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In this lesson, strategies for adding, subtracting, and multiplying integers will be expanded to include decimal numbers. Although performing such operations on decimals is similar to performing them on integers, there are certain rules that need to be followed. These strategies will be developed in this lesson.

### Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

## Zosia's Rectangular Model

Zosia draws a rectangle to represent the product of two whole numbers. The rectangle has a width of units and a length of units. Additionally, the area of the rectangle is the same as the product of the two numbers, which is Zosia then wondered, What would the product be if the side length of each square was unit long? In that case, could the rectangle be used to represent Help Zosia answer the following questions.
a What is the area of a single square in the new rectangle?
b Use the small squares to find the product of and

## Place Value Chart

A place value chart is a table that displays the correct place of a digit in a number. Place value charts are useful for ensuring that digits are aligned correctly. For example, consider a decimal number.
The digits in this decimal number can be written in the place value chart to check their position. The aim here is to identify the positional values of different digits in the number accurately. It can be easier to start by writing the digits around the decimal point. Place value charts can be used when adding and subtracting decimals.

### Discussion

The rules for adding and subtracting decimals are similar to the general rules for adding and subtracting integers. The most important point is to align the decimal points correctly. For example, consider adding the following decimal numbers.
There are three steps to finding this sum.
1
Line up the Decimal Points
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The first step is to line up the decimal points. The sum is written so that the decimal points are on top of each other. The numbers are now written vertically. A place value chart can also be used to align the decimal places. 2
Place Zeros at the End of a Decimal If Needed
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The first number does not have the same number of decimal places as the second number. Place zeros so that the numbers have the same number of decimal places. Now that the decimal digits in each of the numbers are equal, it will be easier to add or subtract these numbers.
3
Add Digits with the Same Place Value
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Temporarily ignore the decimal point and add the digits that have the same place value. This is the same as adding two integers. After that, bring down the decimal point. The sum of the given numbers is
Note that the subtraction of decimals is done in the same way.

## How Long Is Zosia's Video?

Zosia likes singing. She decides to record a video of herself singing.

She records two video clips. The first video clip is minutes long and the second is minutes long.

a Zosia merges these two video clips. What is the total length of the video?
b Zosia removes minutes from the video after reviewing it. What is the final length of the video?

### Hint

a Line up the decimal points so that place-value positions correspond to each other before adding any decimal numbers.
b Line up the decimal points so that place-value positions correspond to each other before doing the subtraction. If necessary, add zeros so that the numbers have the same number of decimal places.

### Solution

a The total length of the video is the sum of the lengths of the clips. The given numbers are decimals, so the first step is to line up the decimal points so that place-value positions correspond to each other.
There is no need to add zeros to the end of either number here because they both have the same number of decimal places. Ignore the decimal point for the moment and continue adding as usual. Then, finally, add the decimal point to the sum. The sum of the decimals is or just This means that Zosia's video is minutes long.
b In the previous part it was found that the video is minutes long. Now Zosia wants to remove some parts from the video. The final length of the video will be the difference between and
Notice that the numbers do not have the same number of decimal places. To make the subtraction easier, add one zero to the end of the first number so that both numbers have an equal number of decimal places.
Now ignore the decimal points and continue subtracting as usual, then place the decimal point in the difference. The difference of the decimals is This means that the final length of the video is minutes.

## Finding the Sum and Difference of Decimals

Recall the steps to follow to add and subtract decimals. First, align the decimal numbers by their place values, one below the other. Then, perform the operation just like as if the numbers were integers. Finally, place the decimal point to the answer. Follow these steps to find the sum or difference shown in the applet. ## Estimating Products by Rounding

Estimation is used to find a value that is close to the exact value of a product. This can be done by rounding the factors in the product to the greatest place value — the first non-zero digit on the left in a number. For example, consider the product of two decimal numbers.
Estimating this product can be done in two steps.
1
Round The Factors
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The factors in the product will be rounded to the greatest place value to make it easier to compute mentally. In the first factor the digit with the greatest place value is In the other factor the digit with the greatest place value is
Next, look at the digit to the right of the greatest place.
• If the digit is less than the underlined number remains the same.
• If the digit is or greater, add to the underlined number.

Next, make all digits to the right of the greatest place value zero. For the digit to the right of the greatest place value is For the digit to the right of the greatest place value is

Estimate
Factor Greatest Place Value Digit to the Right
Tens
Hundredths
The first factor will be rounded to the nearest ten. Since is greater than or equal to add to the number in the rounded place All place values to the right of are replaced with zeros.
Similarly, for the digit to the right of the greatest place value is This is less than so remains the same and all place values to the right of have a value of zero.
The numbers have now been rounded to their greatest place values. The procedure is visualized in the following applet. 2
Multiply The Rounded Factors
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Now multiply the rounded numbers. Since is it will move the decimal point in the other factor two places to the left.

## Multiplying Decimals

Multiplying decimals has the same procedure as multiplying integers. The only difference is where the decimal point is placed in the product. The procedure will be demonstrated with an example.
The following steps can be followed to multiply decimals.
1
Multiply Ignoring the Decimal Point
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Ignore the decimal points for now and multiply the numbers as if multiplying integers. 2
Count the Number of Decimal Places in the Factors
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After the numbers are multiplied, count the total number of decimal places in each factor. The sum of the decimal places is
3
Locate the Decimal Point in the Product
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The number of decimal places in the product is the sum of the decimal places in the factors. In the previous step, it was found that there were decimal places in the example factors. This means that the product will have decimal places as well. The product of and is

## How Much Will Zosia's Brother Pay Her?

Zosia shares her video on a video sharing platform. External credits: upklyak

Mark, Zosia's brother, promises to pay Zosia for each view of her video. So far, people have watched the video.

a Estimate the amount that Mark will pay.
b Find the exact amount that Mark will pay.

