Mental division

Mental division relies a great part on whether you have completely memorised your standard times tables. It is suggested you get better at these first before spending time trying to do mental division.

There are several different ways to do mental division:


First of all check the divisibility rules of a number. If any of these fail for your divisor, the number is not divisible by it - you will get a fraction.

The easier divisibility rules

• If a number ends on 0, it will divide by ten, for example 450. Just take the zero off to give 45.
• If a number ends on 5, or 0, it will divide by 5, for example, 105.
• if the last two digits of a number can be divided by 4, then the whole thing can be divided by 4.
• If the digits in a number add up to a number that can be divided by three, the original number can also be divided by three.
  For example: 123411 will divide by 3, because 1+2+3+4+1+1 = 12, which divides by 3.
• If a number is even, it will divide by two. In other words, any number ending with the digits 0, 2, 4, 6 or 8 will divide by 2.

The harder divisibility rules

• If the number is even, and the digits sum to a number divisible by 3, then the number can be divided by 6.
• If the last three numbers are zero or divisible by 8, then the original number is divisible by 8.
• If the sum of digits of the number is divisible by 9, then the original number is divisible by 9.

---

Common factors
Now you know the divisibility rules, then if both numbers in a division ( the dividend and the divisor ) can be divided by either 10, 5, 4, 3 or 2, Then you can divide both numbers by their common factor to make a simpler division which will give the same result at the end. If you have learned the slightly harder process for 6, 8 and 9 you can test for those those as well.

Example for 10: 4500 / 20 - both numbers end with zero, so just remove a zero from both to give a simpler division: 450 / 2.  In addition, divide by two is the same as 'halve' so we just need to halve 450 to get the final answer: 225.

Example for 5: 420 / 15 - both numbers end with zero or five, so divide both by 5 to give a simpler division: 84 / 3.   Now just divide 84 by 3 to get the final answer: 28. **

Example for 4: 3264 / 4 - The last two digits are 64, which can be divided by 4 because 8 x 8 is 64, and 8 divides by 4. Because 4 is half of 8, then 64 / 4 must be 8 x 2 which is 16.   Now just divide the remaining 3200 by simply dividing 32 by 4 and adding two zeros: 800. Add the 16 we got previously to get the final answer: 816.

Example for 3: 162 / 9 - both numbers can be divided by 3, as per the rule given above.   Divide 3 to get a simpler division: 54 / 3 . Now just divide 54 by 34 to get the final answer: 18.

Example for 2: 272 / 16 - both numbers can be divided by 2, as they are both even:   136 / 8.

Both are even, so keep going, halve again to give : 68 / 4. These are both still even, so just keep going: 34 / 2 which gives a final answer of 17.

** There is another trick to make dividing by 5 easier: double For example, if you want to divide 720 by 24, you can halve both numbers until you get an easier division: Double the dividend, and divide by 10 instead - which just involves taking a zero off. (Whether you want to do this or not depends on whether you are happier dividing by 5, or with doubling.) Some examples:

165 / 5 = 330 / 10 = 33
85 / 5 = 170 / 10 = 17
235 / 5 = 470 / 10 = 47

---

Chunking

Principle: Break down the large number into smaller parts or chunks that are easier to divide by the divisor. For example, if you want to divide:

1560 / 12 then because it is a large number, it is obvious that 1200 divides by 12 to make 100. Take this away from 1560 to leave 360. You now have 360 / 6 and because both numbers are even, can now use the 'divide by 2' rule to whittle this down to : 180 / 6 and again to 90 / 3 which is 30. Add 30 to the 100 we got previously to give a final answer of 130.

---

Divide into a higher, easier number, then subtract or add

Principle: Round the dividend to a nicer number, do the division, then subtract or add the adjustment.

Examples:

156 ÷ 4 156 is just 4 less than 160, which divides easily by 4 to give 40. Now just take the 'extra 4' away from 40 to give 39.

117 ÷ 3 117 is just 3 less than 120, which divides easily by 3 to give 40. Now just take the 'extra 3' away from 40 to give 37. 155 ÷ 5 155 is just 5 more than 150, which divides easily by 5 to give 30 (300/10). Now just add the 'extra 5' away to 30 to give 31.

---

Bus stop with memory images

( This section assumes you already know how bus stop works. If not, see the bus stop option in the 'paper maths' module of this website.)

This works very well for single-digit divisors, for example 345726 / 6



Proceed as follows:

• Remember the number. 345726 is marrow, log, niche. So imagine a marrow showing a log into a dog kennel ( niche is French for kennel. )
• 6 does not go into 3 but goes into 34, giving 5 remainder 4. Think of a hook as the running answer.
• Now the 4 carry makes 45 with the next digit, the 5 of the 'log.' 6 into 45 goes 7 remainder 3. Think of 'ship' for the 7, so you now see a giant hook trying to lift up a ship.
• The remainder 3 joined onto the next digit of log, 7, makes 37. 6 into 37 goes 6 remainder 1. The 6 is an elephants trunk, so now you need to see an elephant trying to pull the ship back to shore, battling the hook.
• The carry of 1 joins onto the first digit of niche, which is 2, making 12. 6 divides into 12 exactly making 2. So now a swan (2) is tugging on the elephant's tail, giving him some help to pull the ship back in.
• Finally 6 goes into the final 6 exactly, giving 1, which is a pencil.
• Your final answer is hook, ship, elephant, swan, pencil, which is 57621.

Difficulty Levels - typical questions:

1: 32 / 2 ( easy halving )
2: 3464 / 4 (divide by 4 rule, with chunking )
3: 45639 / 3 ( bus stop )