4. Add, Subtract, and Multiply Decimals
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Chapter 3
4.
Add, Subtract, and Multiply Decimals
This lesson offers a comprehensive guide to understanding how to add, subtract, and multiply decimals. It uses everyday scenarios, such as calculating video lengths and estimating payments, to make the concepts relatable and practical. The lesson emphasizes the importance of aligning decimal points correctly when adding or subtracting and counting the number of decimal places when multiplying. It also provides tips on rounding numbers to the greatest place value for easier mental calculations. The aim is to equip you with the skills to handle decimal operations in various contexts, from academic exercises to real-world applications like budgeting or measuring.
Show less Show more expand_more| Student Learning Objectives: |
|---|
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| | 12 Theory slides |
| | 13 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Add, Subtract, and Multiply Decimals
Slide of 12
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.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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Challenge
A Rectangular Model
A rectangle can be used to represent the product of two whole numbers. The rectangle has a width of 2 units and a length of 3 units. The area of the rectangle is the product of the two numbers, 6.
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b
Use the small squares to find the product of 0.2 and 0.3.
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Discussion
Place Value Chart
The 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. Decimal Number 673.452 The digits of the number are written in the place value chart below.
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Discussion
Adding and Subtracting Decimals
When adding and subtracting decimals, line up the decimal points before adding adding and subtracting as with integer numbers. Consider the following sum. 3.4 + 12.802 There are three steps to finding this sum.
1
Line up the Decimal Points
expand_more 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
expand_more The first number does not have the same number of visible 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
expand_more Temporarily ignore the decimal point and add the digits that have the same place value. This is the same as adding two integers. After completing all the calculations, bring down the decimal point.
The sum of the given numbers is 16.202.
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Example
How Long Is Zosia's Video?
Zosia recorded a video of herself singing.
She recorded two video clips. The first video clip is 4.56 minutes long and the second is 5.54 minutes long.
a
Zosia merged these two video clips. What is the total length of the video?
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b
If Zosia removes 1.27 minutes from the video, what is the final length of the video?
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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.
& 4.56 + & 5.54 There is no need to add zeros to the end of either number 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 10.10, or just 10.1. This means that Zosia's video is 10.1 minutes long.
b
In the previous part we found that the video is 10.1 minutes long. Now Zosia wants to remove some parts from the video. The final length of the video will be the difference between 10.1 and 1.27.
& 10 .1 - & 1 .27 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. & 10 .1 0 - & 1 .27 Now ignore the decimal points and continue subtracting as usual, then place the decimal point in the difference.
The difference of the decimals is 8.83. This means that the final length of the video is 8.83 minutes.
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Pop Quiz
Finding the Sum and Difference of Decimals
Find the sum or difference shown of the given decimal numbers.
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Discussion
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. 35.92 * 0.014 We can estimate this product in two steps.
1
Round The Factors
expand_more The factors in the product will be rounded to the greatest place value to make it easier to compute mentally. In the first factor 35.92, the digit with the greatest place value is 3. In the other factor 0.014, the digit with the greatest place value is 1. 35.92 * 0.014
Next, look at the digit to the right of the greatest place.
- If the digit is less than 5, the underlined number remains the same.
- If the digit is 5 or greater, add 1 to the underlined number.
Next, make all digits to the right of the greatest place value 0. For 35.92, the digit to the right of the greatest place value is 5. For 0.014, the digit to the right of the greatest place value is 4.
| Estimate 35. 92 * 0.014 | ||
|---|---|---|
| Factor | Greatest Place Value | Digit to the Right |
| 3 5.92 | Tens | 5 |
| 0.01 4 | Hundredths | 4 |
The first factor will be rounded to the nearest ten. Since 5 is greater than or equal to 5, add 1 to the number in the rounded place 3. All place values to the right of 3 are replaced with 0. 35.92 ≈ 40.00 (or 40) Similarly, for 0.014, the digit to the right of the greatest place value is 4. This is less than 5, so the 1 remains the same and all place values to the right of 1 become 0. 0.014 ≈ 0.010 (or 0.01) The numbers have now been rounded to their greatest place values. The procedure is visualized in the following applet.
