Starter Tricks for Mental Math Division
mental math series, part 8
Learning division was exciting for me. Multiplication had been mostly memorization, but division… now that separated the real 4th grade mathletes from the rest of the lot.
I think that often when we’re called to do a division problem in adulthood, that little 4th grader in us either jumps for joy or shrinks in terror.
If you shrink in terror, let’s change that.
Today we’ll learn tricks for:
- Dividing by 10, 100, 1000, …
- Dividing by 5, 50, 500, …
Dividing by 10s
Division asks us how many groups of a certain size we can make from another group. Suppose we’d like to know how many tens are in 850?
According to the decimal place-value system this is
To determine the quantity of 10's in a number ending in zero simply look at the ten’s place-value and include all values to the left. For example, 850 has 85 tens.
We can also write our division problem like this:
This is useful notation to learn early on for two reasons:
- it demonstrates using a number’s factors, e.g. 85 x 10
- it demonstrates the canceling technique
Canceling is a method of simplifying an expression by dividing out a common factor. For example, we can cancel out the 10's in the following expression because 10 ÷ 10 = 1, confirming our answer of 85.
Dividing by 10s, 100s, 1000s, …
In Lesson Thirteen we explored the decimal system beyond the decimal point. Each time we moved right a position, the place’s value reduces by a factor of ten. We can harness this valuable insight to divide any number by ten.
Dividing by ten is essentially the same as shifting our digits over an entire place-value to the right.
For example, 1550.75 ÷ 10 results in the following place-value shift.
You can think of this as shifting the digits one place value right, or moving the decimal point 1 place left.
Let’s extend this idea. Suppose we wanted to divide 1550.75 by 100 this time. Because 100 equals 10 x 10, we’ll shift two place-values.
We could divide by 1000 by shifting three place values since 10 x 10 x 10 = 1000.
Notice that the number of place values we shift is equivalent to the number of zeros in the divisor.
Double and Divide Method
To divide by five mentally:
- Double the number.
- Divide by ten (i.e. shift decimal point one place left).
For example, suppose we wished to find the quantity of 5's in 625.
Step One: Double 625.
Step Two: Divide by ten (move decimal point 1 digit left).
This trick works because we are multiplying the divisor and dividend both by 2. Since 2 ÷ 2 = 1, they essentially cancel out leaving us with 625 ÷ 5 as desired.
To divide by 50, multiply by 2 and then divide by 100. For example, if we want to divide 1075 by 50, begin by doubling 1075.
And then divide by 100.
Feel free to extend this rule even further to divide by 500, 5000, …
I hope you are feeling very accomplished!
You have learned so many mental math techniques in the eight lessons we’ve had so far as a part of this series. I also hope that by this point you’ve had some opportunities to practice them out and about in the real world.
Next Lesson: Can You Solve Einstein’s Puzzle?
Thanks for reading!
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