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Quick Calendar Month Creation Quiz

Published on Sunday, May 25, 2014 in , , , , ,

Learn to perform the Quick Calendar Month Creation feat here.

Quiz will appear below:



2-Digit Year Key

Full Key 1700-2099

Month and Day Association

Calendar Quiz
Note: The Calendar Quiz opens full screen in a new window. Instructions on how to use it can be found in the Calendar Tab of the Quick Calendar Month Creation tutorial, in the Practice Creating Calendars section.

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Quick Calendar Month Creation

Published on Sunday, May 25, 2014 in , , , ,

Introduction

I have a fascination with the day of the week for any date feat. I've posted a detailed tutorial for it here in the Mental Gym, and I've also released a commercially available version called Day One.

My fascination is two-fold: How simple can it be made? How powerful can it be made? After researching several methods, many obscure even to those who enjoy calendar calculating, I've managed to merge two astounding versions that bring together the best combination of power and simplicity I've ever run across.


Effect

You start by bringing out a business card with this calendar design:



You then ask someone for the year, and then the month, of their birthday. With no further questions, you're instantly able to fill in the calendar for that month and year. For example, if the person said they were born in 1979 in September, you could quickly fill in the calendar from memory like this:



They can verify your calendar with any outside source, such as Wolfram|Alpha, and see that the calendar you created from memory is 100% accurate!

In addition, you'll even learn to handle Julian calendar dates all the way back to 45 B.C. almost as easily as our modern Gregorian calendar dates!

Even if you've become frustrated trying to master other day of the day of the week for any date formulas, I think you'll like the simplicity and flexibility of this version. You can start learning the method in the next tab!

Year Key

With this method, you ask for the year first, and you're going to calculate a date from it, ranging from 0 to 6.

In many methods, calculating a number from the year is the biggest hurdle, both because of the number of calculations involved, and the fact that many calculations involve increasing the amount represented by the year. In this method, however, the steps are few, and the method always involves making the year number smaller.

When given a year, the first thing you need to do is to break it up into the last 2 digits, which will be referred to as the year from here on out, and the remaining digits, which will be referred to as the century. For exam, if you're given the year 1975, you would think of 75 as the year and 19 as the century.

First, I'll explain the calculations to perform on the 2-digit year, and then I'll explain how to handle the century.

Step 1 - Partition: Split the 2-digit year into a sum, consisting of the largest multiple of 4 equal to or less than the given year, plus a remainder of 1, 2, or 3, if needed.

Example: 75 is split into 72 + 3. 72 is the largest number equal to or less than 75, and 3 is the remainder needed to make it a sum that totals 75.

Step 2 - Modify & solve: Divide the multiple of 4 by 2, and change the plus sign to a minus sign. Once you've modified the problem, the next step is to solve it.

Example: 72 (from 72 + 3 above) ÷ 2 = 36, so 72 + 3 becomes 36 - 3. Solving this, we get 36 - 3 = 33.

Step 3 - Cast out 7s: From the total in the previous step, subtract the largest multiple of 7 equal to or less than the total. For the reasoning behind this step, see the Modular Arithmetic section of my Day for any Date Online Toolbox post.

Example: The largest multiple of 7 equal to or less than 33 is 28, so we subtract 33 - 28 to get 5.

This may seem like a lot of work at first, but it can be done surprisingly quickly, once you become familiar with it.

Below is a simple widget to show you how this is done with various years. Simply select the first digit of the 2-digit year with the left menu, and the second digit with the right menu. You'll automatically see how the calculations below are affected by each change.

  • Starting year: 75
  • Partition: 72 + 3
  • Modify & solve: 36 - 3 = 33
  • Cast out 7s: 33 - 28 = 5
Note that years which are evenly divisible by 4 are easy to calculate. You'll quickly learn that multiples of 4 allow you to skip the partitioning step, and go straight to dividing a single number by 2.

Also, pay particular attention to the years 01, 02, 03, and 07. When these are partitioned, divided, and subtracted, they all result in negative numbers. In these few cases, you need to add 7 to make them positive. If you can remember that -1 + 7 = 6, -2 + 7 = 5, and -3 + 7 = 4, you can handle all of these situations.

You can practice calculating this key in the Quick Calendar Month Creation Quiz by clicking on the 2-Digit Year Key button.

Century Key

The number you calculated from the year above will now be modified to take the century into account. Adjusting for the century only requires a little memory and a single addition.

If the requested century is in the 1700s, you don't need to add anything. For the 1800s, you'll add 2 to your running total. For the 1900s, you'll add 4, and for the 2000s, you'll add 5. For years after that, the pattern repeats every 400 years:

+ 0+ 2+ 4+ 5
------15001600
1700180019002000
2100220023002400

This table only goes back to 1500 because the calendar we currently use, the Gregorian calendar, was first adopted on October 15, 1582. In later lessons, you'll learn how to handle earlier dates.

To help you recall the right number, remember the nonsense mnemonic, “No tuna for Friday”. No represents +0, tuna represents +2, for represents +4, and Friday represents +5, since Friday is the 5th and last day of the traditional 5-day Monday through Friday work week.

How do you know which numbers go with which years? From right to left, they cover the 1700s, the 1800s, the 1900s, and the 2000s respectively. Bob Goddard (see Credits & Notes below) suggesting remembering that the mnemonics apply to the first 400 years of American history.

Once you can recall which modification goes with which year, you simply add the appropriate number for the century. Don't forget to cast out 7s if you get a number that's 7 or larger.

Going back to our 1975 example from above, we worked out a key number of 5 for 1975. To modify this number for the century, we add 4 for the 1900s, so 5 + 4 = 9. Casting out 7s, we subtract 9 - 7 = 2. So, 2 is our final key number for 1975.

What about, say, 1776? 76 doesn't need to be partitioned, since it's a multiple of 4. 76 ÷ 2 = 38, and casting out 7s we get 38 - 35 = 3. Since the 1700s have a century modification of +0, we simply don't add anything, and leave the total as 3. So, the year key for 1776 is 3.

Next Step

Practice calculating the full key over in the Quick Calendar Month Creation Quiz by clicking on the Full Key 1700-2099 button. Ideally, you want to be able to run through the calculation process for a given year in about 5-6 seconds.

Once you have this year key, what do you do with it? The next step is to take the key number you just calculated, and combine it with a particular month. You'll learn how to do that in the next tab.

Credits & Notes

The mathematical method for handling the 2-digit year comes directly from an article titled, “How to find the day of the week on which any particular date falls” by E. Rogent and W.W. Durbin in the August 1927 issue of The Linking Ring.

