}@font-face{font-family:'KaTeX_Math';src:url(/katex-fonts/KaTeX_Math-Italic.woff2) format('woff2'),url(/katex-fonts/KaTeX_Math-Italic.woff) format('woff'),url(/katex-fonts/KaTeX_Math-Italic.ttf) format('truetype');font-weight:normal;font-style:italic;}@font-face{font-family:'KaTeX_SansSerif';src:url(/katex-fonts/KaTeX_SansSerif-Bold.woff2) format('woff2'),url(/katex-fonts/KaTeX_SansSerif-Bold.woff) format('woff'),url(/katex-fonts/KaTeX_SansSerif-Bold.ttf) format('truetype');font-weight:bold;font-style:normal;}@font-face{font-family:'KaTeX_SansSerif';src:url(/katex-fonts/KaTeX_SansSerif-Italic.woff2) format('woff2'),url(/katex-fonts/KaTeX_SansSerif-Italic.woff) format('woff'),url(/katex-fonts/KaTeX_SansSerif-Italic.ttf) format('truetype');font-weight:normal;font-style:italic;}@font-face{font-family:'KaTeX_SansSerif';src:url(/katex-fonts/KaTeX_SansSerif-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_SansSerif-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_SansSerif-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Script';src:url(/katex-fonts/KaTeX_Script-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Script-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Script-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Size1';src:url(/katex-fonts/KaTeX_Size1-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Size1-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Size1-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Size2';src:url(/katex-fonts/KaTeX_Size2-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Size2-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Size2-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Size3';src:url(/katex-fonts/KaTeX_Size3-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Size3-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Size3-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Size4';src:url(/katex-fonts/KaTeX_Size4-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Size4-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Size4-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Typewriter';src:url(/katex-fonts/KaTeX_Typewriter-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Typewriter-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Typewriter-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}.css-1h42f0z .katex{font:normal 1.21em KaTeX_Main,Times New Roman,serif;line-height:1.2;text-indent:0;text-rendering:auto;}.css-1h42f0z .katex *{-ms-high-contrast-adjust:none!important;border-color:currentColor;}.css-1h42f0z .katex .katex-version::after{content:'0.13.13';}.css-1h42f0z .katex .katex-mathml{position:absolute;clip:rect(1px,1px,1px,1px);padding:0;border:0;height:1px;width:1px;overflow:hidden;}.css-1h42f0z .katex .katex-html>.newline{display:block;}.css-1h42f0z .katex .base{position:relative;display:inline-block;white-space:nowrap;width:-webkit-min-content;width:-moz-min-content;width:-webkit-min-content;width:-moz-min-content;width:min-content;}.css-1h42f0z .katex .strut{display:inline-block;}.css-1h42f0z .katex .textbf{font-weight:bold;}.css-1h42f0z .katex .textit{font-style:italic;}.css-1h42f0z .katex .textrm{font-family:KaTeX_Main;}.css-1h42f0z .katex .textsf{font-family:KaTeX_SansSerif;}.css-1h42f0z .katex .texttt{font-family:KaTeX_Typewriter;}.css-1h42f0z .katex .mathnormal{font-family:KaTeX_Math;font-style:italic;}.css-1h42f0z .katex .mathit{font-family:KaTeX_Main;font-style:italic;}.css-1h42f0z .katex .mathrm{font-style:normal;}.css-1h42f0z .katex .mathbf{font-family:KaTeX_Main;font-weight:bold;}.css-1h42f0z .katex .boldsymbol{font-family:KaTeX_Math;font-weight:bold;font-style:italic;}.css-1h42f0z .katex .amsrm{font-family:KaTeX_AMS;}.css-1h42f0z .katex .mathbb,.css-1h42f0z .katex .textbb{font-family:KaTeX_AMS;}.css-1h42f0z .katex .mathcal{font-family:KaTeX_Caligraphic;}.css-1h42f0z .katex .mathfrak,.css-1h42f0z .katex .textfrak{font-family:KaTeX_Fraktur;}.css-1h42f0z .katex .mathtt{font-family:KaTeX_Typewriter;}.css-1h42f0z .katex .mathscr,.css-1h42f0z .katex .textscr{font-family:KaTeX_Script;}.css-1h42f0z .katex .mathsf,.css-1h42f0z .katex .textsf{font-family:KaTeX_SansSerif;}.css-1h42f0z .katex .mathboldsf,.css-1h42f0z .katex .textboldsf{font-family:KaTeX_SansSerif;font-weight:bold;}.css-1h42f0z .katex .mathitsf,.css-1h42f0z .katex .textitsf{font-family:KaTeX_SansSerif;font-style:italic;}.css-1h42f0z .katex .mainrm{font-family:KaTeX_Main;font-style:normal;}.css-1h42f0z .katex .vlist-t{display:inline-table;table-layout:fixed;border-collapse:collapse;}.css-1h42f0z .katex .vlist-r{display:table-row;}.css-1h42f0z .katex .vlist{display:table-cell;vertical-align:bottom;position:relative;}.css-1h42f0z .katex .vlist>span{display:block;height:0;position:relative;}.css-1h42f0z .katex .vlist>span>span{display:inline-block;}.css-1h42f0z .katex .vlist>span>.pstrut{overflow:hidden;width:0;}.css-1h42f0z .katex .vlist-t2{margin-right:-2px;}.css-1h42f0z .katex .vlist-s{display:table-cell;vertical-align:bottom;font-size:1px;width:2px;min-width:2px;}.css-1h42f0z .katex .vbox{display:-webkit-inline-box;display:-webkit-inline-flex;display:-ms-inline-flexbox;display:inline-flex;-webkit-flex-direction:column;-ms-flex-direction:column;flex-direction:column;-webkit-align-items:baseline;-webkit-box-align:baseline;-ms-flex-align:baseline;align-items:baseline;}.css-1h42f0z .katex .hbox{display:-webkit-inline-box;display:-webkit-inline-flex;display:-ms-inline-flexbox;display:inline-flex;-webkit-flex-direction:row;-ms-flex-direction:row;flex-direction:row;width:100%;}.css-1h42f0z .katex .thinbox{display:-webkit-inline-box;display:-webkit-inline-flex;display:-ms-inline-flexbox;display:inline-flex;-webkit-flex-direction:row;-ms-flex-direction:row;flex-direction:row;width:0;max-width:0;}.css-1h42f0z .katex .msupsub{text-align:left;}.css-1h42f0z .katex .mfrac>span>span{text-align:center;}.css-1h42f0z .katex .mfrac .frac-line{display:inline-block;width:100%;border-bottom-style:solid;}.css-1h42f0z .katex .mfrac .frac-line,.css-1h42f0z .katex .overline .overline-line,.css-1h42f0z .katex .underline .underline-line,.css-1h42f0z .katex .hline,.css-1h42f0z .katex .hdashline,.css-1h42f0z .katex .rule{min-height:1px;}.css-1h42f0z .katex .mspace{display:inline-block;}.css-1h42f0z .katex .llap,.css-1h42f0z .katex .rlap,.css-1h42f0z .katex .clap{width:0;position:relative;}.css-1h42f0z .katex .llap>.inner,.css-1h42f0z .katex .rlap>.inner,.css-1h42f0z .katex .clap>.inner{position:absolute;}.css-1h42f0z .katex .llap>.fix,.css-1h42f0z .katex .rlap>.fix,.css-1h42f0z .katex .clap>.fix{display:inline-block;}.css-1h42f0z .katex .llap>.inner{right:0;}.css-1h42f0z .katex .rlap>.inner,.css-1h42f0z .katex .clap>.inner{left:0;}.css-1h42f0z .katex .clap>.inner>span{margin-left:-50%;margin-right:50%;}.css-1h42f0z .katex .rule{display:inline-block;border:solid 0;position:relative;}.css-1h42f0z .katex .overline .overline-line,.css-1h42f0z .katex .underline .underline-line,.css-1h42f0z .katex .hline{display:inline-block;width:100%;border-bottom-style:solid;}.css-1h42f0z .katex .hdashline{display:inline-block;width:100%;border-bottom-style:dashed;}.css-1h42f0z .katex .sqrt>.root{margin-left:0.27777778em;margin-right:-0.55555556em;}.css-1h42f0z .katex .sizing.reset-size1.size1,.css-1h42f0z .katex .fontsize-ensurer.reset-size1.size1{font-size:1em;}.css-1h42f0z .katex .sizing.reset-size1.size2,.css-1h42f0z .katex .fontsize-ensurer.reset-size1.size2{font-size:1.2em;}.css-1h42f0z .katex .sizing.reset-size1.size3,.css-1h42f0z .katex .fontsize-ensurer.reset-size1.size3{font-size:1.4em;}.css-1h42f0z .katex .sizing.reset-size1.size4,.css-1h42f0z .katex .fontsize-ensurer.reset-size1.size4{font-size:1.6em;}.css-1h42f0z .katex .sizing.reset-size1.size5,.css-1h42f0z .katex .fontsize-ensurer.reset-size1.size5{font-size:1.8em;}.css-1h42f0z .katex .sizing.reset-size1.size6,.css-1h42f0z .katex .fontsize-ensurer.reset-size1.size6{font-size:2em;}.css-1h42f0z .katex .sizing.reset-size1.size7,.css-1h42f0z .katex .fontsize-ensurer.reset-size1.size7{font-size:2.4em;}.css-1h42f0z .katex .sizing.reset-size1.size8,.css-1h42f0z .katex .fontsize-ensurer.reset-size1.size8{font-size:2.88em;}.css-1h42f0z .katex .sizing.reset-size1.size9,.css-1h42f0z .katex .fontsize-ensurer.reset-size1.size9{font-size:3.456em;}.css-1h42f0z .katex .sizing.reset-size1.size10,.css-1h42f0z .katex .fontsize-ensurer.reset-size1.size10{font-size:4.148em;}.css-1h42f0z .katex .sizing.reset-size1.size11,.css-1h42f0z .katex .fontsize-ensurer.reset-size1.size11{font-size:4.976em;}.css-1h42f0z .katex .sizing.reset-size2.size1,.css-1h42f0z .katex .fontsize-ensurer.reset-size2.size1{font-size:0.83333333em;}.css-1h42f0z .katex .sizing.reset-size2.size2,.css-1h42f0z .katex .fontsize-ensurer.reset-size2.size2{font-size:1em;}.css-1h42f0z .katex .sizing.reset-size2.size3,.css-1h42f0z .katex .fontsize-ensurer.reset-size2.size3{font-size:1.16666667em;}.css-1h42f0z .katex .sizing.reset-size2.size4,.css-1h42f0z .katex .fontsize-ensurer.reset-size2.size4{font-size:1.33333333em;}.css-1h42f0z .katex .sizing.reset-size2.size5,.css-1h42f0z .katex .fontsize-ensurer.reset-size2.size5{font-size:1.5em;}.css-1h42f0z .katex .sizing.reset-size2.size6,.css-1h42f0z .katex .fontsize-ensurer.reset-size2.size6{font-size:1.66666667em;}.css-1h42f0z .katex .sizing.reset-size2.size7,.css-1h42f0z .katex .fontsize-ensurer.reset-size2.size7{font-size:2em;}.css-1h42f0z .katex .sizing.reset-size2.size8,.css-1h42f0z .katex .fontsize-ensurer.reset-size2.size8{font-size:2.4em;}.css-1h42f0z .katex .sizing.reset-size2.size9,.css-1h42f0z .katex .fontsize-ensurer.reset-size2.size9{font-size:2.88em;}.css-1h42f0z .katex .sizing.reset-size2.size10,.css-1h42f0z .katex .fontsize-ensurer.reset-size2.size10{font-size:3.45666667em;}.css-1h42f0z .katex .sizing.reset-size2.size11,.css-1h42f0z .katex .fontsize-ensurer.reset-size2.size11{font-size:4.14666667em;}.css-1h42f0z .katex .sizing.reset-size3.size1,.css-1h42f0z .katex .fontsize-ensurer.reset-size3.size1{font-size:0.71428571em;}.css-1h42f0z .katex .sizing.reset-size3.size2,.css-1h42f0z .katex .fontsize-ensurer.reset-size3.size2{font-size:0.85714286em;}.css-1h42f0z .katex .sizing.reset-size3.size3,.css-1h42f0z .katex .fontsize-ensurer.reset-size3.size3{font-size:1em;}.css-1h42f0z .katex .sizing.reset-size3.size4,.css-1h42f0z .katex .fontsize-ensurer.reset-size3.size4{font-size:1.14285714em;}.css-1h42f0z .katex .sizing.reset-size3.size5,.css-1h42f0z .katex .fontsize-ensurer.reset-size3.size5{font-size:1.28571429em;}.css-1h42f0z .katex .sizing.reset-size3.size6,.css-1h42f0z .katex .fontsize-ensurer.reset-size3.size6{font-size:1.42857143em;}.css-1h42f0z .katex .sizing.reset-size3.size7,.css-1h42f0z .katex .fontsize-ensurer.reset-size3.size7{font-size:1.71428571em;}.css-1h42f0z .katex .sizing.reset-size3.size8,.css-1h42f0z .katex .fontsize-ensurer.reset-size3.size8{font-size:2.05714286em;}.css-1h42f0z .katex .sizing.reset-size3.size9,.css-1h42f0z .katex .fontsize-ensurer.reset-size3.size9{font-size:2.46857143em;}.css-1h42f0z .katex .sizing.reset-size3.size10,.css-1h42f0z .katex .fontsize-ensurer.reset-size3.size10{font-size:2.96285714em;}.css-1h42f0z .katex .sizing.reset-size3.size11,.css-1h42f0z .katex .fontsize-ensurer.reset-size3.size11{font-size:3.55428571em;}.css-1h42f0z .katex .sizing.reset-size4.size1,.css-1h42f0z .katex .fontsize-ensurer.reset-size4.size1{font-size:0.625em;}.css-1h42f0z .katex .sizing.reset-size4.size2,.css-1h42f0z .katex .fontsize-ensurer.reset-size4.size2{font-size:0.75em;}.css-1h42f0z .katex .sizing.reset-size4.size3,.css-1h42f0z .katex .fontsize-ensurer.reset-size4.size3{font-size:0.875em;}.css-1h42f0z .katex .sizing.reset-size4.size4,.css-1h42f0z .katex .fontsize-ensurer.reset-size4.size4{font-size:1em;}.css-1h42f0z .katex .sizing.reset-size4.size5,.css-1h42f0z .katex .fontsize-ensurer.reset-size4.size5{font-size:1.125em;}.css-1h42f0z .katex .sizing.reset-size4.size6,.css-1h42f0z .katex .fontsize-ensurer.reset-size4.size6{font-size:1.25em;}.css-1h42f0z .katex .sizing.reset-size4.size7,.css-1h42f0z .katex .fontsize-ensurer.reset-size4.size7{font-size:1.5em;}.css-1h42f0z .katex .sizing.reset-size4.size8,.css-1h42f0z .katex .fontsize-ensurer.reset-size4.size8{font-size:1.8em;}.css-1h42f0z .katex .sizing.reset-size4.size9,.css-1h42f0z .katex .fontsize-ensurer.reset-size4.size9{font-size:2.16em;}.css-1h42f0z .katex .sizing.reset-size4.size10,.css-1h42f0z .katex .fontsize-ensurer.reset-size4.size10{font-size:2.5925em;}.css-1h42f0z .katex .sizing.reset-size4.size11,.css-1h42f0z .katex .fontsize-ensurer.reset-size4.size11{font-size:3.11em;}.css-1h42f0z .katex .sizing.reset-size5.size1,.css-1h42f0z .katex .fontsize-ensurer.reset-size5.size1{font-size:0.55555556em;}.css-1h42f0z .katex .sizing.reset-size5.size2,.css-1h42f0z .katex .fontsize-ensurer.reset-size5.size2{font-size:0.66666667em;}.css-1h42f0z .katex .sizing.reset-size5.size3,.css-1h42f0z .katex .fontsize-ensurer.reset-size5.size3{font-size:0.77777778em;}.css-1h42f0z .katex .sizing.reset-size5.size4,.css-1h42f0z .katex .fontsize-ensurer.reset-size5.size4{font-size:0.88888889em;}.css-1h42f0z .katex .sizing.reset-size5.size5,.css-1h42f0z .katex .fontsize-ensurer.reset-size5.size5{font-size:1em;}.css-1h42f0z .katex .sizing.reset-size5.size6,.css-1h42f0z .katex .fontsize-ensurer.reset-size5.size6{font-size:1.11111111em;}.css-1h42f0z .katex .sizing.reset-size5.size7,.css-1h42f0z .katex .fontsize-ensurer.reset-size5.size7{font-size:1.33333333em;}.css-1h42f0z .katex .sizing.reset-size5.size8,.css-1h42f0z .katex .fontsize-ensurer.reset-size5.size8{font-size:1.6em;}.css-1h42f0z .katex .sizing.reset-size5.size9,.css-1h42f0z .katex .fontsize-ensurer.reset-size5.size9{font-size:1.92em;}.css-1h42f0z .katex .sizing.reset-size5.size10,.css-1h42f0z .katex .fontsize-ensurer.reset-size5.size10{font-size:2.30444444em;}.css-1h42f0z .katex .sizing.reset-size5.size11,.css-1h42f0z .katex .fontsize-ensurer.reset-size5.size11{font-size:2.76444444em;}.css-1h42f0z .katex .sizing.reset-size6.size1,.css-1h42f0z .katex .fontsize-ensurer.reset-size6.size1{font-size:0.5em;}.css-1h42f0z .katex .sizing.reset-size6.size2,.css-1h42f0z .katex .fontsize-ensurer.reset-size6.size2{font-size:0.6em;}.css-1h42f0z .katex .sizing.reset-size6.size3,.css-1h42f0z .katex .fontsize-ensurer.reset-size6.size3{font-size:0.7em;}.css-1h42f0z .katex .sizing.reset-size6.size4,.css-1h42f0z .katex .fontsize-ensurer.reset-size6.size4{font-size:0.8em;}.css-1h42f0z .katex .sizing.reset-size6.size5,.css-1h42f0z .katex .fontsize-ensurer.reset-size6.size5{font-size:0.9em;}.css-1h42f0z .katex .sizing.reset-size6.size6,.css-1h42f0z .katex .fontsize-ensurer.reset-size6.size6{font-size:1em;}.css-1h42f0z .katex .sizing.reset-size6.size7,.css-1h42f0z .katex .fontsize-ensurer.reset-size6.size7{font-size:1.2em;}.css-1h42f0z .katex .sizing.reset-size6.size8,.css-1h42f0z .katex .fontsize-ensurer.reset-size6.size8{font-size:1.44em;}.css-1h42f0z .katex .sizing.reset-size6.size9,.css-1h42f0z .katex .fontsize-ensurer.reset-size6.size9{font-size:1.728em;}.css-1h42f0z .katex .sizing.reset-size6.size10,.css-1h42f0z .katex .fontsize-ensurer.reset-size6.size10{font-size:2.074em;}.css-1h42f0z .katex .sizing.reset-size6.size11,.css-1h42f0z .katex .fontsize-ensurer.reset-size6.size11{font-size:2.488em;}.css-1h42f0z .katex .sizing.reset-size7.size1,.css-1h42f0z .katex .fontsize-ensurer.reset-size7.size1{font-size:0.41666667em;}.css-1h42f0z .katex .sizing.reset-size7.size2,.css-1h42f0z .katex .fontsize-ensurer.reset-size7.size2{font-size:0.5em;}.css-1h42f0z .katex .sizing.reset-size7.size3,.css-1h42f0z .katex .fontsize-ensurer.reset-size7.size3{font-size:0.58333333em;}.css-1h42f0z .katex .sizing.reset-size7.size4,.css-1h42f0z .katex .fontsize-ensurer.reset-size7.size4{font-size:0.66666667em;}.css-1h42f0z .katex .sizing.reset-size7.size5,.css-1h42f0z .katex .fontsize-ensurer.reset-size7.size5{font-size:0.75em;}.css-1h42f0z .katex .sizing.reset-size7.size6,.css-1h42f0z .katex .fontsize-ensurer.reset-size7.size6{font-size:0.83333333em;}.css-1h42f0z .katex .sizing.reset-size7.size7,.css-1h42f0z .katex .fontsize-ensurer.reset-size7.size7{font-size:1em;}.css-1h42f0z .katex .sizing.reset-size7.size8,.css-1h42f0z .katex .fontsize-ensurer.reset-size7.size8{font-size:1.2em;}.css-1h42f0z .katex .sizing.reset-size7.size9,.css-1h42f0z .katex .fontsize-ensurer.reset-size7.size9{font-size:1.44em;}.css-1h42f0z .katex .sizing.reset-size7.size10,.css-1h42f0z .katex .fontsize-ensurer.reset-size7.size10{font-size:1.72833333em;}.css-1h42f0z .katex .sizing.reset-size7.size11,.css-1h42f0z .katex .fontsize-ensurer.reset-size7.size11{font-size:2.07333333em;}.css-1h42f0z .katex .sizing.reset-size8.size1,.css-1h42f0z .katex .fontsize-ensurer.reset-size8.size1{font-size:0.34722222em;}.css-1h42f0z .katex .sizing.reset-size8.size2,.css-1h42f0z .katex .fontsize-ensurer.reset-size8.size2{font-size:0.41666667em;}.css-1h42f0z .katex .sizing.reset-size8.size3,.css-1h42f0z .katex .fontsize-ensurer.reset-size8.size3{font-size:0.48611111em;}.css-1h42f0z .katex .sizing.reset-size8.size4,.css-1h42f0z .katex .fontsize-ensurer.reset-size8.size4{font-size:0.55555556em;}.css-1h42f0z .katex .sizing.reset-size8.size5,.css-1h42f0z .katex .fontsize-ensurer.reset-size8.size5{font-size:0.625em;}.css-1h42f0z .katex .sizing.reset-size8.size6,.css-1h42f0z .katex .fontsize-ensurer.reset-size8.size6{font-size:0.69444444em;}.css-1h42f0z .katex .sizing.reset-size8.size7,.css-1h42f0z .katex .fontsize-ensurer.reset-size8.size7{font-size:0.83333333em;}.css-1h42f0z .katex .sizing.reset-size8.size8,.css-1h42f0z .katex .fontsize-ensurer.reset-size8.size8{font-size:1em;}.css-1h42f0z .katex .sizing.reset-size8.size9,.css-1h42f0z .katex .fontsize-ensurer.reset-size8.size9{font-size:1.2em;}.css-1h42f0z .katex .sizing.reset-size8.size10,.css-1h42f0z .katex .fontsize-ensurer.reset-size8.size10{font-size:1.44027778em;}.css-1h42f0z .katex .sizing.reset-size8.size11,.css-1h42f0z .katex .fontsize-ensurer.reset-size8.size11{font-size:1.72777778em;}.css-1h42f0z .katex .sizing.reset-size9.size1,.css-1h42f0z .katex .fontsize-ensurer.reset-size9.size1{font-size:0.28935185em;}.css-1h42f0z .katex .sizing.reset-size9.size2,.css-1h42f0z .katex .fontsize-ensurer.reset-size9.size2{font-size:0.34722222em;}.css-1h42f0z .katex .sizing.reset-size9.size3,.css-1h42f0z .katex .fontsize-ensurer.reset-size9.size3{font-size:0.40509259em;}.css-1h42f0z .katex .sizing.reset-size9.size4,.css-1h42f0z .katex .fontsize-ensurer.reset-size9.size4{font-size:0.46296296em;}.css-1h42f0z .katex .sizing.reset-size9.size5,.css-1h42f0z .katex .fontsize-ensurer.reset-size9.size5{font-size:0.52083333em;}.css-1h42f0z .katex .sizing.reset-size9.size6,.css-1h42f0z .katex .fontsize-ensurer.reset-size9.size6{font-size:0.5787037em;}.css-1h42f0z .katex .sizing.reset-size9.size7,.css-1h42f0z .katex .fontsize-ensurer.reset-size9.size7{font-size:0.69444444em;}.css-1h42f0z .katex .sizing.reset-size9.size8,.css-1h42f0z .katex .fontsize-ensurer.reset-size9.size8{font-size:0.83333333em;}.css-1h42f0z .katex .sizing.reset-size9.size9,.css-1h42f0z .katex .fontsize-ensurer.reset-size9.size9{font-size:1em;}.css-1h42f0z .katex .sizing.reset-size9.size10,.css-1h42f0z .katex .fontsize-ensurer.reset-size9.size10{font-size:1.20023148em;}.css-1h42f0z .katex .sizing.reset-size9.size11,.css-1h42f0z .katex .fontsize-ensurer.reset-size9.size11{font-size:1.43981481em;}.css-1h42f0z .katex .sizing.reset-size10.size1,.css-1h42f0z .katex .fontsize-ensurer.reset-size10.size1{font-size:0.24108004em;}.css-1h42f0z .katex .sizing.reset-size10.size2,.css-1h42f0z .katex .fontsize-ensurer.reset-size10.size2{font-size:0.28929605em;}.css-1h42f0z .katex .sizing.reset-size10.size3,.css-1h42f0z .katex .fontsize-ensurer.reset-size10.size3{font-size:0.33751205em;}.css-1h42f0z .katex .sizing.reset-size10.size4,.css-1h42f0z .katex .fontsize-ensurer.reset-size10.size4{font-size:0.38572806em;}.css-1h42f0z .katex .sizing.reset-size10.size5,.css-1h42f0z .katex .fontsize-ensurer.reset-size10.size5{font-size:0.43394407em;}.css-1h42f0z .katex .sizing.reset-size10.size6,.css-1h42f0z .katex .fontsize-ensurer.reset-size10.size6{font-size:0.48216008em;}.css-1h42f0z .katex .sizing.reset-size10.size7,.css-1h42f0z .katex .fontsize-ensurer.reset-size10.size7{font-size:0.57859209em;}.css-1h42f0z .katex .sizing.reset-size10.size8,.css-1h42f0z .katex .fontsize-ensurer.reset-size10.size8{font-size:0.69431051em;}.css-1h42f0z .katex .sizing.reset-size10.size9,.css-1h42f0z .katex .fontsize-ensurer.reset-size10.size9{font-size:0.83317261em;}.css-1h42f0z .katex .sizing.reset-size10.size10,.css-1h42f0z .katex .fontsize-ensurer.reset-size10.size10{font-size:1em;}.css-1h42f0z .katex .sizing.reset-size10.size11,.css-1h42f0z .katex .fontsize-ensurer.reset-size10.size11{font-size:1.19961427em;}.css-1h42f0z .katex .sizing.reset-size11.size1,.css-1h42f0z .katex .fontsize-ensurer.reset-size11.size1{font-size:0.20096463em;}.css-1h42f0z .katex .sizing.reset-size11.size2,.css-1h42f0z .katex .fontsize-ensurer.reset-size11.size2{font-size:0.24115756em;}.css-1h42f0z .katex .sizing.reset-size11.size3,.css-1h42f0z .katex .fontsize-ensurer.reset-size11.size3{font-size:0.28135048em;}.css-1h42f0z .katex .sizing.reset-size11.size4,.css-1h42f0z .katex .fontsize-ensurer.reset-size11.size4{font-size:0.32154341em;}.css-1h42f0z .katex .sizing.reset-size11.size5,.css-1h42f0z .katex .fontsize-ensurer.reset-size11.size5{font-size:0.36173633em;}.css-1h42f0z .katex .sizing.reset-size11.size6,.css-1h42f0z .katex .fontsize-ensurer.reset-size11.size6{font-size:0.40192926em;}.css-1h42f0z .katex .sizing.reset-size11.size7,.css-1h42f0z .katex .fontsize-ensurer.reset-size11.size7{font-size:0.48231511em;}.css-1h42f0z .katex .sizing.reset-size11.size8,.css-1h42f0z .katex .fontsize-ensurer.reset-size11.size8{font-size:0.57877814em;}.css-1h42f0z .katex .sizing.reset-size11.size9,.css-1h42f0z .katex .fontsize-ensurer.reset-size11.size9{font-size:0.69453376em;}.css-1h42f0z .katex .sizing.reset-size11.size10,.css-1h42f0z .katex .fontsize-ensurer.reset-size11.size10{font-size:0.83360129em;}.css-1h42f0z .katex .sizing.reset-size11.size11,.css-1h42f0z .katex .fontsize-ensurer.reset-size11.size11{font-size:1em;}.css-1h42f0z .katex .delimsizing.size1{font-family:KaTeX_Size1;}.css-1h42f0z .katex .delimsizing.size2{font-family:KaTeX_Size2;}.css-1h42f0z .katex .delimsizing.size3{font-family:KaTeX_Size3;}.css-1h42f0z .katex .delimsizing.size4{font-family:KaTeX_Size4;}.css-1h42f0z .katex .delimsizing.mult .delim-size1>span{font-family:KaTeX_Size1;}.css-1h42f0z .katex .delimsizing.mult .delim-size4>span{font-family:KaTeX_Size4;}.css-1h42f0z .katex .nulldelimiter{display:inline-block;width:0.12em;}.css-1h42f0z .katex .delimcenter{position:relative;}.css-1h42f0z .katex .op-symbol{position:relative;}.css-1h42f0z .katex .op-symbol.small-op{font-family:KaTeX_Size1;}.css-1h42f0z .katex .op-symbol.large-op{font-family:KaTeX_Size2;}.css-1h42f0z .katex .op-limits>.vlist-t{text-align:center;}.css-1h42f0z .katex .accent>.vlist-t{text-align:center;}.css-1h42f0z .katex .accent .accent-body{position:relative;}.css-1h42f0z .katex .accent .accent-body:not(.accent-full){width:0;}.css-1h42f0z .katex .overlay{display:block;}.css-1h42f0z .katex .mtable .vertical-separator{display:inline-block;min-width:1px;}.css-1h42f0z .katex .mtable .arraycolsep{display:inline-block;}.css-1h42f0z .katex .mtable .col-align-c>.vlist-t{text-align:center;}.css-1h42f0z .katex .mtable .col-align-l>.vlist-t{text-align:left;}.css-1h42f0z .katex .mtable .col-align-r>.vlist-t{text-align:right;}.css-1h42f0z .katex .svg-align{text-align:left;}.css-1h42f0z .katex svg{display:block;position:absolute;width:100%;height:inherit;fill:currentColor;stroke:currentColor;fill-rule:nonzero;fill-opacity:1;stroke-width:1;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;}.css-1h42f0z .katex svg path{stroke:none;}.css-1h42f0z .katex img{border-style:none;min-width:0;min-height:0;max-width:none;max-height:none;}.css-1h42f0z .katex .stretchy{width:100%;display:block;position:relative;overflow:hidden;}.css-1h42f0z .katex .stretchy::before,.css-1h42f0z .katex .stretchy::after{content:'';}.css-1h42f0z .katex .hide-tail{width:100%;position:relative;overflow:hidden;}.css-1h42f0z .katex .halfarrow-left{position:absolute;left:0;width:50.2%;overflow:hidden;}.css-1h42f0z .katex .halfarrow-right{position:absolute;right:0;width:50.2%;overflow:hidden;}.css-1h42f0z .katex .brace-left{position:absolute;left:0;width:25.1%;overflow:hidden;}.css-1h42f0z .katex .brace-center{position:absolute;left:25%;width:50%;overflow:hidden;}.css-1h42f0z .katex .brace-right{position:absolute;right:0;width:25.1%;overflow:hidden;}.css-1h42f0z .katex .x-arrow-pad{padding:0 0.5em;}.css-1h42f0z .katex .cd-arrow-pad{padding:0 0.55556em 0 0.27778em;}.css-1h42f0z .katex .x-arrow,.css-1h42f0z .katex .mover,.css-1h42f0z .katex .munder{text-align:center;}.css-1h42f0z .katex .boxpad{padding:0 0.3em 0 0.3em;}.css-1h42f0z .katex .fbox,.css-1h42f0z .katex .fcolorbox{box-sizing:border-box;border:0.04em solid;}.css-1h42f0z .katex .cancel-pad{padding:0 0.2em 0 0.2em;}.css-1h42f0z .katex .cancel-lap{margin-left:-0.2em;margin-right:-0.2em;}.css-1h42f0z .katex .sout{border-bottom-style:solid;border-bottom-width:0.08em;}.css-1h42f0z .katex .angl{box-sizing:border-box;border-top:0.049em solid;border-right:0.049em solid;margin-right:0.03889em;}.css-1h42f0z .katex .anglpad{padding:0 0.03889em 0 0.03889em;}.css-1h42f0z .katex .eqn-num::before{counter-increment:katexEqnNo;content:'(' counter(katexEqnNo) ')';}.css-1h42f0z .katex .mml-eqn-num::before{counter-increment:mmlEqnNo;content:'(' counter(mmlEqnNo) ')';}.css-1h42f0z .katex .mtr-glue{width:50%;}.css-1h42f0z .katex .cd-vert-arrow{display:inline-block;position:relative;}.css-1h42f0z .katex .cd-label-left{display:inline-block;position:absolute;right:calc(50% + 0.3em);text-align:left;}.css-1h42f0z .katex .cd-label-right{display:inline-block;position:absolute;left:calc(50% + 0.3em);text-align:right;}.css-1h42f0z .katex-display{display:block;margin:1em 0;text-align:center;}.css-1h42f0z .katex-display>.katex{display:block;white-space:nowrap;}.css-1h42f0z .katex-display>.katex>.katex-html{display:block;position:relative;}.css-1h42f0z .katex-display>.katex>.katex-html>.tag{position:absolute;right:0;}.css-1h42f0z .katex-display.leqno>.katex>.katex-html>.tag{left:0;right:auto;}.css-1h42f0z .katex-display.fleqn>.katex{text-align:left;padding-left:2em;}.css-1h42f0z body{counter-reset:katexEqnNo mmlEqnNo;}.css-1h42f0z .link-like{color:#007bff;-webkit-text-decoration:underline;text-decoration:underline;}.css-1h42f0z .text-overline{-webkit-text-decoration:overline;text-decoration:overline;}.css-1h42f0z code,.css-1h42f0z kbd,.css-1h42f0z pre,.css-1h42f0z samp{font-family:monospace;}.css-1h42f0z code{padding:2px 4px;color:#444;background:#ddd;border-radius:4px;}.css-1h42f0z figcaption,.css-1h42f0z caption{text-align:center;}.css-1h42f0z figcaption{font-size:12px;font-style:italic;overflow:hidden;}.css-1h42f0z h3{font-size:1.75rem;}.css-1h42f0z h4{font-size:1.5rem;}.css-1h42f0z .mathBlock{font-size:24px;-webkit-padding-start:4px;padding-inline-start:4px;}.css-1h42f0z .mathBlock .katex{font-size:24px;text-align:left;}.css-1h42f0z .math-inline{background-color:#f0f0f0;display:inline-block;font-size:inherit;padding:0 3px;}.css-1h42f0z .videoBlock,.css-1h42f0z .imageBlock{margin-bottom:16px;}.css-1h42f0z .imageBlock__image-align--left,.css-1h42f0z .videoBlock__video-align--left{float:left;}.css-1h42f0z .imageBlock__image-align--right,.css-1h42f0z .videoBlock__video-align--right{float:right;}.css-1h42f0z .imageBlock__image-align--center,.css-1h42f0z .videoBlock__video-align--center{display:block;margin-left:auto;margin-right:auto;clear:both;}.css-1h42f0z .imageBlock__image-align--none,.css-1h42f0z .videoBlock__video-align--none{clear:both;margin-left:0;margin-right:0;}.css-1h42f0z .videoBlock__video--wrapper{position:relative;padding-bottom:56.25%;height:0;}.css-1h42f0z .videoBlock__video--wrapper iframe{position:absolute;top:0;left:0;width:100%;height:100%;}.css-1h42f0z .videoBlock__caption{text-align:left;}.css-1h42f0z table{width:-webkit-max-content;width:-moz-max-content;width:max-content;}.css-1h42f0z .tableBlock{max-width:100%;margin-bottom:1rem;overflow-y:scroll;}.css-1h42f0z .tableBlock thead,.css-1h42f0z .tableBlock thead th{border-bottom:1px solid #333!important;}.css-1h42f0z .tableBlock th,.css-1h42f0z .tableBlock td{padding:10px;text-align:left;}.css-1h42f0z .tableBlock th{font-weight:bold!important;}.css-1h42f0z .tableBlock caption{caption-side:bottom;color:#555;font-size:12px;font-style:italic;text-align:center;}.css-1h42f0z .tableBlock caption>p{margin:0;}.css-1h42f0z .tableBlock th>p,.css-1h42f0z .tableBlock td>p{margin:0;}.css-1h42f0z .tableBlock [data-background-color='aliceblue']{background-color:#f0f8ff;color:#000;}.css-1h42f0z .tableBlock [data-background-color='black']{background-color:#000;color:#fff;}.css-1h42f0z .tableBlock [data-background-color='chocolate']{background-color:#d2691e;color:#fff;}.css-1h42f0z .tableBlock [data-background-color='cornflowerblue']{background-color:#6495ed;color:#fff;}.css-1h42f0z .tableBlock [data-background-color='crimson']{background-color:#dc143c;color:#fff;}.css-1h42f0z .tableBlock [data-background-color='darkblue']{background-color:#00008b;color:#fff;}.css-1h42f0z .tableBlock [data-background-color='darkseagreen']{background-color:#8fbc8f;color:#000;}.css-1h42f0z .tableBlock [data-background-color='deepskyblue']{background-color:#00bfff;color:#000;}.css-1h42f0z .tableBlock [data-background-color='gainsboro']{background-color:#dcdcdc;color:#000;}.css-1h42f0z .tableBlock [data-background-color='grey']{background-color:#808080;color:#fff;}.css-1h42f0z .tableBlock [data-background-color='lemonchiffon']{background-color:#fffacd;color:#000;}.css-1h42f0z .tableBlock [data-background-color='lightpink']{background-color:#ffb6c1;color:#000;}.css-1h42f0z .tableBlock [data-background-color='lightsalmon']{background-color:#ffa07a;color:#000;}.css-1h42f0z .tableBlock [data-background-color='lightskyblue']{background-color:#87cefa;color:#000;}.css-1h42f0z .tableBlock [data-background-color='mediumblue']{background-color:#0000cd;color:#fff;}.css-1h42f0z .tableBlock [data-background-color='omnigrey']{background-color:#f0f0f0;color:#000;}.css-1h42f0z .tableBlock [data-background-color='white']{background-color:#fff;color:#000;}.css-1h42f0z .tableBlock [data-text-align='center']{text-align:center;}.css-1h42f0z .tableBlock [data-text-align='left']{text-align:left;}.css-1h42f0z .tableBlock [data-text-align='right']{text-align:right;}.css-1h42f0z .tableBlock [data-vertical-align='bottom']{vertical-align:bottom;}.css-1h42f0z .tableBlock [data-vertical-align='middle']{vertical-align:middle;}.css-1h42f0z .tableBlock [data-vertical-align='top']{vertical-align:top;}.css-1h42f0z .tableBlock__font-size--xxsmall{font-size:10px;}.css-1h42f0z .tableBlock__font-size--xsmall{font-size:12px;}.css-1h42f0z .tableBlock__font-size--small{font-size:14px;}.css-1h42f0z .tableBlock__font-size--large{font-size:18px;}.css-1h42f0z .tableBlock__border--some tbody tr:not(:last-child){border-bottom:1px solid #e2e5e7;}.css-1h42f0z .tableBlock__border--bordered td,.css-1h42f0z .tableBlock__border--bordered th{border:1px solid #e2e5e7;}.css-1h42f0z .tableBlock__border--borderless tbody+tbody,.css-1h42f0z .tableBlock__border--borderless td,.css-1h42f0z .tableBlock__border--borderless th,.css-1h42f0z .tableBlock__border--borderless tr,.css-1h42f0z .tableBlock__border--borderless thead,.css-1h42f0z .tableBlock__border--borderless thead th{border:0!important;}.css-1h42f0z .tableBlock:not(.tableBlock__table-striped) tbody tr{background-color:unset!important;}.css-1h42f0z .tableBlock__table-striped tbody tr:nth-of-type(odd){background-color:#f9fafc!important;}.css-1h42f0z .tableBlock__table-compactl th,.css-1h42f0z .tableBlock__table-compact td{padding:3px!important;}.css-1h42f0z .tableBlock__full-size{width:100%;}.css-1h42f0z .textBlock{margin-bottom:16px;}.css-1h42f0z .textBlock__text-formatting--finePrint{font-size:12px;}.css-1h42f0z .textBlock__text-infoBox{padding:0.75rem 1.25rem;margin-bottom:1rem;border:1px solid transparent;border-radius:0.25rem;}.css-1h42f0z .textBlock__text-infoBox p{margin:0;}.css-1h42f0z .textBlock__text-infoBox--primary{background-color:#cce5ff;border-color:#b8daff;color:#004085;}.css-1h42f0z .textBlock__text-infoBox--secondary{background-color:#e2e3e5;border-color:#d6d8db;color:#383d41;}.css-1h42f0z .textBlock__text-infoBox--success{background-color:#d4edda;border-color:#c3e6cb;color:#155724;}.css-1h42f0z .textBlock__text-infoBox--danger{background-color:#f8d7da;border-color:#f5c6cb;color:#721c24;}.css-1h42f0z .textBlock__text-infoBox--warning{background-color:#fff3cd;border-color:#ffeeba;color:#856404;}.css-1h42f0z .textBlock__text-infoBox--info{background-color:#d1ecf1;border-color:#bee5eb;color:#0c5460;}.css-1h42f0z .textBlock__text-infoBox--dark{background-color:#d6d8d9;border-color:#c6c8ca;color:#1b1e21;}.css-1h42f0z.css-1h42f0z{color:#2B3148;background-color:transparent;font-family:var(--calculator-ui-font-family),Verdana,sans-serif;font-size:20px;line-height:24px;overflow:visible;padding-top:0px;position:relative;}.css-1h42f0z.css-1h42f0z:after{content:'';-webkit-transform:scale(0);-moz-transform:scale(0);-ms-transform:scale(0);transform:scale(0);position:absolute;border:2px solid #EA9430;border-radius:2px;inset:-8px;z-index:1;}.css-1h42f0z .js-external-link-button.link-like,.css-1h42f0z .js-external-link-anchor{color:inherit;border-radius:1px;-webkit-text-decoration:underline;text-decoration:underline;}.css-1h42f0z .js-external-link-button.link-like:hover,.css-1h42f0z .js-external-link-anchor:hover,.css-1h42f0z .js-external-link-button.link-like:active,.css-1h42f0z .js-external-link-anchor:active{text-decoration-thickness:2px;text-shadow:1px 0 0;}.css-1h42f0z .js-external-link-button.link-like:focus-visible,.css-1h42f0z .js-external-link-anchor:focus-visible{outline:transparent 2px dotted;box-shadow:0 0 0 2px #6314E6;}.css-1h42f0z p,.css-1h42f0z div{margin:0;display:block;}.css-1h42f0z pre{margin:0;display:block;}.css-1h42f0z pre code{display:block;width:-webkit-fit-content;width:-moz-fit-content;width:fit-content;}.css-1h42f0z pre:not(:first-child){padding-top:8px;}.css-1h42f0z ul,.css-1h42f0z ol{display:block margin:0;padding-left:20px;}.css-1h42f0z ul li,.css-1h42f0z ol li{padding-top:8px;}.css-1h42f0z ul ul,.css-1h42f0z ol ul,.css-1h42f0z ul ol,.css-1h42f0z ol ol{padding-top:0;}.css-1h42f0z ul:not(:first-child),.css-1h42f0z ol:not(:first-child){padding-top:4px;} Speaking

Speaking speed

Speaking time

Reading speed

Reading time