Print bookPrint book

LESSON: Dividing by Decimals

Site: MN Partnership for Collaborative Curriculum
Course: Mathematics Essentials Q1
Book: LESSON: Dividing by Decimals
Printed by: Guest user
Date: Thursday, November 21, 2024, 2:54 PM

Description

Dividing by Decimals

Table of contents

INTRODUCTION

Dividing by Decimals


READ: Dividing Decimals by Whole Numbers

Divide Decimals By Whole Numbers

To divide means to split up into equal parts. You have learned how to divide whole numbers in an earlier lesson. Now we are going to learn how to divide decimals by whole numbers.When we divide a decimal by a whole number, we are looking at taking that decimal and splitting it up into sections.

Example

4.64 \div 2 = ______

The first thing that we need to figure out when working with a problem like this is which number is being divided by which number. In this problem, the two is the divisor. Remember that the divisor goes outside of the division box. The dividend is the value that goes inside the division box. It is the number that you are actually dividing.

2 \overline{)4.64 \;}

We want to divide this decimal into two parts. We can complete this division by thinking of this problem as whole number division. We divide the two into each number and then we will insert the decimal point when finished. Here is our problem.

& \overset{232}{2\overline{ ) 4.64 \;}}

Finally, we can insert the decimal point into the quotient. We do this by bringing up the decimal point from its place in the division box right into the quotient. See the arrow in this example to understand it better, and here are the numbers for each step of the division.

& \overset{\overset{ \ 2.32} {\uparrow}}{2 \overline{ ) 4.64 \;}}\\ & \quad \underline{4 \quad }\\ & \quad \ 0 6\\ & \quad \ \underline{ \ \ 6 \ }\\ & \qquad 04

Our answer is 2.32.


How do we divide decimals by whole numbers when there is a remainder?

Example

14.9 \div 5 = ______

The first thing that we can do is to set up this problem in a division box. The five is the divisor and the 14.9 is the dividend.

5 \overline{)14.9 \;}

Next we start our division. Five goes into fourteen twice, with four left over. Then we bring down the 9. Five goes into 49, 9 times with four left over. That four is our remainder.

& \overset{2.9 \ \ } { \ 5 \overline{ ) {14.9}} \ {r \ 4} \;}\\ & \underline{- \ 10 \ \; \;}\\ & \quad \ 49\\ & \ \underline{- \ 45 \; \;}\\ & \quad \ \ \ 4

However, when we work with decimals, we don’t want to have a remainder. We can use a zero as a placeholder. In this example, we can add a zero to the dividend and then see if we can finish the division. We add a zero and combine that with the four so we have 40. Five divides into forty eight times.

Here is what that would look like.

& \overset{ \quad 2.98}{5 \overline{ ) {14.90 \;}}}\\ & \underline{-10 \ \ }\\ & \quad \ 49\\ & \ \ \underline{-45 \ }\\ & \qquad 40

Our final answer is 2.98.

When working with decimals, you always want to add zeros as placeholders so that you can be sure that the decimal is as accurate as it can be. Remember that a decimal shows a part of a whole. We can make that part as specific as we can possible make it.

READ: Divide Decimals by Decimals by Rewriting Divisors

Divide Decimals by Decimals by Rewriting Divisors as Whole Numbers

In our introductory problem, Miles is working on dividing up sand. If you were going to complete this problem yourself, you would need to know how to divide decimals by decimals.

How can we divide a decimal by a decimal?

To divide a decimal by a decimal, we have to rewrite the divisor. Remember that the divisor is the number that is outside of the division box. The dividend is the number that is inside the division box.

Let’s look at an example.

Example

2.6 \overline{)10.4 \;}

In this problem, 2.6 is our divisor and 10.4 is our dividend. We have a decimal being divided into a decimal. Whew! This seems pretty complicated. We can make our work simpler by rewriting the divisor as a whole number.

How can we do this?

Think back to the work we did in the last section when we multiplied by a power of ten. When we multiply a decimal by a power of ten we move the decimal point one place to the right.

We can do the same thing with our divisor. We can multiply 2.6 times 10 and make it a whole number. It will be a lot easier to divide by a whole number.

2.6 \times 10 = 26

What about the dividend?

Because we multiplied the divisor by 10, we also need to multiply the dividend by 10. This is the only way that it works to rewrite a divisor.

10.4 \times 10 = 104

Now we have a new problem to work with.

& \overset{ \qquad 4}{26\overline{ ) 104 \;}}

Our answer is 4.

What about if we have two decimal places in the divisor?

Example

.45 \overline{)1.35 \;}In this example, we want to make our divisor .45 into a whole number by multiplying it by a power of ten. We can multiply it by 100 to make it a whole number. Then we can do the same thing to the dividend.

Here is our new problem and quotient.

& \overset{ \qquad 3}{45\overline{ ) 135 \;}}

Now it is time for you to practice a few. Rewrite each divisor and dividend by multiplying them by a power of ten. Then find the quotient.

  1. 1.2 \overline{)4.8 \;}
  2. 5.67 \overline{)11.34 \;}
  3. 6.98 \overline{)13.96 \;}


Take a minute to check your rewrite and quotient. Is your work accurate?

READ: Using Additional Zero Placeholders

Find Quotients of Decimals by Using Additional Zero Placeholders

The decimals that we divided in the last section were all evenly divisible. This means that we had whole number quotients. We didn’t have any decimal quotients.

What can we do if a decimal is not evenly divisible by another decimal?

If you think back, we worked on some of these when we divided decimals by whole numbers. When a decimal was not evenly divisible by a whole number, we had to use a zero placeholder to complete the division.

Here is a blast from the past problem.

Example

5 \overline{)13.6 \;}

When we divided 13.6 by 5, we ended up with a 1 at the end of the division. Then we were able to add a zero placeholder and finish finding a decimal quotient. Here is what this looked like.

& \overset{ \quad 2.72}{5 \overline{ ) {13.60 \;}}}\\ & \underline{-10 \;\;}\\ & \quad \ 36\\ & \ \ \underline{-35\;\;}\\ & \qquad 1 - \ \text{here is where we added the zero placeholder}\\ & \qquad 10\\ & \quad \ \underline{-10}\\ & \qquad \ \ 0

We add zero placeholders when we divide decimals by decimals too.

Example

1.2 \overline{)2.79 \;}

The first thing that we need to do is to multiply the divisor and the dividend by a base ten number to make the divisor a whole number. We can multiply both by 10 to accomplish this goal.

12 \overline{)27.9 \;}

Now we can divide.

& \overset{ \quad \ \ 2.3}{12 \overline{ ) {27.9 \;}}}\\ & \ \underline{-24 \;\;}\\ & \quad \ \ 39\\ & \quad \underline{-36}\\ & \qquad \ 3

Here is where we have a problem. We have a remainder of 3. We don’t want to have a remainder, so we have to add a zero placeholder to the problem so that we can divide it evenly.

& \overset{ \quad \ \ 2.32}{12 \overline{ ) {27.90 \;}}}\\ & \ \underline{-24\;\;}\\ & \quad \ \ 39\\ & \quad \underline{-36\;\;}\\ & \qquad \ 30\\ & \quad \ \ \underline{-24}\\ & \qquad \quad 6

Uh Oh! We still have a remainder, so we can add another zero placeholder.

& \overset{ \quad \ 2.325}{12 \overline{ ) {27.900 \;}}}\\ & \ \ \underline{-24\;\;}\\ & \ \quad \ \ 39\\ & \ \quad \underline{-36\;\;}\\ & \ \qquad \ 30\\ & \ \quad \ \ \underline{-24\;\;}\\ & \ \qquad \quad 60\\ & \ \qquad \ \underline{-60\;\;}\\ & \ \qquad \quad \ \ 0

Sometimes, you will need to add more than one zero. The key is to use the zero placeholders to find a quotient that is even without a remainder.


EXAMPLE 1

EXAMPLE 2

EXAMPLE 3

CHECK Yourself! Dividing Decimals