The modification of the century, as well as the “No tuna for Friday” mnemonic, is from step 2 of Bob Goddard's First Sunday Doomsday Algorithm.

Months

Now that you've calculated a key number based on the year and century, which will be treated as a date from here on out. The next step is to ask for a month. Each month is associated with a day of the week, and all you'll do to create the correct calendar for the month is to put the date you calculated from the year and century together with the day associated with the given month!

Months and Days

To help you recall with days are associated with which months, I developed 7 short rhymes to help make the connections memorable:

February, March, and November,
Three on Sunday to remember.


February, March, and November are all associated with Sunday. Sunday also happens to be the only day of the week associated with 3 different months. Each of the other days is associated with only 1 or 2 months each.

June day?
Moon-day!


This short rhyme makes it easy to remember that June is associated with Monday.

A skeptic needs clues,
but Sept./Dec. needs Tues.


Sept. is Septemeber, and Dec. is short for December, and they're both associated with Tues., or Tuesday.

Winds in April, Winds in July,
“Winds”-day’s the day, wet or dry.


This is simple enough. April and July are associated with Wednesday.

Jan’s and October’s 31,
Reminds us that Thursday won.


Jan, of course, refers to January. Not only are January and October both associated with Thursday, but they're the only set of months in a group which all happen to share the same length.

Month 5 is May,
Day 5 is Friday.


May being associate with Friday made for an easy rhyme, of course. Since May actually rhymes with every day of the week, I thought the additional association of the 5th month and the 5th and final day of the work week strengthened the association.

August’s few holidays make you yearn,
For the day off named after Saturn.


In most English-speaking countries, August tends to be the month with the fewest nation-wide holidays. This made for an interesting rhyme to help you memorize that August is associated with “Saturn-day”, or Saturday.

Leap Year Exceptions

Leap years stand out in this method because they don't need partitioning like most other numbers. If you're given a leap year, work out the key as taught previously, and then make a mental note that it's a leap year.

January and February both change the days with which they are associated in leap years. January in a leap year moves 1 day back from Thursday to Wednesday. In other words, in a leap year, January, April, and July are all associated with Wednesday, and October is the only month associated with Thursday.

Similarly, February also moves 1 day back from Sunday to Saturday in a leap year. In leap years, then, February and August are both associated with Saturday, and only March and November are associated with Sunday.

These two leap year exceptions may seem difficult to manage at first, but if you just think of January and February moving their day associations back by 1 day, it quickly becomes natural.

Remember that any year which is evenly divisible by 4 is a leap year, except for years ending in 00. Years ending in 00 are only leap years if they're evenly divisible by 400. For example, 2000 was a leap year, but 1900 and 2100 are not.

Practice your association of days with months by going to the Quick Calendar Month Creation Quiz and clicking on the Month and Day Association button.

Once you're comfortable with the month-to-day associations and calculating the key from the year, you're ready to quickly create an entire calendar month, as explained in the next tab.

Credits & Notes

The approach of associating each year and century with a date, and each month with a day of the week is also from “How to find the day of the week on which any particular date falls” by E. Rogent and W.W. Durbin in the August 1927 issue of The Linking Ring.

In the original article, however, Durbin and Rogent's month and day associations were based on dates in the 1800s. In order to make the math work properly with Bob Goddard's First Sunday Doomsday Algorithm, however, the month and date associations have been changed so that they're now based in the 1700s.

Since the month and day association have been changed, the rhymes for remembering each association are original with me.

Creating the Calendar

At this point, many calendar calculation methods would have you do one last calculation to work out the day of the week for the given date.

Since my goal here is to minimize the calculation, this step simply involves creating a calendar for the given month and year. Here's a PDF template for a set of blank calendars, sized for printing on the back of business cards. When you bring out a single one to use, it should appear like this:



This step is where all of your calculation and memory work come together. To show you how, let's say you ask for a year, and you're given 1982. As you write down 1982 in the Date section of the calendar, you run through the calculations in your head: 82 = 80 + 2, which becomes 40 - 2 = 38, cast out 7s so we have 3, add 4 for the 1900s, giving 7, casting out 7s, leaves us with 0. We now know 1982 is a 0 year.

For the month, we'll say that you're given April, which you should immediately associate with Wednesday (Remember? “Wind in April...”), as you write down April in the Date section.

At this point, you know that 1982 is a 0 year and April is associated with a Wednesday, yet the calendar looks like this:



Generally, the rule is to put the day of the week you recalled above the number you were given. If you have 5 and Monday, you'd put Monday directly above the 5 on the calendar. In this case, though, we have 0 and Wednesday, so what do you do?

Simple, think of the “0th” of a month as being 1 week before the 7th. In the case of 0, you always put the day of the week above the 7. Since April is associated with Wednesday, we put the abbreviation Wed above the 7th, like this:



From here, it's easy to see the remaining days of the week should be placed:



Don't forget to give the month the correct number of days. In our example, April is a 30-day month, so we add a 29 and 30 in the bottom row:



At this point, you're done creating the calendar! If this was someone's birthday, and they were born on April 17, 1982, a quick look at the calendar is all you need to see that they were born on a Saturday. You can circle that date and the day of the week, and now they have a souvenir of their birthday, and your amazing calendar skills!

Another Leap Year Reminder

In the previous tab, you learned that when working with leap years, and you're given either January or February as the month, you need to move the day associate back by 1 day. Let's see how this works in practice.

For this example, we'll choose 1992 as the year. Multiples of 4 aren't partitioned, so we start with 92 ÷ 2 = 46, and cast out 7s (46 - 42) to get 4. Adding 4 for the 1900s to that, we get 8, and we cast out 7s to get 1. You've worked out that 1992 is a 1 year, and also realize you need to keep alert if they ask for January and February.

Continuing with this example, we'll imagine that February is chosen as the month. When you hear this, you need to recall 2 concepts. First, February is associated with Saturday in a leap year, and second, that a leap-year February has 29 days. Putting all that together, the resulting calendar should appear like this when finished:



Leap-year Januarys and Februarys are probably the worst-case scenario you'll face in this method. Once you're comfortable with those, you're ready to perform this feat.

Verification by the Spectators

For this feat to really be impressive, they need to know that the calendar you created is correct. In the past, this meant carrying around cumbersome almanacs. Today, though, the handiest and most convenient way is carry an internet-connected mobile device around.

They can verify it with any kind of search, but I usually have them point their browser to Wolfram|Alpha (http://www.wolframalpha.com). Once you have the calendar created, have them enter the year and month followed by the word calendar, and Wolfram|Alpha will generate the calendar for that month!