2
Multiply The Rounded Factors
expand_more Now multiply the rounded numbers. Since 0.01 is 10^(- 2), it will move the decimal point in the other factor two places to the left. cr & 40 * & 0.01 & 0.40
The given product is about 0.4. 35.92 * 0.014 ≈ 0.4
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Discussion
Multiplying Decimals
Multiplying decimals is similar to multiplying integers. Multiply the numbers first, then place the decimal point in the product. Consider the example product. 1.69 * 3.7 The following steps can be followed to multiply decimals.
1
Multiply Ignoring the Decimal Point
expand_more Ignore the decimal points for now and multiply the numbers as if they were integers.
2
Count the Number of Decimal Places in the Factors
expand_more Count the total number of decimal places in each factor.
Add these numbers to find the total number of decimal places in the product. The sum of the decimal places for this product is 3.
3
Locate the Decimal Point in the Product
expand_more The number of decimal places in the product is the sum of the decimal places in the factors. There are a total of 3 decimal places in the example factors, so the product will have 3 decimal places as well.
The product of 1.69 and 3.7 is 6.253.
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Example
How Much Has Zosia Made?
Zosia designs and sells digital sticker packs.
External credits: upklyak
She earns $1.75 for each sticker pack she sells. So far, 59 people purchased a sticker pack.
a
Estimate how much money Zosia has made.
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b
Find the exact amount of money that Zosia has made.
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Hint
a
Round the numbers to the greatest place value to make it easier to calculate mentally. Round 59 to the nearest ten and 1.75 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
We can estimate the amount of money Zosia has earned by rounding the given numbers.
59 * 1.75 Let's round each factor to the greatest place value to estimate the product. For the number 59, the digit with the greatest place value is 5. In the other factor, the digit with the greatest place value is 1. 59 * 1.75 Now let's take a look at the digit to the right of the greatest place values.
- If the digit is less than 5, the underlined number remains the same.
- If the digit is 5 or greater, add 1 to the underlined number.
After that, make all digits to the right of the greatest place value 0. Starting with the first factor, the digit to the right of the underlined number is 9. This is greater than 5, so the digit in the rounded place goes up by 1 and the other digit is changed to a 0. ccc 5 9 & * &1.75 ↓ & & 60 && For 1.75, the number in the tenths place is 7. Since 7 is greater than 5, the number in the rounded place 1 is increased by 1 and the other digits are converted to 0. Note that we can ignore these zeros because they come at the end of the decimal and do not change the value of the product. ccc 59 & * &1. 75 ↓ & & ↓ 60 & * &2 The product of 60 and 2 is 120. Zosia has earned about $120 from her sticker packs.
b
Let's multiply the whole number and the decimal number to find the exact amount of money that Zosia has made.
59 * 1.75 The steps for multiplying these numbers is the same as multiplying two decimal numbers.
- Multiply the numbers as if the decimal factor were a whole number.
- Count the number of decimal places in the decimal factor.
- The number of decimal places in the product is the same as the number of decimal places in the decimal factor.
Start by multiplying 59 and 175.
The product of the numbers is 10 325. Since there are two decimal places in 1.75, the product of 59 and 1.75 must be a decimal number with two decimal places. r 59 * 1. 75 103. 25 l→ → ← r 0 decimal places + 2 decimal places 2 decimal places The product is 103.25. This means that Zosia has earned $103.25 from her digital sticker packs.
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Example
How Much Does The Platform Pay?
Zosia posted a video of herself singing on a video sharing platform. She earns money each time someone watches it.
This week, the platform paid her 0.113 times the money she earned for selling digital sticker packs.
a
Recall that Zosia made $103.25 on selling sticker packs. Estimate the amount that the she earned on the video sharing platform this week by rounding the factors to their greatest place values.
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b
Find the exact amount that she made from the video sharing platform.
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Hint
a
Round the numbers to the greatest place value to make it easier to calculate mentally. In this case, round 103.25 to the nearest hundred and 0.113 to the nearest tenth.
Solution
a
This week, Zosia earned $103.25 * 0.113 from her video. Let's round the factors to the greatest place value to find an estimate for the product.
103.25 * 0.113 The digit to the right of the greatest place value determines how the number is rounded.
- If the digit is less than 5, the number in the rounded place remains the same.
