Decimal division

We continue practicing multiplying and dividing decimals by powers of ten. This time, the practice problems use an exponent, such as 0.245 x 10^5. We also look briefly at whole-number divisions where the divisor is a power of ten.

We continue practicing multiplying and dividing decimals by powers of ten. This time, the practice problems use an exponent, such as 0.245 x 10^5. We also look briefly at whole-number divisions where the divisor is a power of ten.

The rule or shortcut for multiplying and dividing decimals by 10, 100, and 1000 is really easy: you just move the decimal point as many steps as you have zeros in the power of ten. But do you know what this rule is based on? We look at that concept using PLACE VALUE CHARTS. The rule says that if you're multiplying, you move the decimal point to the right (so as to make the number bigger), and vice versa for division. But in reality, it's not the POINT that is moving, but the NUMBER itself is moving within the different places. We can see this clearly by placing the number in a place value chart or table and considering what happens in the multiplication or division, place by place. This lesson is meant for 5th grade and onwards. See the 2nd part of the lesson at Check out also my free books and worksheets at

The rule or shortcut for multiplying and dividing decimals by 10, 100, and 1000 is really easy: you just move the decimal point as many steps as you have zeros in the power of ten. But do you know what this rule is based on? We look at that concept using PLACE VALUE CHARTS. The rule says that if you're multiplying, you move the decimal point to the right (so as to make the number bigger), and vice versa for division. But in reality, it's not the POINT that is moving, but the NUMBER itself is moving within the different places. We can see this clearly by placing the number in a place value chart or table and considering what happens in the multiplication or division, place by place. This lesson is meant for 5th grade and onwards. See the 2nd part of the lesson at Check out also my free books and worksheets at

A lesson for about 5th grade... first we solve a multi-step word problem that could be from real life: if you know the total of 7 bottles of water and 15 blueberry muffins (Mathy's favorite), and you know the cost of one bottle of water, then how do you find the cost of one muffin? The solution involves several operations with decimals. Secondly, I present a multi-step calculation and the task is to find a situation that fits it. I suggest one, Mathy has another, and maybe you can find a third. :)

A lesson for about 5th grade... first we solve a multi-step word problem that could be from real life: if you know the total of 7 bottles of water and 15 blueberry muffins (Mathy's favorite), and you know the cost of one bottle of water, then how do you find the cost of one muffin? The solution involves several operations with decimals. Secondly, I present a multi-step calculation and the task is to find a situation that fits it. I suggest one, Mathy has another, and maybe you can find a third. :)

To divide by a decimal, first change the ENTIRE division problem to another with a whole-number divisor, yet with the same answer. How? Simply multiply both numbers in the division problem (the dividend and the divisor) by 10 repeatedly, until the divisor is a whole number. (The dividend doesn't matter; it can be a decimal). The shortcut for multiplying by 10 is of course that you move the decimal point, and most school books only give you the shortcut, without explaining this principle for division of decimals that in reality, we are MULTIPLYING both the dividend & divisor by 10, 100, 1000 (powers of ten). We can do this process, because in it, the answer (the quotient) does not change. Think about it: 0.2 goes into 0.7 as many times as 2 goes into 7, or 20 goes into 70. Similarly, if you have a division problem with a decimal divisor, such as 4.392 / 0.7, you're solving how many times 0.7 goes into 4.392. But that is the same as finding how many times 7 goes into 43.92. This process

To divide by a decimal, first change the ENTIRE division problem to another with a whole-number divisor, yet with the same answer. How? Simply multiply both numbers in the division problem (the dividend and the divisor) by 10 repeatedly, until the divisor is a whole number. (The dividend doesn't matter; it can be a decimal). The shortcut for multiplying by 10 is of course that you move the decimal point, and most school books only give you the shortcut, without explaining this principle for division of decimals that in reality, we are MULTIPLYING both the dividend & divisor by 10, 100, 1000 (powers of ten). We can do this process, because in it, the answer (the quotient) does not change. Think about it: 0.2 goes into 0.7 as many times as 2 goes into 7, or 20 goes into 70. Similarly, if you have a division problem with a decimal divisor, such as 4.392 / 0.7, you're solving how many times 0.7 goes into 4.392. But that is the same as finding how many times 7 goes into 43.92. This process

Dividing a decimal by a decimal can seem counter-intuitive to students. What can 0.6 divided by 0.2 signify? To see that, we first take a look at FACT FAMILIES involving two multiplications & two divisions that use the same numbers. For example, if 7 x 0.04 = 0.28, then 0.28 / 0.04 must equal 7. The quotient is LARGER than either of the numbers in the division! This can seem baffling and confusing, but it is true. The easy way to think about such divisions is to consider them as "measurement divisions": how many times does the DIVISOR (the 2nd number) FIT into the DIVIDEND (the 1st number). For example, to solve 0.36 divided by 0.04, think how many times 4 hundredths "fits into" or goes into or divides into 36 hundredths. That is the same number of times as 4 goes into 36, so the answer is plain 9 (the whole number 9; not any decimal). We look at several such decimal divisions that can be solved with mental math, and lastly solve a word problem that applies this exact concept. Check ou

Dividing a decimal by a decimal can seem counter-intuitive to students. What can 0.6 divided by 0.2 signify? To see that, we first take a look at FACT FAMILIES involving two multiplications & two divisions that use the same numbers. For example, if 7 x 0.04 = 0.28, then 0.28 / 0.04 must equal 7. The quotient is LARGER than either of the numbers in the division! This can seem baffling and confusing, but it is true. The easy way to think about such divisions is to consider them as "measurement divisions": how many times does the DIVISOR (the 2nd number) FIT into the DIVIDEND (the 1st number). For example, to solve 0.36 divided by 0.04, think how many times 4 hundredths "fits into" or goes into or divides into 36 hundredths. That is the same number of times as 4 goes into 36, so the answer is plain 9 (the whole number 9; not any decimal). We look at several such decimal divisions that can be solved with mental math, and lastly solve a word problem that applies this exact concept. Check ou