The following is a base-10 number.
Convert this number to base-60 and then write it
in modern sexagesimal form.
Place the appropriate numbers in the boxes given.
We are given the base-10 number 3,386.65
or 3,386 and 65/100.
The semicolon in this form represents where the decimal
would be and therefore we'll convert 3,386 to base-60
using these two place values and then we will convert
65/100 or .65 to base-60 using this place value.
We know in base-10 that place values are powers of 10
and in base-60, the place values are powers of 60.
First step is to determine the highest power of 60,
then we'll divide the whole number of 3,386 at least once.
Because 60 squared is equal to 3,600, which is greater
than 3,386, the highest power of 60 that will divide
3,386 at least once is 60 to the first,
which means this place value is the 60 to the first
or 60's place value.
This place value is 60 to the zero or the one's place value
and this place value to the right of the decimal point
is 60 raised to the power of negative one
or the 1/60 place value.
We begin by determining the whole number in the 60's
place value by dividing the whole number of 3,386 by 60.
We first determine how many 60's in 338 which is five.
Five times 60 is 300.
We subtract, the difference is 38,
bring down the last digit of six.
Determine how many 60's in 386 which is six.
Six times 60 is 360.
We subtract, the different is 26.
The quotient of 56 is the whole number
in the 60's place value.
The next step is to take the remainder and divide
it by the value of the next place value,
which is the one's place value, but we know that 26
divided by one is going to be 26 and therefore,
26 is the whole number in the one's place value.
So now we know that 3,386 in base-10 is 56, 26 base-60,
but we still need to determine what .65 is in base-60.
We need to determine how many 1/60 are in 0.65
which is another quotient.
We would have 0.65 divided by 1/60, but this time,
let's write this quotient in a different form.
Let's write 0.65 as 65/100
and then we have divided by 1/60.
Dividing by 1/60 is equivalent to
multiplying by the reciprocal.
This is equivalent to 65/100 times 60 over one
and now we can simplify before multiplying.
60 and 100 share a common factor of 20.
There are three 20's in 60 and five 20's in 100
and five and 65 share a common factor of five.
There's one five in five and 13 fives in 65.
Notice now this is just 13 times three which equals 39
and now we know there are 39 1/60 in 0.65
which means 39 is the whole number in the 1/60 place value.
3,386.65 in base-10 is equal to 56, 26;39 in base-60.
Let's verify this before we go.
Let's verify that 3,386.65 is equal to 56 times 60,
plus 26 times one,
plus 39 times 1/60.
Let's verify this on the calculator.
56 times 60,
plus 26 times one is 26,
plus 39 times 1/60,
does give us 3,386.65 in base-10.
I hope you found this helpful.