- If the digit is 5 or greater, add 1 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 0. Since the digit with the greatest place value is the leftmost non-zero digit int he number, 103.25 has its greatest place value in the hundreds place and 0.113 in the tenths place. 103.25 * 0.113 In 103.25, the digit in the tens place is 0. Since 0 is less than 5, the digit in the hundreds place 1 remains the same and the other digits are replaced with 0. This results in the estimated number 100. ccc 1 03.25 & *& 0.113 ↓ && 100 && In 0.113, the number in the hundredths place is 1. Since 1 is less than 5, the number in the tenth place 1 is kept unchanged. Then, the other digits are replaced with 0. Since these zeros do not change the value of the number, we can ignore them. ccc 103.25 & *& 0.1 13 ↓ &&↓ 100 & * & 0.1 Multiplying a number by 0.1 means dividing the number by 10 because 0.1 = 110. This means that the product of 100 and 0.1 is 10.
100 * 0.1
▼
Evaluate
10
The platform pays about $10 for 59 views.
b
Recall the steps to follow when multiplying two decimal numbers.
- Multiply as if the factors were whole numbers.
- Count the number of decimal places in each factor.
- The number of decimal places in the product is the sum of the number of decimal places in each factor.
Start by multiplying 10325 and 113.
There are 2 decimal places in the first factor 103.25, and there are 3 decimal places 0.113. This means that the product of the decimal numbers must be a decimal number with 2+3=5 decimal places.
The product of the numbers is 11.66725. Since money is represented by numbers with two decimal places, let's round 11.66275 to two decimal places. 11.66275 & Round ⟶ & 11.67 Zosia earned $11.67 from view of her singing video last week. This is close the estimate we found in Part A, so this answer is reasonable.
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Pop Quiz
Finding the Product of Decimals
Multiply the decimal numbers. Make sure that the decimal point is placed correctly!
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Closure
Modeling the Product of Numbers
Adding, subtracting, and multiplying decimal numbers works much like it does with whole numbers. The key is placing the decimal point correctly. Let's take another look at the rectangle model for multiplying two numbers.
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b
Use the small squares to find the product of 0.2 and 0.3.
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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 0.1 unit.
Since the area of a square is the product of its two sides, this square has an area of 0.1* 0.1 square units. 0.1 * 0.1 Now let's multiply the decimals as if they were whole numbers. 1 * 1 =1 Then we count the number of decimal places in each factor and add them. rr 0. 1 * 0. 1 & l→ → & r 1 decimal place + 1 decimal place 2 decimal places The number of decimal places in the product must be 2. Let's add zeros to the left of 1 and move the decimal point two places to the left. rr 0. 1 * 0. 1 0. 01 & l→ → ← & r 1 decimal place + 1 decimal place 2 decimal places The area of a single square is 0.01 square unit.
b
Notice that the area of the rectangle is the product of 0.2 and 0.3. This means that finding the area of the rectangle is the same as finding 0.2 * 0.3.
In the previous part, we found that the area of a each square is 0.01 square unit.
Since there are 6 small squares, the area of the rectangle can be found by multiplying 0.01 by 6. Multiply the numbers as if they were whole numbers. 1 * 6 = 6 The sum of the number of decimal places in each factor is 2. This means that the number of decimal places in the product must be 2. rr 0. 01 * 6 0. 06 & l→ → ← & rr & 2 decimal places + & 0 decimal places & 2 decimal places The area of the rectangle is 0.06 square units. This is also equal to the product of 0.2 and 0.3. Alternatively, we can use the Commutative and Associative Properties of Multiplication to find the product.
0.2 * 0.3
▼
Rewrite
Rewrite
Rewrite 0.2 as 2 * 0.1
(2* 0.1) * 0.3
Rewrite
Rewrite 0.3 as 3 * 0.1
(2* 0.1) * (3* 0.1)
AssociativePropMult
Associative Property of Multiplication
2* (0.1 * 3* 0.1)
CommutativePropMult
Commutative Property of Multiplication
2* (3 * 0.1 * 0.1)
AssociativePropMult
Associative Property of Multiplication
(2* 3) * (0.1* 0.1)
Multiply
Multiply
6 * 0.01
Multiply
Multiply
0.06
Either way, the answer is the same!
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Add, Subtract, and Multiply Decimals
Exercises
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We want to add a whole number and a decimal number. c 11.22 + 12 We can do this by lining up the digits according to their place values. Since the numbers do not have the same number of decimal places, we need to insert zeros so that each place has a digit. cr & 11.22 + & 12. 00 Now we can add the numbers just like we do with whole numbers. Then we will bring down the decimal point and place it into the answer.