For our April 1982 example, entering the phrase April 1982 calendar into Wolfram|Alpha will display that calendar, and you can see that both start on the same day and end on the same month. You can also verify the February 1992 calendar in the same way.

Practice Creating Calendars

Over in the Quick Calendar Month Creation Quiz, you can practice creating calendars by clicking on the Calendar Quiz button. This quiz is different than the previous quizzes, however, as it opens up full-screen in a new window.

There are two pull-down menus near the top. By default, the one on the top-right reads Gregorian: 1700-2099, which are the years you'll probably want to start with when you're first learning this feat.

Clicking on this menu will bring up other selections, which are: Gregorian: 1582-9999, which tests on years in the current calendar, but in a much wider range (the first 2 items on the tips page can help here), Julian: 1 AD - 1582 AD, and Julian: 45 BC - 1 BC. You won't need to worry about these until you've read the next page.

To use this quiz, select a range from the top-right menu, and then click on the New Date button on the top-left of the screen. Down by the word “Date:”, at the bottom, a random year from the selected range will appear, followed shortly by a random month.

The days of the week towards the top of the screen can be scrolled left or right by using the left and right arrow keys on your keyboard, or by swiping them left or right, if you're using a mobile device. What you want to do is use this ability to scroll the days to align the key you calculated from the given year with the day you associate with the given month.

For example, if you're given 2014, and you quickly work out a key of 2, and then you're given the month of May, which is associated with Friday, you want to scroll the days left or right, so that Friday is right above the 2.

In order to make sure the number of days are right, you use the pull-down menu at the top-left of the screen, which defaults to 28 days. You can also select 29 days, 30 days, and 31 days, and the added days will appear in the bottom row. Continuing with our 2014 May example, you'd want to select 31 days, since that's the length of May, so that 29, 30, and 31 appear in the bottom row.

Once you're satisfied that you've created the month, click the Verify button at the top right to see whether you are correct. It can inform you whether the number of days in the month is wrong, whether the days aren't labeled correctly, or whether you got it correct. If nothing is correct, it will simply say, “Oops. Try again, please.”

To try again, simply click the New Date button, and you're given a brand new random year and month!

Once you're comfortable working with years in the Gregorian calendar, the next tab will show you how to handle years in the Julian calendar, all the way back to 45 B.C.!

Credits & Notes

Creating a calendar month as an alternative to doing calculations involving an individual date is an approach I developed when I created Day One. In fact, the calendar PDF linked in the top of this tab was originally developed for Day One.

The quiz seen here is also a modified version of the calendar quiz I originally developed and included with Day One.

Julian Calendar

The calendar we currently use, known as the Gregorian calendar, was first used by Spain, Portugal, and most of Italy on the Gregorian date of October 15, 1582. The date immediately prior on the Julian calendar was October 4, 1582.

Other countries didn't adopt the Gregorian calendar until later. Britain and its colonies, including America at the time, adopted it in 1752, jumping from the Julian date of September 2 to the Gregorian date of September 14 on the following day.

In the Julian calendar, leap years happened every 4 years, including years ending in 00. With the Gregorian calendar, years ending in 00 are only leap years if they were evenly divisible by 400. For example, 2000 was a leap year, but 1900 was not and 2100 will not be.

With this shifting ahead of days, and different leap year rules, you might think it would be difficult to account for this change in a calendar calculation. I have good news for you. If you want to calculate a Julian date with this method, it's astonishingly simple.

To adjust for a Julian date, the only modification you need to do concerns the century. Instead of using the 0-2-4-5 “No tuna for Friday” approach, simply add the century number itself, casting out 7s if needed.

For example, if someone asked about 1066, you would start with 66 just as before. 66 = 64 + 2, which is modified and solved as 32 - 2 = 30. Casting out 7s from 30 leaves 2. For the Julian calendar, we just add the century number (10): 2 + 10 = 12. Casting out 7s from 12 leaves us with 5.

Next, if someone asks for the month of October, we recall that October is associated with Thursday (“Jan’s and October’s 31...“), so we can create a Julian calendar month for October 1066, by starting with a Thursday the 5th. Yes, Wolfram|Alpha can verify Julian calendar dates, too.

As long as you don't accept dates in 1582 from October 5th to October 14th, you can now handle dates as far back as 1 A.D., and as far into the future as you like!

To practice creating calendar for Julian years and months from 1 A.D. to 1582 A.D., you can go the the Quick Calendar Month Creation Quiz page, and click on the Calendar Quiz button. When the Calendar Quiz opens, select Julian: 1 AD - 1582 AD from the top-right menu to practice these dates.

B.C. Dates

The Julian calendar was first put in place by Julius Caesar in 45 B.C., so wouldn't it be nice to handle those first few dates, as well?

The first thing to remember is that the calendar we use today went from 1 B.C. directly to 1 A.D., so there was no year 0.

Handling dates such as these only requires one additional adjustment. When given a B.C. date, subtract the year from 57, and then treat it like any other Julian year.

If you're asked to create a calendar in, say, 25 B.C., the first step is to subtract it from 57: 57 - 25 = 32. Now you can handle that as a Julian year. 32 doesn't need to be partitioned, so we just do 32 ÷ 2 = 16, and casting out 7s from 16 leaves 2. Since the century is effectively 0, we just leave the key at 2.

Now, if someone asks for June, we recall that June is associated with a Monday (“June day? Moon-day!”), so June in 25 B.C. must've had a Monday the 2nd, which Wolfram|Alpha verifies for us!

When dealing with Julian and/or B.C. dates, don't forget the leap year adjustment, if needed. For example, since 25 B.C. is a leap year, you'd still work out that the key number for it is 2, but if someone asked for January, you need to recall that it's associated with Wednesday in a leap year. That way, you know January in 25 B.C. had a Wednesday the 2nd.

You can also practice BC dates by going to the Quick Calendar Month Creation Quiz page, and clicking on the Calendar Quiz button. After the Calendar Quiz opens, select Julian: 45 BC - 1 BC from the top-right menu to practice these dates.

Once you're comfortable working with Julian calendars and calendars back to 45 B.C., check out these tips to help your performance along.

Credits & Notes

The Julian and B.C. date modifications both come directly from the approach used in Bob Goddard's First Sunday Doomsday Algorithm. Basing the Gregorian century modifications on the 1700s is what makes this possible.

Tips

• Only the last 4 digits of a year matter. Working with the year 15739562 is no different than working with the year 9562.

• Another way to think about the centuries, especially when dealing with years far away from the 1700s, 1800s, 1900s, and 2000s:
  • If it's divisible by 400, like the 2000s, add 5.
  • If it's even, but not evenly divisible by 400, like the 1800s, add 2.
  • If it's just after a multiple of 400, like the 1700s, don't add anything.
  • If it's just before a multiple of 400, like the 1900s, add 4.

