Multiplying Decimals Cheat Sheet? The 135 Top Answers

You should become efficient at using the four basic operations with decimal numbers—addition, subtraction, multiplication, and division.

Add and subtract decimals

example 1

Add 23.6 + 1.75 + 300.002.

Adding zeros can solve the problem more easily.

example 2

Subtract 54.26 – 1.1.

Example 3

Subtract 78.9 – 37.43.

An integer has an understood decimal point to the right of it.

example 4

Subtract 17 – 8.43.

Multiply decimals

To multiply decimals, simply multiply as usual. Then count the total number of digits above the line, to the right of all decimal points. Put your decimal point in your answer with the same number of digits to the right of it as above the line.

Example 5

Multiply 40.012 × 3.1.

Divide decimals

Dividing decimals is the same as dividing other numbers, except that if the divisor (the number you’re dividing by) is a decimal, shift it right as many places as necessary until it’s a whole number. Then move the decimal point in the dividend (the number to divide into) the same number of places. Sometimes you need to add zeros to the dividend (the number in the division bracket). Note that the decimal point in the quotient (answer) is above that in the dividend.

Example 6

Split

Example 7

Split

example 8

Split

Multiplying and dividing is a very important and fundamental math skill that every student should master. However, problems that many encounter in reality, and not all questions that are asked of you, are nice and even. Although calculators have made our jobs easier, knowing how to multiply and divide decimals is a skill that all students should master.

Multiply decimals:

Multiplying decimals is actually very similar to multiplying whole numbers, except that you have to remember to take into account all the decimal places that exist in the 2 numbers you are multiplying together.

Let’s work on an example. When multiplying decimal points, do not align the numbers with the decimal points. Instead, align the numbers all the way from the right and multiply each number as you would if you were multiplying whole numbers. Once you’ve summarized your answers, it’s important to put the decimal point back into the answer. Since both of the numbers you multiplied are decimal, you can assume that the product of the two numbers is a decimal. To determine the decimal place, count the sum of the decimal places in the 2 numbers to be multiplied and shift the decimal place to the right of the product. The number of digits to be shifted should equal the sum of the decimal places of the two numbers to be multiplied.

Dividing decimals:

Likewise, dividing decimal numbers is like dividing whole numbers. When dividing decimals, remember that the position of the decimal in the dividend determines the decimal point in the result.

If the divisor is not a whole number, shift the decimal place in the divisor to the right until it becomes a whole number. Likewise, move the same number of decimal places in the dividend. (If your divisor is an integer, you can skip this step and divide normally.) If the divisor doesn’t go into dividends evenly, add a 0 to the right of the dividend and add a decimal to the quotient of 2 numbers (This Decimal determines where the decimal is in the quotient!). Continue dividing as usual until the result is consistent or a repeating pattern is noticed.

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Article overview

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To multiply decimals, align the numbers vertically so the decimals are in the same position. Then multiply as usual, temporarily ignoring the decimals. When you have your answer, count how many numbers are in each of the original factors to the right of the decimal point. For example, 0.02 is 2 places to the right of the decimal point. If you multiplied 0.02 × 0.4, the total to the right of the decimal point is 3. Take that total, and then move the decimal point in your answer that number of places. If you move the decimal point past the first number of the answer, add zeros after the decimal point until you reach the numbers. If you want to learn how to check your answer to see if it’s correct, keep reading the article!

When multiplying a decimal by an integer, removing the decimal point can make multiplying easier, but where does the decimal point go when you want to get your result? In this tutorial you will learn how to estimate the position of the decimal point after multiplying!