The sum is 23.22.
This time we will add two numbers with the same number of decimal places.
c
39.59 + 41.41
Let's line up the digits according to their place values. We do not need to add any zeros this time.
cr
& 39. 59
+ & 41.41
Now we will add the numbers as if they are whole numbers, then place the decimal point in the answer at the end of the process.
The sum of the numbers is 81.00. The zeros to the right of the decimal point do not affect the value of the number, so we can remove them. The answer is then 81.
We are given two decimal numbers that have a different number of decimal places.
c
5.7 + 89.332
Like before, let's start by lining up the digits according to their place values. We need to insert zeros so that each place has a digit because these numbers do not have the same number of decimal places.
cr
& 5.7 00
+ & 89.332
Now we add the numbers as if they are whole numbers, then place the decimal point in the answer at the end of the process.
The numbers add up to 95.032.
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Solution
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We want to subtract a decimal number from a whole number. c 29-9.23 The numbers do not have the same number of decimal places. Recall that placing zeros to the right of a decimal point does not affect the value of the number. This is why we can insert zeros so that each place has a digit. Let's line up the numbers. cr & 29. 00 - & 9.23 Now we can subtract the numbers just like we do with whole numbers. Then we can place the decimal point in the answer.
The difference is 19.77.
We are asked to find the following difference.
c
4.567 - 2.725
Let's line up the digits according to their place values. This time we will not need to add any zeros because the numbers have the same number of decimal places.
cr
& 4.567
- & 2.725
Let's subtract the numbers as if they are whole numbers. We will place the decimal point in the answer at the end of the process.
The difference of the numbers is 1.842.
Let's find the given difference.
c
8.83 - 2.7
We will start by lining up the digits according to their place values. We need to add a zero so that each place has a digit since the numbers do not have the same number of decimal places.
cr
& 8. 83
- & 2.7 0
Now that the numbers have the same number of decimal places, we can subtract them!
The difference is 6.13.
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Solution
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We want to evaluate the given expression. 25.92 + 17.25- 9.243 We can add the first two decimals, then subtract the third decimal from their sum. (25.92 + 17.25)- 9.243 Let's find the sum of the first two decimals. We do this by lining up the decimal points before adding the numbers as usual.
The sum of the first two numbers is 43.17. Now we will subtract 9.243 from 43.17. First we write the numbers vertically so that the place values line up. Let's also insert a zero to the right of 43.17 so that the numbers have the same number of decimal places.
As a result, we found that the given expression is 33.927.
Consider the next expression. 33.06 - 7.3 + 9.24 Let's start by subtracting 7.3 from 33.06. Then we will add the third number to the difference. (33.06 - 7.3 ) + 9.24 We will line up the decimal points. Since the number of decimal places are different for these numbers, we insert a zero to the right of 7.3. This will ensure that the digits with the same place value come under each other. Let's subtract 7.30 from 33.06!
Now we will add 25.76 to 9.24. Just like before, we will start by writing the numbers vertically so that the decimal values line up.
We found that the given expression is 35.00, or 35.
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We want to estimate the given product. 12.31 * 6 We can see that the second factor is already a whole number. We will round the first factor to the greatest place value to make our estimation. This will make it easier to use mental math. The first non-zero digit on the left in 12.31 is the 1 in the tens place. This is the greatest place value of the number. 12.31 * 6 Now we look at the digit to the right of the place that is being rounded.
- If the digit is less than 5, then the number in the rounded place remains the same.
- If the digit is 5 or greater, add 1 to the number in the rounded place.
After that, we remove any decimal digits and convert the non-decimal digits to the right of the greatest place value into 0. Let's round 12.31!
We can now multiply 10 and 6. 10 * 6 = 60 Since 10* 6 is 60, we can say that the product of 12.31 and 6 is about 60.
Let's check our answer with a calculator. We will multiply 12.31 by 6 and compare the product with our estimation.
The product of 12.31 and 6 is 73.86. This is pretty close to our approximation, so we can say that our answer makes sense.
We want to estimate the given product. 36.47 * 22.98 To make our estimation, we will round each factor to the greatest place value. The digit with the greatest place value in 36.47 is 3, and it is 2 in the second factor. 36.47 * 22.98 The greatest place value of each number is the tens place, so we need to look at the digits in the ones places.
- If the digit is less than 5, then the underlined number remains the same.
- If the digit is 5 or greater, add 1 to the underlined number.
After that, we remove any decimal digits and convert the non-decimal digits to the right of the greatest place value into 0.
We can now multiply 40 and 20. Notice that both of these numbers are multiples of 10. We can use the Commutative and Associative Properties of Multiplication to mentally find this product.
We found that the product of 36.47 and 22.98 is about 800.
We can check our answer with a calculator. We will multiply 13.92 by 2.7 and compare the product with our estimation.
The product of the numbers is 837.8508. We can say that our answer is correct because our estimation is close to the actual product.
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We want to multiply a decimal number by a whole number. 0.017 * 8 Let's recall the steps we follow when multiplying decimals.
- Multiply the numbers as if the decimal factors were whole numbers.
- Count the number of decimal places in each factor.
- The number of decimal places in the product is the sum of the number of decimal places in the factors.
We can start by multiplying 17 and 8. cr & 1 7 * & 8 & 1 3 6 The product of the numbers is 136. Now let's count the number of decimal places in each factor. cr & 0. 017 * & 8 & 136 l→ → r 3 decimal places + 0 decimal places 3 decimal places The sum is equal to 3. This means that the product must have three decimal places. cr & 0. 017 * & 8 & 0. 136 l→ → ← r 3 decimal places + 0 decimal places 3 decimal places The product of the given numbers is 0.136.
Let's find the given product.
0.65 * 2.22
We first ignore the decimal points and multiply, just like we do with two whole numbers.
Then we place the decimal point in the product by counting the number of decimal places in each factor.
The product of 0.65 and 2.22 is 1.4430. Since the zeros to the right of a decimal point do not affect the value of the number, we can ignore the zero at the end of the number. This means that 1.443 is also an answer.
We want to calculate the product of 17.1 and 3.012.
17.1 * 3.012
The first thing we will do is to ignore the decimal points for now to multiply the numbers just as we do with two whole numbers.
Now we place the decimal point in the product by counting the number of decimal places in each factor.
The product of 3.012 and 17.1 is 51.5052.
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{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["13.25"]}}
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Our expression includes multiple operations, so let's start by recalling the order of operations. Note that expressions inside parentheses are evaluated first, followed by exponents, then multiplication and division, and then finally, addition and subtraction. Let's consider our expression again. 20.23(8.55-3.25) For the given expression, we will evaluate the expression inside the parentheses completely before multiplying. Let's first subtract 3.25 from 8.55. We can do this by lining up the decimal points so that place-value positions correspond to each other. cr & 8 . 5 5 -& 3 . 2 5 & 5 . 3 0 The difference in the parentheses is equal to 5.30, or simply 5.3. We will now multiply this number by 20.23. Let's ignore the decimal points and multiply the numbers. cr & 2 0 2 3 * & 5 3 & 6 0 6 9 + & 1 0 1 1 5 0 & 1 0 7 2 1 9 Next we will place the decimal point in the product by counting the number of decimal places in each factor. cr & 20. 23 * & 5. 3 & 6069 + & 101150 & 107. 219 l ⟶ ⟶ ⟵ cl & 2 decimal places + & 1 decimal place & 3 decimal places The expression is equal to 107.219.
Once again, let's examine the given expression to decide which operation will be done first. 0.25 + 2.5* 5.2 For this expression, we will first perform the multiplication, then the addition. Let's multiply 2.5 by 5.2. Recall that we ignore the decimal points and multiply the numbers as usual. cr & 2 5 * & 5 2 & 5 0 + & 1 2 5 0 & 1 3 0 0 Next we place the decimal point in the product by counting the number of decimal places in each factor. cr & 2. 5 * & 5. 2 & 50 + & 1250 & 13. 00 l ⟶ ⟶ ⟵ cl & 1 decimal place + & 1 decimal place & 2 decimal places The product of the numbers is 13.00. We will now add 0.25 to this number. We line up the decimal points so that place-value positions line up as well. cr & 1 3 . 0 0 +& 0 . 2 5 & 1 3 . 2 5 The expression is equal to 13.25.
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Add, Subtract, and Multiply Decimals
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