• If you're comfortable with using the Mnemonic Major System, you can speed up this feat even further. Since dates within a century repeat in a pattern every 28 years, all you have to do is memorize the 2-digit year keys for the years 00-27. You not only have quick recall of those years, but of all the other years as well:
  • Years ending in 28 through 55: Subtract 28, then recall the corresponding key.
  • Years ending in 56 through 83: Subtract 56, then recall the corresponding key.
  • Years ending in 84 through 99: Subtract 84, then recall the corresponding key.

• You don't have to use the business card calendar PDF. Any calendar that allows you to display an entire month AND adjust which days go with which months will work. Some examples:

• There's a simple formula for working out how many days needed to be added to the Julian calendar to adjust it to the Gregorian calendar. You can learn it from my Changing Calendars Mentally post. When working with the Julian calendar, this can make an impressive additional detail.

• Which months have a Friday the 13th? Once you've taken the year and century into account and you have your key number, ask yourself which day of the week is that many days after Saturday, and then ask yourself which months are associated with that day of the week. The months will be the only ones with a Friday the 13th!

For example, if you're given a year whose key you calculate as 3, ask yourself what day is 3 days after Saturday? The answer, of course, is Tuesday. Which months are associated with Tuesday? September and December are associated with that day, so you can state that September and December are the only months that year with a Friday the 13th!

This makes an impressive extra feat to add in to your performance. Note: As always, don't forget that January's and February's associations move 1 day back in leap years!

• What do you do if someone asks for a year before 45 B.C.? Explain that the pre-Julian Roman calendars were just too different. For example, one version had only 304 days each year, and another had 355 days each year, with leap years making some years as long as 378 days. The last pre-Julian year, 46 B.C., was adjusted to 445 days long!

• Hans-Christian Solka, author of the Day of Week Calculation Blog, also suggests the following related online resources for further reading:

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Moon Phase For Any Date

Published on Thursday, January 03, 2013 in , , ,

Basics

Before you learn how to determine the moon's phases in your head, you'll need to learn and remember some important moon-related terms, so you can describe each phase.

Moon's Age

No, this doesn't refer to the moon being roughly 4.5 billion years old. It refers to the number of days since the most recent new moon, and ranges from 0 to about 30 days.

Over at cycletourist.com, there's a Moon age calculator that uses the date and your time zone to give the exact moon age. Try a few different dates on that calculator, so you can get an idea of the different ages.

Northern vs. Southern Hemisphere

The way the moon proceeds through its phases appears differently, depending on whether you're in the Northern or Southern hemisphere. Here's how the moon's cycle appears in the Northern Hemisphere:



To make this progression more memorable, we'll just focus on stages 3, 5, and 8 from the above diagram:

These phases look like the word “DOC”. The D is shaped like the half-moon, the O is shaped like the full moon, and the C is shaped like the crescent.

In the Southern hemisphere, however, the moon starts its phases from the left side, and proceeds until it disappears on the right, like this:



Looking at only stages 2, 5, and 7 from this picture, we get:

You've probably already guessed that this progression can be remembered using the word COD.

To remember that DOC applies to the Northern hemisphere, use the fact that north is up on most maps, and just think of Bugs Bunny's classic saying, "What's up, Doc?" For COD, recall that a codpiece is designed to be worn in a man's “Southern hemisphere.”

Events vs. Phases

In the 8 stages shown the above pictures, 4 are really only short one-day events, while the rest are part of phases, each of which are about a week long. I'll start with the events, since most people learn about those first.

Events: New Moon vs. Full Moon

The full moon is simple enough to remember. It's when the moon is as full, or as brightly lit as its going to get in its current cycle, as shown in stage 5 of either of the above graphics.

Some people confuse the concept of a new moon with a full moon. If you think of NEW MOON as being the same as NO MOON, that shouldn't be a problem. Image 1 in either of the above graphics shows a new moon.

Events: First Quarter vs. Third Quarter Moon

The term quarter moon often seems confusing, especially since they refer to what most people think of as a half-moon. Quarter, in this context, refers to the point in the cycle, not the shape itself.

Think of quarters in a moon cycle like quarters of a year. After a new year, just like a new moon, comes the first quarter of the year. The third quarter of a year is the last to happen before you head into the next new year, so a third quarter moon is the last you see before the next new moon.

In the above graphics, stage 3 is the first quarter moon, and stage 7 is the third quarter moon. Since the side that's lit in either quarter varies with the hemisphere from which it's being viewed, the DOC and COD mnemonics are very important.

Phases: Waxing vs. Waning

Waxing is the phase in which the moon is getting brighter from night to night. Stages 2, 3, and 4 all show a moon in a waxing phase. If you think of either ear-wax buildup, or waxy yellow buildup on a linoleum floor, this will help you remember that waxing refers to the moon building up, or getting brighter.

Waning, on the other hand, refers to the moon getting darker and darker, as in stages 6, 7, and 8 above. To help you remember this, you can think of Lil Wayne, and use Lil to help you recall that waning means that the light on the moon is getting “littler.” Alternatively, you might think of Wayne Newton as “Wayne Newmoon,” to remind you that waning moons are getting closer and closer to the new moon.

Phases: Crescent vs. Gibbous

The crescent phase is easy to recall, especially if you're familiar with crescent rolls or crescent wrenches. When use to refer to the moon, crescent simply means that less than half the moon is illuminated, as in stages 8, 1, and 2 above.

Gibbous, on the other hand, is an unfamiliar word to many. It refers to the phase when half or more of the moon is illuminated, as in stages 3, 4, 5, 6, and 7 above. Imagine you're trying to negotiate a deal with someone, and they keep saying, “Gibbous more than half or no deal!”

Quick Notes Before Moving On

Waxing or waning, when combined with crescent or gibbous, identifies the exact phase of the moon at any given time. The waxing crescent phase runs from the new moon to the first quarter moon. Next comes the waxing gibbous moon, which runs from the first quarter moon to the full moon.

After that, there's the waning gibbous, from the full moon to the third quarter moon, and the waning crescent, from the third quarter moon back to the new moon.

The line dividing the dark part of the moon from the light part is known as the terminator, which can make from some amusing sci-fi movie references.

Once you're comfortable with all the terms, you're ready to start learning the formula for determining the moon phase in your head, which begins in the next tab.

About The Formula

The moon phase calculation formula has much in common with the Day of the Week For Any Date method taught here in the Mental Gym, including the developer of the formula, John Conway.

Just like the day of the week feat, you're going to translate the year, date (1-31), century, and year into key numbers, and add them up.

In the day of the week feat, you then performed a modulo 7 operation on this number, but for the roughly 30-day moon cycle, you're going to perform modulo 30. For a clear explanation of this concept, read BetterExplained's post, Fun With Modular Arithmetic.

Unlike the calendar formula, the moon phase formula is not exact. The real moon's orbit is quite complex, and exact calculations include corrections for every detail.

The formula you'll learn is simple enough to do in your head, and will be accurate to within about a day or so. You'll learn how to deal with this in the Presentation section of the Practice tab.

Month Key

The month key is very easy to remember. For March to December, you simply use the standard month number. For example, March's key number is 3, April's key number is 4, and so on up to December's key number, which is 12.

January and February aren't hard to recall. January's key number is 3, and February's key number is 4. If you can remember the month numbers, and these numbers for the first two months, you'll be able to remember the year key without trouble.

Date Key

If you thought the month key was easy, the date key is even easier! All you use is the number of the date, 1 through 31, as the date key.

So, your first step is to look at the month and date, and add the two keys together. July 12th? That's 7 + 12 = 19. February 7th? That's 4 + 7 = 11. January 19th? That's 3 + 19 = 22.

Once you have a total of the month and day keys, it's usually a good idea to add 30 to the total before working with the century and year keys. With those, you're going to be doing some subtracting, and adding 30 will help ensure that the running total remains a positive number.

When you're comfortable getting and adding the month and day keys, it's time to learn the year and century keys.

Century Key

When performing this feat, it's best to use a date that's meaningful, such as a birthday or anniversary. So, all you really need to know are keys for the 1900s and 2000s.

If you're given a date in the 1900s, you simply subtract 4 from the running total. If you're given a date in the 2000s, you'll subtract 8 instead. That's it!

Year Key

Unlike the previous keys, there are a few steps to working out the year key. Don't worry, they can still be done in your head.

First, you'll need to know your multiples of 19: 0, 19, 38, 57, 76, and 95.

This is because the first step in working out the year key is finding out how close you are to the nearest multiple of 19, whether that multiple is more or less than the year number.

Let's use 2008 as an example, focusing on the year as simply 8. 8 is between 0 and 19, but obviously closer to 0. To get 8 starting from 0, you'd add 0 + 8, so think only of +8 at this stage.

It's important to note that this was just the first step to finding 2008's key, so +8 is the number we need for the 2nd step, not the key itself.

Before moving on to the 2nd step, though, let's try another example year, such as 2013. Working with 13, we see that it's between 0 and 19, but closer to 19. Starting from 19, you'd get to 13 by working our 19 - 6, so 13's number at this stage should be thought of as -6.

This is what I mean about finding the distance to the closest multiple of 19. 8 was 8 MORE than 0, so we think of it as PLUS 8. 13 was 6 LESS than 19, so we think of 13 as MINUS 6.

What about, say, 59? That would be +2, because 57 is the nearest multiple of 19, and 57 + 2 = 59. How about 88? That becomes -7. Do you understand why?

This first step seems complicated when you first learn about it, but with practice, it's not hard to determine the number you need quickly and easily. Regardless of the year, notice that you'll always get a number ranging from -9 to +9 as a result.

Now it's time to explain the second step. First, I'll explain the long way of working through this step, then I'll give you a shortcut that gives the same result.

LONG VERSION: Whether the number you got in the first step was positive or negative, you're going to work through this step as if it's a positive number, and then restore the proper sign at the end.

Going back to our 2008 example, we got +8 in the first step, and we remember that plus sign. Next, we multiply that by 11, and get 88. Finally, 88 modulo 30 is 28, and we recall that the original number was positive, so we know that the 28 is positive, as well. This is the final number we need, so we would add 28 to our running total.

What happens with 2013? Applying the first step to 13 gave us -6 (remember?), so we'll work through this process as just 6, remembering the minus sign for later. 6 × 11 = 66, and 66 modulo 30 is 6. Restoring the minus sign from earlier, we get -6. This simply means that we'd subtract 6 from our running total.

SHORTCUT: Let's run all the numbers from -9 to +9 through the above process, and see if we can find some patterns:



The first thing to notice here is that neither the sign nor the ones digit (the rightmost digit) change. It's also easy to see that the multiples of 3 don't change at all.

If you think of the multiples of 3 as being 2-digit numbers (±03, ±06, and ±09), then you can also see that the tens digit of each number is the ones digit modulo 3. For example, 7 modulo 3 is 1, so we just put a 1 in front of a 7 to make 17.

Just like in the long version, when you do the modulo 3, you need to temporarily drop the sign, then put it back afterwards, for this formula to work out properly.

Let's work through the shortcut method with a year ending in 53, as an example. In the first step, 53 becomes -4 (you should still understand why). We temporarily drop the minus sign, and perform 4 mod 3 to get 1, so we put a 1 in there to make -14. This means we need to subtract 14 from our running total.

The more you practice and perform this feat, the more quickly these numbers will come to mind. Sooner than you expect, the numbers will come from your memory quicker than from calculation.

Now that you understand how to properly figure the year, date, century, and year keys, it's time to put them all together and practice, which we'll do in the next tab!

Getting Results

Now that you understand how to find each key, it's time to put them all together and get the final result!

Simply add each of the key numbers together. If the number is greater than 30, take the answer modulo 30.

You'll wind up with a number ranging from 0 to 29, and this number represents the moon's age in days. The chart below shows how the age corresponds to the four major moon events:



To see how this works, let's work through an example date, April 1, 1994. April gives 4 and the 1st gives 1, so that's a total of 5. As I mentioned earlier, I usually like to add 30 at this point to prevent working with negative numbers, so 5 + 30 = 35.

The key for the 1900s is -4, so 35 - 4 = 31. Running 94 through the year key process gives us -11 (Because 95 - 1 = 94, and -1 * 11 = -11, remember?), so 31 - 11 = 20. In this case, 20 is below 30, so we don't have to work out modulo 30.

Our estimate, then, is that the moon's age is about 20 days old (±1 day). That suggests we're about 5 days past the full moon, so the moon would be getting darker each night, but a majority of the moon's face would still be illuminated. In technical terms, the moon was in a waning gibbous phase.

If you enter April 1, 1994 into Wolfram|Alpha and scroll down to the bottom, you can verify that the moon was indeed in a waning gibbous phase on that date!

As a matter of fact, Wolfram|Alpha provides a great way to test your ability to work through this formula. Get a random date in the 20th or 21st century by clicking here, then use that date to work through the moon phase formula in your head. When you've got an answer, click the date in the Result pod, and scroll to the bottom to see the actual phase of the moon on that date.