Multiply decimals

23.8×4.1

16.903×2.2

Step 1:

Step 2:

Step 3:

Step 4:

To multiply decimals, we use different steps than adding or subtracting decimals. In fact, you don’t need to align the decimals at all. 1) To solve this problem, we will first pretend that there are no decimals at all. Start by solving the question 238 x 41. Now we need to put the decimal back in. Let’s think about it: 23.8 is about 20. And 4.1 is about 4. So we could estimate that the answer should be about 20 x 4 = 80. That would mean that the decimal should be after the 97, to give us a score of 97.58. Therefore 23.8 x 4.1 = 97.58. However, there is a rule you can use to know where to put the decimal without having to guess. Look at how many numbers are behind the decimal point in each of the numbers we multiplied. There is a digit after each decimal point, resulting in two numbers. This tells us that there must also be two numbers after the decimal point in our answer. Ex. 2) Start by multiplying 16903 x 22 Now look at the number of digits after the decimal point. So 16.903 x 2.2 = 37.1866 Let’s check our answer against an estimate: 16.903 is about 20 2.2 is about 220 x 2 = 40 We can see from our estimate that 37.1866 is a reasonable answer. Multiply the numbers and ignore the decimals. Add up how many decimal places come in both factors Digits come after the decimal point in the answer. (Optional) Estimate the answer to see if your answer and the placement of your decimal point are appropriate.

Multiplication of a decimal by 10, 100, 1000

The working rule of multiplying a decimal number by 10, 100, 1000, etc. is:

If the multiplier is 10, 100, or 1000, we shift the decimal point to the right by as many places as the number of zeros after 1 in the multiplier.

1. To multiply a decimal number by 10, move the decimal point in the multiplicand one place to the right.

For example:

(i) 834.7×10

Here we’ve multiplied the number 834.7 by 10, so we’re shifting 1 place to the right.

Or,

834.7×10

= (8347/10) × 10

= 8347/1

= 8347

(ii) 73.5 × 10 = 735

(iii) 100.9 × 10 = 1009

2. To multiply a decimal number by 100, move the decimal point in the multiplicand two places to the right.

For example:

(i) 98.26×100

Here we’ve multiplied the number 98.26 by 100, so we’re moving 2 places to the right.

Or,

98.26×100

= (9826/100) × 100

= 9826/1

= 9826

(ii) 6.006 × 100 = 600.6

(iii) 0.77 × 100 = 77

3. To multiply a decimal number by 1000, shift the decimal point in the multiplicand three places to the right.

For example:

(i) 793.41×1000

Here we’ve multiplied the number 793.41 by 1000, so we’re moving 3 places to the right.

Or,

793.41×1000

= (79341/100) × 1000

= 79341 × 10

= 793410

(ii) 9.15 × 1000 = 9150

(iii) 0.017 × 1000 = 17

4. To multiply a decimal number by 10, 100, 1000, etc., shift the decimal point of the multiplicand to the right as many places as there are zeros in the multiplier.

For example:

(i) 1854.347×10

Here we’ve multiplied the number by 10, so we’re moving 1 place to the right.

(ii) 72.4×100

Here there is only one decimal place and 100 has two zeros, so we put a zero at the end of the number.

(iii) 887.43×1000

There are only 2 digits after the decimal point, but 1000 has 3 zeros, so we put a zero at the end of the number.

Note: Remember that when a decimal number is multiplied by 10, 100, 1000, etc., the decimal is shifted to the right by as many places as the number of zeros in the multiplier and if the number of zeros is greater than the digits after it the decimal number, then additional zeros must be added to the product.

● Decimal.

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Why decimals are difficult

A couple of elementary school teachers recently surreptitiously admitted to me that they “never got decimals.” I was wondering what is so difficult about decimals. For people who “get” decimals, they’re just another number with the decimal point showing. This was obviously not the case for everyone.

So, 21st-century style, I Googled “Why are decimals difficult?”

I got some wonderfully interesting results, including a review by Hugues Lortie-Forgues, Jing Tian, ​​and Robert S. Siegler called “Why Is Learning Fractional and Decimal Arithmetic So Difficult?”, which I refer to in this post.

You must know

This is important for statistics teachers. In particular, students learning about statistics sometimes have trouble discerning whether a p-value of 0.035 is less than or greater than the alpha value of 0.05. In this post, I talk about why that might be. I will also provide links to a few videos that you might find helpful. There might be some useful insights for math teachers.

