How to divide decimal places

Decimal numbers and multiplying and dividing by 10's

Multiplying or dividing by 10s does not change the sequence of digits of the numbers. The position of the comma is changed.

Examples:

\$\$0,05 * 1 = 0,05\$\$
\$\$0,05 * 10 = 0,5\$\$
\$\$0,05 * 100 = 5\$\$
\$\$0,05 * 1000 = 50\$\$

The comma is shifted to the right by as many places as the 10's number has 0s.

Examples:

\$\$0,05 : 1 = 0,05\$\$
\$\$0,05 : 10 = 0,005\$\$
\$\$0,05 : 100 = 0,0005\$\$
\$\$0,05 : 1000 = 0,00005\$\$

The comma is shifted to the left by as many places as the 10's number has 0s.

This means that in the example there are more and more 0s between the comma and the 5.

Numbers with 10s are numbers with a 1 and various 0s.

Now you can ask yourself what happens when you calculate \$\$ 0.05: 0.1 \$\$. The result is \$\$ 0.5 \$\$. So the number is getting bigger.

The explanation can easily be derived from the fractions. \$\$ 0.05: 1/10 \$\$ is the same as \$\$ 0.05 * 10/1 \$\$.
\$\$ 10/1 \$\$ is the reversal break.

Multiply by decimal numbers

Multiplying by decimal numbers is easy because you are doing the math as if the numbers given didn't contain a comma.

Only when you know the result of the multiplication do you put the decimal point in the result.

Where does the comma belong in the result number?

You count all the digits of the two factors of the task after the decimal point. Your result has just as many decimal places.

Examples:

If you are unsure whether you have placed the decimal point correctly, you can skip your result.

Example:

In your head, calculate \$\$ 7 * 7 = 49 \$\$. So the result should be a bit bigger. You got the comma right.

Division with decimal numbers

When dividing with decimal numbers, there are three different cases for the written division.

1. There is a comma in the first number.

You do the math as you did with the written division up to now. As soon as you exceed the comma (there is a green arrow here), you put the comma in the quotient. Rolling over helps here too. The 15 only fits once into the 28. So the comma must be placed after the 1 in the result.

There are tasks in which you immediately exceed the decimal point. In the lower case, the result is 0.19.

The 15 also fits 0 times into the 2.

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2. There is a comma in the second number.

For these tasks you first apply the multiplication by a number 10 so that the comma is omitted. You have to multiply both numbers by the number 10.
This exercise becomes \$\$ 2850: 15 = \$\$.
Now you can solve the problem again as before.

3. Both numbers have commas.

In these tasks you move the comma until the divisor no longer contains a comma. You apply the multiplication by 10s again. Here you have to calculate both numbers \$\$ * 10 \$\$. The task becomes:

You already learned how to deal with the comma in the first number in the first case.

\$\$285 * 10 = 2850\$\$

\$\$1,5 * 10 = 15\$\$

Dividend \$\$: \$\$ Divisor \$\$ = \$\$ quotient

Get the 0 "from above"

When dividing it happens that you cannot get by with the digits of the number in order to be able to give a solution. You can get any number of zeros "from above" to solve the problem.

Example:

You don't see how many 0's you will need from the task. That's why you don't write them right away in the assignment.

The number without end

By dividing you can get numbers that never end.

Example:

Numbers like \$\$ 1.66666… \$\$ are called pure period numbers. You don't write down all 6s because there would be an infinite number. Instead you write \$\$ 1, bar (6) \$\$. The line is always exactly above the sequence of digits that is repeated.

You can cancel the written invoice if you see that you always get the same value when you subtract.

The saying “Everything has an end, only the sausage has two” is refuted by dividing numbers. There are numbers that never end.

For the number \$\$ 3.8756565656… \$\$ you write \$\$ 3.87 bar (56) \$\$. The numbers 5 and 6 are repeated here. This number is called mixed periodicbecause before the period there are still numbers that are not repeated.

Periodic numbers are also called non-terminating.

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If you do a task with \$\$: 7 \$\$, the period is very long.

\$\$ 1: 7 = 0. bar (1428571) \$\$

\$\$ 2: 7 = 0. bar (285714) \$\$

You can see here that the sequence of digits is repeated in the two division problems. It's always like that. If you knew the sequence of digits by heart, you could stop calculating after the first decimal place. You would then have to add more digits out of your head. All six digits always appear when calculating \$\$: 7 \$\$.

What do you need decimal numbers for?

The most common use of \$\$: \$\$ and \$\$ * \$\$ with decimal numbers when shopping.

Example:

You have € 3.56 in your wallet. You would like to buy surprise eggs from it. An egg costs just € 0.44. How many eggs can you buy?

Invoice:

3,56 \$\$:\$\$ 0,44 \$\$=\$\$

356 \$\$: \$\$ 44 \$\$ = \$\$ 8 remainder 4
352

4

You don't have to calculate any further here, because you now know that you can buy 8 eggs. The correct result is \$\$ 8, bar (09) \$\$.

Even if you want to convert a fraction into a decimal number, you do the math with decimal numbers.

Example:

\$\$2/5 = 2:5 = 0,4\$\$