﻿ ▷ Sexagesimal system: Conversion of units, operations and exercises

Sexagesimal system: Conversion of units, operations and exercises resolved

Next I’m going to explain what the sexagesimal system is and how we can pass from one unit to another in its simple form (later I’ll also explain what this is).

I’ll also show you how to move from simple to complex form and from complex to simple form, with exercises solved step by step.

What is the sexagesimal system?

The sexagesimal system is a numbering system in which every 60 units is changed.

Time and angles are measured with the sexagesimal system.

The time units that are measured with this system are hours, minutes and seconds:

The units for the angles are degrees minutes and seconds:

Both time measurements and angle measurements can be expressed in simple form or in complex form.

Let’s see what each of these forms means.

Measurements in simple form and in complex form

Simple measures are those that are expressed in a single unit, such as for example:

Measurements in complex form are expressed in more than one unit, for example:

In this lesson, I am going to teach you how to pass from one unit to another in its simple form and the complex form will be left for the next lesson.

How to go from seconds to minutes

The smallest unit within the sexagesimal system is the seconds, both for time and for angles.

60 seconds are equivalent to 1 minute, both for time:

As for the angles:

That means that after 60 seconds, I can change the unit and express it in minutes.

How do I change from seconds to minutes?

To change the seconds to minutes, you have to divide the seconds by 60, because I am changing to a larger unit. The result of the division will be in minutes.

For example: How many minutes is 225 seconds? Express in a simple way:

How many minutes is 87 seconds? Express in a simple way:

I could express the result in minutes and seconds, but that would be complex.

Can I change the seconds to minutes if I have less than 60 seconds?

Yes you can, but the result would be a number smaller than 1.

For example: How many minutes is 38 seconds? Express in a simple way:

As you can see, the result is less than 1 minute.

How to move from minutes to hours or from minutes to degrees

How to move from minutes to hours or from minutes to degrees

How to move from minutes to hours or from minutes to degrees.
As we have just seen, the next largest unit that seconds are minutes, both for time and for angles.

60 minutes equals 1 hour for time:

And 60 minutes equals 1 degree for angles:

This means that after 60 minutes, I can change the unit and express it in hours for time or in degrees for angles.

How do I move from minutes to hours or from minutes to degrees?

To pass minutes to hours or degrees, divide the minutes by 60. The result of the division will be in hours or degrees, depending on whether it is for time or for angles.

For example: How many hours is 287 minutes? Express in a simple way:

How many degrees is 155 minutes? Express in a simple way:

Can I turn minutes into hours in degrees if I have less than 60 minutes?

As with seconds, you can, but the result would be a number smaller than 1.

For example, how many degrees is 46 minutes?

We divide 46 minutes by 60 and we are left:

How to go from seconds to hours or from seconds to degrees

How to go from seconds to hours or from seconds to degrees

To pass seconds to hours, first I have to pass seconds to minutes and then those minutes to seconds, in the case of time.

In the case of angles, it would be the same: first you have to change the seconds to minutes and then the minutes to degrees.

In other words, divide by 60 twice.

For example, how many hours will be 5268 seconds?

First I divide the seconds by 60 to calculate the minutes:

And I divide these minutes again by 60 to calculate the hours:

Another example: How many degrees is 684 seconds?

I divide the seconds by 60:

And I divide the minutes again by 60:

Whenever you go from a smaller unit to a larger unit, you have to divide by 60 [/box].

How to move from hours to minutes or from degrees to minutes

How to move from hours to minutes or from degrees to minutes.

To move from hours to minutes, since I am moving from a larger unit to a smaller unit, I have to multiply by 60.

For example, how many minutes is 3 hours?

I multiply the hours by 60 and I’m left:

How many minutes is 54º?

For degrees we do the same:

How to go from minutes to seconds

In the same way, to move from minutes to seconds, I’m moving to a smaller unit, so I also have to multiply by 60.

For example, how many seconds is 15 minutes?

I multiply the minutes by 60 and I’m left:

How to go from hours to seconds or from degrees to seconds

How to go from hours to seconds or from degrees to seconds

To move from hours or degrees to seconds, I’m moving to a unit twice smaller, so I have to multiply by 60 twice. First I pass the hours to minutes and then the minutes to seconds.

For example, how many seconds is 4 hours?

I multiply the hours by 60 and have the minutes:

And now I multiply the minutes by 60 and have the seconds:

How many seconds is 25º?

First I multiply the degrees by 60 to get the minutes:

And then I multiply the minutes by 60 to get the seconds:

Whenever you go from a larger unit to a smaller unit, you have to multiply by 60 [/box].