Presentation

Due to the margin of error of this formula, you can't pin down the specific events (new moon, 1st quarter moon, full moon, and 3rd quarter moon) exactly. Since the various phases are each about a week long, and overlap, you can use this to your advantage.

I often use the difference in technical terms used by astronomers (and on Wolfram|Alpha) in contrast with an informal description of the moon, including how average people would refer to the moon. This is usually enough to be considered accurate and verifiable with an outside source.

Don't forget the importance of having an outside source to verify your claim on the moon. If you can't have someone look up the date online, at least carry an almanac or other reference in which your audience can verify the moon phase.

If you make sure people verify your claim with Wolfram|Alpha, you can also usually state your estimate of the moon's age openly. Wolfram|Alpha, at least at this writing, doesn't include this information.

Here's a general idea of what to say for any given estimate:

29, 0, or 1 days: “On that date, the moon was in a crescent phase, as the 3rd quarter moon happened about a week ago. In fact, so little of the moon is lit, most people would just refer to it as a new moon.”

2 through 6 days: “The new moon was just about (2, 3, 4, 5, or 6) days before that date, so it's getting brighter every night, but less than half the moon is lit on the right side. It's known as the waxing crescent phase.”

7 or 8 days: “It's been about a week since the new moon, so the moon is in a waxing phase - in other words, getting brighter each night. The average person might even call it a half-moon or 1st-quarter moon.”

9 through 13 days: “The 1st quarter moon was about (1 to 2, 2 to 3, 3 to 4, 4 to 5, or 5 to 6) days before that date, so the moon is getting brighter each night, and more than half is lit. This is called the waxing gibbous phase.”

14, 15, or 16 days: “It's been just about a week since the 1st quarter moon, and so much of the moon is lit that most people will say things like, 'All the crazies are out, it's a full moon!'”

17 through 21 days: “Just about (1 to 2, 2 to 3, 3 to 4, or 4 to 5) days before that was the full moon, so most of the moon is still illuminated, but it's getting darker each night. The technical term for this is the waning gibbous phase.”

22 or 23 days: “On that date, it's been about a week since the full moon, so around half of the left side of the moon is illuminated. Many would refer to this as a half-moon or 3rd quarter moon.”

24 through 28 days: “Around (1 to 2, 2 to 3, 3 to 4, 4 to 5, or 5 to 6) days before that date was the 3rd quarter moon, so the moon is getting darker each night, and less than half the moon is illuminated. Astronomer call this the waning crescent phase.”

The above statements were written with the assumption that the speaker was talking about the Northern hemisphere. For the Southern hemisphere, just switch all the references to left and right.

Once you've practiced both the formula itself and the presentation of your moon estimate, you're ready to perform this feat for someone else.

There are many more ways to use your newfound knowledge of the moon, and you can learn about those in the Tips/Tools tab.

Additionally, you can learn more about the formula itself in the Why? tab.

Tips and Tools

• This routine works well with most versions of the Day of the Week For Any Date feat. I teach one version here in the Mental Gym, as well as a quicker and simpler version called Day One, for sale at Lybrary.com.

• Be alert to key number combinations that total 0, to cut your calculation time:
  • Years ending in a multiple of 19 (00, 19, 38, 57, 76, 95): 0 (year) = 0
  • The 4th of any month in the 1900s: +4 (date) -4 (1900s) = 0
  • The 8th of any month in the 2000s: +8 (date) -8 (2000s) = 0
  • Any February or April in the 1900s: +4 (month) -4 (1900s) = 0
  • Any August in the 2000s: +8 (month) -8 (2000s) = 0
  • Use the Moon Century/Year Combo Key Generator widget below to find years that require no adjustment, such as 2017.

• Instead of trying to describe the moon, you might try having images of 10 to 12 different phases of the moon on your mobile device. Since you have less than half of the 30 different images, it's reasonable that the photo you're showing isn't an exact match to the moon. There's even a moon-shaped toy called Moon In My Room that can display 12 different phases of the moon, which is perfect for this presentation.

• If you want to work with other centuries, I've developed a Wolfram|Alpha widget that will work out the century keys for anywhere from the 1800s to the 3000s. Simply select the century from the menu, then click Submit.

The widget will return 2 numbers, r and s. You can use either one of these as the key for your chosen century, but one if often simpler to use than the other. For example, selecting the 2900s will return r = 6 and s = -24. This means that you can use either +6 or -24 as your key for the 2900s. In this example, adding 6 is the easier way to go.



• Should you decide to focus on a single year, you can use the widget below to get a single key that compensates for the century and the year at once. If you can memorize several keys, you can use the widget to lookup each year and that range.

Use the pull-down menus to select any year from 1800 to 3000, and then click the Submit button. Like the above widget, you're given two choices, r and s, and you can decide which one is easier to use. You can ignore the variable y, as it's simply the last two digits of the year you input.



• If you have Day One, and you're familiar with the Major/Peg System for memorizing numbers, it's not hard to develop a Day One-type approach to the Moon Phase feat.

Use the Moon Century/Year Combo Key Generator widget above to determine keys for each of the years 2000 through 2018, and link from the same year images I use in Day One to the Major/Peg images for each adjustment (I add a spear to the peg image to represent the minus sign, so I know when to subtract). The years 2019 - 2099 can be reduced to these images via modulo 19, similar to the use of modulo 28 in Day One. To adjust for a date the 1900s, simply add 4.

I jokingly refer to this approach as Night One.

• Here are some excellent videos that can provide helpful background knowledge:
• If you'd like to understand more about each part of the estimation formula itself, you can learn more about it in the Why? tab.

Why?

While some may prefer to use the formula without understanding how it works, I've added this section for people who want to understand it better.

Why Modulo 30? If you were to draw a line from the center of the Earth through the center of the moon to a fixed star, it would take about 27.321661 days (27 days, 7 hours, 43 minutes, 11.5 seconds, known as a sidereal month) to do so. However, it wouldn't return to the exact same phase, since the Earth has moved roughly 45 million miles in that same amount of time.

To return to the same moon phase takes longer, an average of about 29.530588 days (29 days, 12 hours, 44 minutes, 2.9 seconds, known as a synodic month).

The phase of the moon in this 29.530588-day cycle is approximated through the use of modulo 30, which is easier to work with mentally.

Why Focus on Groups of 19 Years? When estimating moon phases, it's useful to know how long it will be before the moon will be in the same phase on the same date. As it turns out, the answer is surprisingly accurate 19 years and 2 hours! This is known as the Metonic cycle.

Over a 100-year period, the difference is only about 10.5 hours, so working in groups of 19 works very well in the estimate. The estimation itself adjusts every 100 years, and takes this into account.

Why Multiply the Years by 11? If the moon revolved exactly 12 times around the Earth every tropical year (the technical name for the year as most people know it), we wouldn't need to make any adjustment for years, as the moon would always be in the same phase on the same day.

If it took the moon, say, 2 days short of a tropical year to revolve exactly 12 times, then we'd have to adjust by adding 2 days for every year. This would be done simply by multiplying by 2.

It turns out that the time required for 12 revolutions of the moon is about 10 78 days short of a tropical year. This is approximated in the estimation formula through multiplying by 11.

Why Do Years Range From -9 to 9, Instead of 0 to 18? Over the 19-year Metonic cycle, that small 18 day difference can start to add up. 18 years × 18 day = 188 day = 218 days!

When originally developing the moon phase estimation formula, John Conway worked out the rather ingenious -9 to +9 approach you learned. What's so ingenious about it?

Go to this Desmos Calculator page, and drag the slider back and forth from 0 to 18 years. The red bar shows the 18-day difference building up over that time. The numbers on the y-axis (the vertical axis) represent the total number of days in error.

By comparison the blue bar, representing the -9 to +9 approach, through the first 9 years, and then begins to decrease dramatically as you add more years!

Hopefully, this gives you a better idea of how this reduces the margin of error.

What is the Margin of Error?

With all the approximate numbers we're using, the resulting accuracy of this estimation is quite amazing.

To give you an idea of just how accurate it can be, I've provided the following calendar. You can use the drop-down menus to select any month from January 1900 to December 2099, and the calendar will instantly change.

Est., in each day, represents the estimated age of the moon you get by working through the method in this tutorial.

Real is the actual age of the moon, using calculations from Practical Astronomy with Your Calculator, 3rd edition (The newest edition includes spreadsheet calculations), modified from javascript code by Janet L. Stein Carter.

As I've modified them, these calculations reflect the phase of the moon as it appears at noon in Greenwich Mean Time on each given day. The moon's age in degrees, from 0° to 359°, is converted into its age in days in the synodic month, and then rounded to the nearest whole number.


0

Mental Division: Decimal Accuracy

Published on Sunday, July 29, 2012 in ,

Introduction

While there are many mental math sites and videos freely available on the internet, very few seem to focus on division.

That's likely because division rarely comes out nice and neat, like addition, subtraction, and multiplication do. You have to deal with “leftovers” in the forms of remainders, fractions, or decimals.

Fractions are often seen as the most human-friendly way of expressing left-over numbers. Consequently, being able to give exact decimals, especially for unusual fractions such as 5/7 or 13/15, seems like an amazing feat to most people.

This tutorial will focus solely on division problems whose results are less than 1. For example, you'll learn how to deal with 5 ÷ 7 (or the equivalent fraction 5/7), but not how to deal with, say, 16 ÷ 7 (or 16/7).

Fortunately, there are numerous patterns of which we can take advantage, that will help make the task much easier than many people would suspect. You can start learning about these patterns in the next section.

Fractions to Memorize

When dealing with smaller divisors, such as those from 2 to 11, it's quite useful to have the decimal equivalents of all such problems less than 1 memorized.

You probably already know all the answers for dividing by 2 through 4 already:

1/2=.5

1/3=.333... (the dots are used to refer to the endless repetition of digits)
2/3=.666...

1/4=.25
2/4=1/2=.5
3/4=.75

Dividing by 5 is very easy. You just double the numerator, and put a decimal point in front of it:

1/5=.2
2/5=.4
3/5=.6
4/5=.8

You also already know how to divide most of the numbers by 6, as well:

2/6=1/3=.333...
3/6=1/2=.5
4/6=2/3=.666...

All you have to know is two more 6ths:

1/6=.1666...
5/6=.8333...

I'll come back to the 7th, but for now, I'm going to cover the 8ths, since they're not much more difficult than 4ths. The trick here is to multiply the numerator by 125, and stick the decimal point in front of it:

1/8=.125
2/8=1/4=.250
3/8=.375
4/8=1/2=.500
5/8=.625
6/8=3/4=.750
7/8=.875

The last one almost everyone is familiar with is how to express 10ths:

1/10=.1
2/10=.2
3/10=.3
4/10=.4
5/10=.5
6/10=.6
7/10=.7
8/10=.8
9/10=.9

New Patterns

To be able to memorize every fraction/division problem up to dividing by 11, then, all most people need to learn is how to handle 7ths, 9ths, and 11ths.

9ths are especially easy, as you simply start with the decimal point, and repeat the numerator endlessly:

1/9=.111...
2/9=.222...
3/9=.333...
4/9=.444...
5/9=.555...
6/9=.666...
7/9=.777...
8/9=.888...

If you know your multiples of 9 up to 10, then you can handle 11ths. Simply multiply the numerator by 9, and express that as a 2-digit number (9 × 1 = “09”). Repeat these 2 digits endlessly, and you've got your 11ths:

1/11=.090909...
2/11=.181818...
3/11=.272727...
4/11=.363636...
5/11=.454545...
6/11=.545454...
7/11=.636363...
8/11=.727272...
9/11=.818181...
10/11=.909090...

Finally, there's the 7ths. The 7ths have a very unusual pattern, starting with 1 ÷ 7:

1/7=.142857142857142857...

Notice that same sequence of numbers, 142857, repeats over and over again. To make things even easier, every 7th has this same repeating pattern. The only thing that changes is which number comes immediately after the decimal point. Here are the 7ths:

1/7=.142857142857142857...
2/7=.2857142857142857...
3/7=.42857142857142857...
4/7=.57142857142857...
5/7=.7142857142857...
6/7=.857142857142857...

To determine which number comes first, simply multiply the numerator by 14, and note the digit in the 10s place. That will be the digit that comes first. For example, with 3/7ths, you'd do 3 × 14 = 42, and since 4 is the digit in the tens place, you'd start the pattern with 4, giving you .42857142857... and so on.

Once you're comfortable with these basic memorized patterns, you're ready to move full speed ahead to the next section, where you'll learn to divide by numbers near 100.

Dividing by 100

Dividing a 2-digit number by 100 is easy, of course. Take the numerator, place the decimal point in front of it, and you're done. 46/100? .46! 67/100? .67!

Dividing by 99

Surprisingly, dividing a 2-digit number by 99 is almost as easy as dividing by 100. The only difference is that the numerator repeats without end. 46/99? .464646...! 67/99? It's .676767...!

Think of 99 as being like 100, but since the 9s repeat, so does the numerator.