### Hint

a Round the numbers to the greatest place value to make it easier to calculate mentally. Round to the nearest ten and to the nearest whole number.
b Multiply decimal numbers like multiplying whole numbers. Place the decimal point in the product by counting the number of decimal places in each factor.

### Solution

a The product of the given numbers will be the amount that Mark will pay.
Each factor will be rounded to the greatest place value to estimate this product. In the number the digit with the greatest place value is In the other factor, the digit with the greatest place value is
Now, take a look at the digit to the right of the greatest place value.
• If the digit is less than the underlined number remains the same.
• If the digit is or greater, add to the underlined number.
After that, make all digits to the right of the greatest place value zero. Starting with the first factor, the digit to the right of the underlined number is This is greater than so the digit in the rounded place is increased by and the other digit is changed to a
For the number in the tenths place is Since is greater than the number in the rounded place is increased by and the other digits are converted to In this case, the zeros can be ignored because they will not change the value of the product.
The product of and is Mark will pay about dollars.
b The whole number and the decimal number must be multiplied to find the exact amount that Mark will pay.
The steps for multiplying these numbers is the same as multiplying two decimal numbers.
1. Multiply the numbers as if the decimal factor were a whole number.
2. Count the number of decimal places in the decimal factor.
3. The number of decimal places in the product is the same as the number of decimal places in the decimal factor.
Start by multiplying and The product of the numbers is Since there are two decimal places in the product of and must be a decimal number with two decimal places.
The product is This means that Mark will pay Zosia dollars.

## How Much Does The Platform Pay?

In addition to the amount that Mark gives her, Zosia also earns money per view from the video sharing platform.

The platform pays times what Mark paid for views.

a Recall that Mark paid Estimate the amount that the platform pays by rounding the factors to their greatest place values.
b Find the exact amount that the platform pays.

### Hint

a Round the numbers to the greatest place value to make it easier to calculate mentally. In this case, round to the nearest hundred and to the nearest tenth.
b Multiply as though the decimals were whole numbers. Place the decimal point in the product by counting the number of decimal places in each factor.

### Solution

a The product of and is what the platform pays for views. The factors in this product will be rounded to the greatest place value to find an estimate for the product.
The digit to the right of the greatest place value determines how the number is rounded.
• If the digit is less than the number in the rounded place remains the same.
• If the digit is or greater, add to the number in the rounded place.
After rounding, remove any decimal digits and convert the non-decimal digits to the right of the greatest place value into Since the digit with the greatest place value is the leftmost non-zero digit int he number, has its greatest place value in the hundreds place and in the tenths place.
In the digit in the tens place is Since is less than the digit in the hundreds place remains the same and the other digits are replaced with This results in the estimated number
In the number in the hundredths place is Since is less than the number in the tenth place is kept unchanged. Then, the other digits are replaced with Since these zeros do not change the value of the number, they can be omitted.
Multiplying a number by means dividing the number by because This means that the product of and is
Evaluate
The platform pays about for views.
b Recall the steps when multiplying two decimal numbers.
1. Multiply as if the factors were whole numbers.
2. Count the number of decimal places in each factor.
3. The number of decimal places in the product is the sum of the number of decimal places in each factor.
Start by multiplying and There are decimal places in the first factor and there are decimal places This means that the product of the decimal numbers must be a decimal number with decimal places. The product of the numbers is Since money is represented by numbers with two decimal places, is rounded to two decimal places.
The platform pays dollars for views. This is close the estimate found in Part A, so this answer is reasonable.

## Finding the Product of Decimals

Multiply the decimal numbers. Make sure that the decimal point is placed correctly! ## Modeling the Product of Numbers

Mathematical operations such as addition, subtraction, and multiplication are performed on decimal numbers in a similar way that they are performed on integers. The most important point here is where to place the decimal point. Now consider Zosia's rectangle that can be used to represent the product of two numbers. Zosia changes the side length of each square to Answer the following questions according to the given model.
a What is the area of a single square in the new rectangle?
b Use the small squares to find the product of and

### Hint

a The area of a square is the product of its two sides.
b Use the area of the squares to find the area of the rectangle.

### Solution

a The given rectangle consists of six squares. Each square has a side length of unit. Since the area of a square is the product of its two sides, this square has an area of square units.
Now multiply the decimals as if they were whole numbers.
Then, count the number of decimal places in each factor and add them.
The number of decimal places in the product must be Insert zeros to the left of and move the decimal point two places to the left.
The area of a single square is square unit.
b Notice that the area of the rectangle is the product of and This means that finding the area of the rectangle is the same as finding In the previous part, it is found that the area of a each square is Since there are small squares, the area of the rectangle can be found by multiplying by Multiply the numbers as if they were whole numbers.
The sum of the number of decimal places in each factor is This means that the number of decimal places in the product must be
The area of the rectangle is This is also equal to the product of and Alternatively, the Commutative and Associative Properties of Multiplication can be used to find the product.
Rewrite
Either way, the answer is the same!
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