Simple to complex measurement step

Next I’m going to explain you how to pass the units from simple to complex form and how to pass from some units to others in complex form.

How to pass seconds to minutes in complex form

To pass the seconds to minutes in complex form, we manually divide the seconds by 60. The quotient will be the minutes and the rest will be the seconds that are left over:

Let’s see an example: Express 145 seconds in minutes and seconds.

We divide 145 by 60:

The result is 2 minutes and the rest 25 seconds, so 145 seconds is 2 minutes and 25 seconds:

If we had less than 60 seconds, we cannot pass them to minutes in complex form, since the result of the division would be less than 1.

How to pass minutes to hours or degrees in a complex way

To pass minutes to hours or degrees in a complex way, that is, in hours and minutes, we manually divide by 60. The result will be the hours and the rest of the minutes that are left over.

For example: How many degrees and minutes are 213′?

We manually divide 213 by 60:

It gives me as result 3, which are degrees and as rest 33 minutes:

As with seconds, to move from minutes to hours or degrees I have to have more than 60 minutes.

Transferring minutes from simple to complex

Now let’s see how to pass minutes from simple to complex form. I will explain it to you with an example:

Express in complex form 125,4′:

First we separate the whole part from the decimal part (don’t forget that the decimal part has 0 in front of it):

If the whole part is equal to or greater than 60, divide it manually by 60, as I explained above. The quotient will be degrees and the rest will be minutes.

As we have 125′, let’s divide it by 60:

Therefore, 125′ is 2nd and 5′:

If the whole part was less than 60, it would stay as it is.

On the other hand we have the decimal part, which we have to pass to seconds. For it, we multiply it by 60:

Therefore, 0.4 minutes is 24 seconds, which we have to add to the result of the integer part and the initial measurement in complex form will be:

Let’s go with another example: Express in complex form 18.95 minutes:

First, we separate the whole part from the decimal part:

This time, the whole part stays as it is because it is less than 60.

We pass the decimal part to seconds multiplying by 60:

That we add it to the 18 minutes of the whole part and therefore 18.98 minutes are:

How to pass hours or degrees from simple to complex form

To pass hours or degrees from simple form to complex form, proceed in a similar way as we have just seen in the previous section.

For example: Express in complex form 5,68 h:

We separate the whole part from the decimal part:

The whole part is left as it is. So we already have 5 hours of the result of the complex form.

The decimal part, we pass it to minutes multiplying it by 60:

Therefore, 5.68 hours will be equal to 5 hour and 40.8 minutes:

Now these 40.8 minutes, we pass them back to complex form, repeating the process.

We separate the whole part from the decimal part:

The whole part is left as it is, since it is less than 60 and the decimal part is multiplied.

Therefore, 40.8 minutes equals 40 minutes and 48 seconds:

And finally, we already have the initial quantity in complex form:

I leave you here as a summary of all the steps we have taken to pass the hours to complex form:

How to pass from complex to incomplete

Next I am going to teach you how to pass measures from their complex form to their simple form, that is, those measures that are expressed in several units, express them in a single unit.

Shift from complex form measurements to simple form

In order to pass the measurements from complex to incomplete form it is necessary to convert each one of the units in which we want to obtain and then add them.

We are going to see several examples.

Express in seconds

Express in seconds 6h 58 min 45 s:

First we have to pass the 6 hours to seconds, the 58 minutes to seconds, the 45 seconds we already have and add all the seconds.

To pass the hours to seconds we have to multiply by 60 twice. The first time we multiply by 60, we pass the hours to minutes:

The second time we multiply by 60 we convert the minutes we have obtained in the previous operation to seconds:

Therefore, 6 hours correspond to 21600 seconds:

Now we spend the minutes to seconds multiplied by 60:

Therefore, 58 minutes correspond to 3480 seconds:

As a summary, we put all units together in seconds:

And finally we add up all the seconds we have:

Therefore, we have passed the measurement in hours, minutes and seconds to seconds:

Express in minutes

Express in minutes 25º 39′ 5”

We pass the 25º to minutes and the 5” to minutes. And we add all the minutes.

To pass the 25º to minutes, multiply by 60:

25º correspond to 1500′:

To pass the 5” to minutes, we divide by 60, since we are passing to a larger unit:

As a summary, we put all the units together in minutes:

And we add up all the minutes we have:

Therefore, 25º 39′ 5” corresponds to 1539,08′:

Express in hours or degrees

To express in hours or degrees any measure, you must bear in mind that both the minutes and seconds are moving to a larger unit and you will have to divide by 60, twice in the case of seconds and once in the case of seconds.