Dividing by 90

The technique for dividing any number from 1 to 89 by 90 takes us back to grade school math. Remember learning long division and getting answers like “3 remainder 7”? That's the type of thinking you'll need here.

To divide a 2-digit number by 90, just divide the numerator by 9, but work out the answer in the “x remainder y” format. Once you know that, x will be the first digit after the decimal point, and y repeats after that.

For example, let's work out 58 ÷ 90. Ask yourself, what is 58 divided by 9? Assuming your know your multiplication tables, you should think of it as 6 remainder 4. That means the decimal equivalent of 58 ÷ 90 is .6444....

Remember this trick for 90 by thinking of the way you divided by 9 when you had 0 knowledge of fractions.

Dividing by 91

To dividing by 91, you'll first need to know how to multiply any number from 1 through 90 by 11. Below is a video to give you a good quick tutorial if you're not already familiar with the technique (If your browser supports Flash, here's another excellent tutorial). There's also a page where you can practice.



To divide by 91, you start by multiplying the numerator by 11, and then subtract 1. This number will likely be 3 digits, but if it's only 2 digits, place a 0 in front of it (76, for example, becomes 076). Put a decimal point in front of this 3 digit number, and you've got the first 3 digits of your answer.

To get the next 3 digits of the answer, subtract each of the digits from the previous step from 9. At this point, the number will repeat with the same 6 digits forever.

As a full example, let's work out 44 ÷ 91. We multiply 44 by 11 to get 484, and subtract 1, so we have 483. The first 3 digits of the answer, then are .483. Next, we work out 9 (always) - 4 (first digit) to get 5. 9 - 8 (second digit) = 1 and 9 - 3 (third digit) = 6, so the next 3 digits are these answers, 5, 1, and 6. Putting that all together, and repeating those same 6 digits, the decimal comes to .483516483516....

If you think of the 6 digits in the answer as two 3-digit numbers that always add up to 999, (Such as 483 and 516 in our previous example) it makes this easier. Since 91 ends in 1 just like 11, it's easy to remember that the technique for dividing by 91 involves multiplying by 11.

Dividing by 98

98 has a particularly surprising pattern. To start, you begin with the numerator, adding a 0 in front if it's a 1-digit number, and keep doubling the number. For example, with 6 ÷ 98, we start with 06 after the decimal point and keep doubling:
06 12 24 48
Checking with Wolfram|Alpha, we see that 6 ÷ 98 does indeed start with .06122448!

There are two more rules you need to know for this technique. First, anytime your doubling sequence gives you a number of 49 or more, you need to add 1 before continuing. Second, when you continue the sequence, double that modified number, but only give the last two digits. Continues the sequence as if these last two digits were just another two digits in the sequence, remembering to follow both of these rules.

To make this clearer, let's work out 16 ÷ 98. We start the series as before:
16 32 64
Stop! That 64 is over 49, so we need to add 1, making it a 65:
16 32 65
From there, we double 65 to get 130, but remember the second rule, that we only include the final two digits, the 30:
16 32 65 30
From here, we continue as if 30 were just another 2-digit number in the sequence. The next number would be 60, which is more than 49, so we add 1:
16 32 65 30 61
Sure enough, 16 ÷ 98 works out to be .1632653061 (and beyond).

Once you get the hang of these rules, you can carry out the decimal equivalent as far as you like. Remember the add 1 rule even when you're starting with a number equal to or more than 49. For 78 ÷ 98, you'd start by adding 1 to 78, and continue from there:
79 59 18 36 73
Do you follow the pattern there? If so, then you can mentally work out that 78 ÷ 98 is .79591836734693877551020408 and even go beyond that if you like!

Since 98 is 2 away from 100, remembering this slightly unusual doubling sequence shouldn't be a problem.

Take some time and master these specific numbers. When you're ready to move on, I'll show you how to use what you already know to automatically handle even more division problems.

Reduce

As with many things in life, simplifying before working on the problem is the first thing you should do. If you get a problem such as 25 ÷ 35, or a number like 25/35ths, reduce the problem by dividing both parts by the same number, and keep doing this until you make the two numbers as small as possible. When you realize this boils down to 5/7, you can use the 7ths technique to realize that this works out to .7142857....

Here's a quick refresher course in reducing division problems and fractions:



Being familiar with the quick divisibility tests for the numbers 2, 3, 4, 5, 6, 9, and 10 will help greatly here.

Enlarge

Attempting to reduce the fraction or division problem should always be your first step, but sometimes you just can't reduce the problem to dividing to 2 through 11. This is where knowing how to divide by numbers near 100 comes in handy. See if you can make the problem larger, into one you're already familiar with.

Let's say you want to know the decimal equivalent of 11/25ths. You don't have a technique for handling 25ths, but if you scale both parts of the problem up so it reads as 44/100, then it's easily apparent that the answer is .44.

What about 13/15ths? Multiply both numbers by 6, and you get 78/90ths, which you should easily work out to .86666... using the strategies you've learned.

Similarly, dividing by 13 can be turned into dividing by 91 by multiplying times 7 (because 91= 13 × 7) and dividing by 33 becomes dividing by 99 just by multiplying by 3. You can also divide by 49 by scaling up by 2, which turns it into a problem of dividing by 98.

If you need to divide by 14, and you can't reduce it to 7ths, you can multiply both numbers by 7 to turn it into a problem of dividing by 98. 11/14ths won't reduce to 7ths, but will scale up to 77/98ths, or .7857142857....

Reduce AND Enlarge

Again, reducing should always be the first step, but sometimes enlarging the problem after reducing can help. For example, here's how to handle 24 ÷ 39. Both numbers are divisible by 3, so the same result can be obtained by reducing it to 8/13. We can then scale this problem up to 56/91, and work out that it's .6153846....

Using this approach, you'll be able to handle any problem involving dividing by 13, 14, 15, 18, 20, 25, 30, 45, 49, and 50. Often, you'll be able to handle many other numbers, as well. 26/28ths is the same as 13/14ths, and thus 91/98ths.

Estimate

Sometimes, however, you're stuck with a division problem or fraction such as 32/43rds. That won't reduce to anything, and scaling the 43rds up to 86ths or 129ths doesn't bring up any familiar divisors.

If you look closely, however, you can see that it's quite close to 33/44, or 3/4, so you can estimate that it's roughly .75. 32/43 works out to roughly .7441860, so you can see that this is a good guess.

Most lessons in estimating fractions, such as this video, focus on rounding to 0, ½, or 1, but if you can round to the nearest 3rd, 4ths and 5ths, your estimated answers will improve greatly.

I hope you've found this tutorial useful and enjoyable!