Integers and rational numbers

Whole numbers are the numbers we start with when we start learning math – 1, 2, 3, 4,… and 0. The zero plays an interesting role as it has no size per se but acts as a placeholder , to ensure that we can recognize the meaning of a number. Without zero, 2001 and 201 and 21 would all look the same! We realize early on that longer numbers represent larger amounts. We know that a salary with a lot of zeros is better than one with a few. $1,000,000 is more than $200 even though 2 is greater than 1.

Rational numbers are those that fall in between, but also include integers. All of the following numbers are considered rational numbers: ½, 0.3, 4/5, 34.87, 3¾, 2000

When we talk about whole numbers, we can say what number comes before and after the number. 35 comes before 36. 37 comes after 36. But we can’t do that with rational numbers. There are infinitely many rational numbers in any given interval. There are infinitely many rational numbers between 0 and 1.

Rational numbers are usually expressed as fractions (½, 3¾) or decimals (0.3, 34.87).

There are several things that make rational numbers (fractions and decimals) difficult. In this post, I’ll focus on decimal numbers

Decimal notation and size of the number

As I explained earlier, when we learn about whole numbers, a useful rule of thumb we learn is that longer strings of digits equal larger numbers. However, the length of the decimal is unrelated to its size. For example, 10045 is greater than 230. The longer number corresponds to a larger order of magnitude. But 0.10045 is less than 0.230. We look at the first digit after the period to see which number is larger. The way you judge which of two decimal places is larger is very different than with integers. The second of my videos illustrates this.

Effect of multiplying by numbers between 0 and 1

The results of multiplying by decimals between 0 and 1 are different than what we are used to.

When we learn about multiplying integers, we realize that when we multiply, the result is always greater than the two numbers we’re multiplying.

3 × 4 = 12. 12 is greater than 3 or 4.

However, if we multiply 0.3 × 0.4, we get 0.12, which is less than 0.3 and 0.4. Or if we multiply 6 by 0.4, we get 2.4, which is less than 6 but greater than 0.4. This can be quite confusing.

Aside for statistics teachers

In statistics, we often report the R-squared value from the regression. To get it, we square r, the correlation coefficient, and what’s a fairly respectable value like 0.6 is reduced to a mere 0.36.

Effect of division by decimals between 0 and 1

Likewise, when we divide integers by integers, the result is less than the number we divide. 100 / 5 = 20. Twenty is less than 100, but in this case greater than 5. But when we divide by a decimal between 0 and 1, everything goes crazy and things get bigger! 100/ 0.5 = 200. People who are at home with all this madness don’t notice it, but I can see how it can alarm the beginner.

Decimal arithmetic does not behave like normal arithmetic

addition and subtraction

When we add or subtract two numbers, we need to line up the decimals so we know we’re adding values ​​with corresponding place values. This looks different than the default algorithm where we align the right side. Actually it’s the same, but because the decimal point is invisible it doesn’t appear to be the same.

Method for multiplying decimal numbers

When multiplying numbers with decimal places, do it like normal multiplication and then count the number of places to the right of the decimal in each of the factors and add them together and that is the number of places to the right of the decimal in the answer! I have a confession here. I know how to do it and have taught it, but I can’t remember ever finding out why we do it or getting students to work it out.

Method for dividing decimal numbers

Is that even a thing? My immediate answer is to use a calculator. I remember being a little more reckless in moving the decimal point so that it disappears from the number we’re dividing by. But who has been dividing by hand for a long time?

Okay, teacher friends – now I understand why you find decimal numbers difficult.

answers

The paper talks about approaches that help. The most important is that students need to spend time understanding the size.

My suggestion is to do a lot of work with money. Somehow we get that in our heads.

And use a calculator, along with a reasonable estimate.

Here are two videos I made to help people get familiar with decimals.

When multiplying a decimal by an integer, removing the decimal point can make multiplying easier, but where does the decimal point go when you want to get your result? In this tutorial you will learn how to estimate the position of the decimal point after multiplying!