Once you have everything in hours or degrees, you only have to add them, as we have done in the previous examples.

Operations in the sexagesimal system. Exercises solved

Next I will explain how to operate in the sexagesimal system with measures in complex form. You will learn how to add, subtract, multiply and divide in the sexagesimal system.

How to add in the sexagesimal system

How to add in the sexagesimal system.

First we add the degrees with the degrees, the minutes with the minutes and the seconds with the seconds, placing one measure under another and placing the result underneath them:

If the seconds add up to more than 60, we have to move them to a complex form, that is, to minutes and seconds, dividing them by 60. The quotient will be minutes and the rest will be seconds.

In our case, we have 81”, which add up to more than 60, so we pass them to complex form dividing by 60:

And we are left:

And the 21 seconds are part of the final result, which is now this, after making the change:

It hasn’t changed, we have only passed the seconds to minutes and seconds so as not to have more than 60 seconds.

If we have more than 60 minutes we have to do the same thing we have done with the seconds, that is to say, to pass them to complex form (hours and minutes) dividing by 60.

In our case we have 84′, so we divide them by 60. The quotient will be degrees and the rest will be minutes:

Therefore, 84′ is equal to 1º and 24′:

And the 24 minutes will be part of the result, which we have left:

Therefore, the sum of the two measures in complex form with their result is:

How to subtract in the sexagesimal system

Now let’s see how to subtract in the sexagesimal system. Let’s do the next subtraction:

We place the measurements vertically to perform the subtraction and place each unit under the same unit: the hours under the hours, the minutes under the minutes and the seconds under the seconds:

Remember that when we do a subtraction, the amount that is placed above is called minuende and the amount that is placed below is called subtracting.

I remind you this so I can refer to each measure while I explain how to subtract in the sexagesimal system.

We start by subtracting the seconds.

If this is not possible because the seconds of the minuendo are less than the seconds of the subtracting, in the minuendo, one of the minutes becomes 60 seconds and are added to the seconds it already has. With this it is obtained that the seconds of the minuendo are greater than the seconds of the subtracting.

In our case, we cannot subtract 47 seconds from 32 seconds. Therefore, we are going to take one of the minutes of the minuendo and we are going to turn it into seconds.

To do this, of the 15 minutes we have, we subtract 1:

That minute that we have subtracted we pass it to seconds:

So the measure is as follows (with 1 minute less and 60 seconds more, which after all is the same):

We place the new measure transformed into the minuendo and now yes, we proceed to subtract the seconds:

We continue subtracting the minutes.

If the number of minutes of the minuendo are less than the number of minutes of the subtracting, we have to do something similar that we have done with the seconds, ie in the minuendo, we have to convert one of the hours into 60 minutes and add those 60 minutes to those we already have.

In doing so, we will have more minutes in the minuendo than in the subtracting and we will be able to subtract them.

In our case, we cannot subtract 39 minutes from 14 minutes, therefore, we are going to take one of the 41 hours, we are going to spend them in minutes and we are going to add them to the 14 minutes we have.

For them, of the 41 hours we have, we subtract 1:

That hour, we pass it to minutes:

And those 60 minutes are added to the 14 minutes we already have:

So the transformed measure is one hour less and 60 minutes more:

We place the new measure in the minuende of the subtraction, and now yes, you can subtract the minutes:

And finally we subtract the hours:

The subtraction of the two measures in complex form with their result is:

How to multiply a sexagesimal measure by a number

How to multiply a sexagesimal measure by a number

I’m going to explain now how to multiply a sexagesimal measure by a number, with this example:

We multiply the hours, minutes, and seconds independently by the number:

And we proceed in the same way as in the sum. If the number of seconds is more than 60, we have to move them to a complex form, that is, to minutes and seconds, dividing them by 60. The quotient will be minutes and the rest will be seconds.

In our case, we have 108 seconds, so we pass them to complex form dividing by 60:

What we have left:

And the 48 seconds become part of the final result, which has remained so:

If we have more than 60 minutes, we have to pass them to complex form, dividing by 60. As we have 137 minutes, we pass them to complex form (hours and minutes) dividing by 60:

What’s left:

The 17 minutes are part of the final result and the 2 hours are added to the 24 hours we already had:

And the final result looks like this:

Therefore, the multiplication of the sexagesimal measure by the number, with its result is:

How to divide a sexagesimal measure by a number

To finish, I’m going to explain how to divide a sexagesimal mean by a number, with this example:

We start by manually dividing the degrees by 13:

The quotient is part of the result and the rest has to be multiplied by 60 to pass it to minutes: