Example 1

11.44_{eight} â€“ 3.65_{eight} + 2.61_{eight}

Apply BODMAS

Rewrite 11.44_{8} â€“ 3.65_{8} + 2.61_{8} as 11.44_{8} + 2.61_{8 }â€“ 3.65_{8}

Answer = 10.4_{8}

Example 2

Work out the base in the following and use it to find the missing figures

The highest number here is 5, therefore the bases can be 6, 7, e.t.c …….

Lets try Base 6

1st Column (from right to left) = 2 + 1 = 3. 3 is less than 6 so we write down 3

2nd Column = x + 5 = 3. 3, in this case, is the remainder. Through trial and error we know that 4 + 5 = 9 9 Ã· 6 = 1 r 3. So 4 in this case is the missing number and we carry 1 to the 3rd column.

3rd Column = The remainder here is 0. Since we are trying base 6 and 6 Ã· 6 = 1 r 0….. 1 + x + 3 = 6. (remember we are carrying 1 from the 2nd column). âˆ´ x = 6 – 4 = 2. Carry 1 to the 4th column

4th column = We are carrying 1 from the 3rd column so 1 + 3 + 2 = 6. Since we are trying base 6, 6 Ã· 6 = 1 r 0. Write 0 and carry 1 to the 5th column

5th column = 1 + 0 = 1

From our analysis we can see that the required base is 6 and the missing numbers are 2, 4

âˆ´ 3242_{6} â€“ 2351_{6} = 10033_{6}

**Class Activity**

Confirm the previous example by solving 10033_{6} – 2351_{6}

Example 3

**Explanation**

The highest number is 4 so we can try 5, 6, 7 etc

1st Column (from right to left) = 1 – 3. We have to borrow. If we borrow 5 from 2nd column we get 5 + 1 – 3 = 1, so we know it can’t be 5 because the result is 2, not 1. Next let’s borrow 6 âˆ´ 6 – 1 + 3 = 2. So we can keep trying 6. Remember we borrowed 1 from 2nd column.

2nd Column = 0 – 1 = -1. – 1 is less than the result 3 so we have to borrow from the 3rd columnâˆ´ 6 + (- 1) – x = 3. We now have 5 – x = 3. The missing number in this column is 2. Remember we borrowed 1 from 2nd column.

3rd Column = 2 -1 = 1. 1 is less than the result 4 so we have to borrow from the 4th column âˆ´ 6 + 1 – x = 4. We now have 7 – x = 4. The missing number in this column is 3.

4th column = 4 – 1 = 3, âˆ´ 3 – 3 = 0

We can see that from the equation the numbers are in base 6 and the missing numbers are 3, 2.

âˆ´ 4201_{6} â€“ 3322_{6} = 432_{6}

Example 4

Find the missing number in the addition of the following numbers, in base seven

**Solution: 1**^{st}** step**

Solve 4321_{7} + 1234_{7}

The resulting numbers are all less than seven so this addition can be done easily.

1st Column (from right to left) = 1 + 4 = 5

2nd Column = 2 + 3 = 5

3rd Column = 3 + 2 = 5

4th column = 4 + 1 = 5

Subtract the result from the 1st step from the final result to get the missing numbers.

**Explanation**

The highest number is 4 so we can try 5, 6, 7 etc

1st Column (from right to left) = 7 + 1 – 5 = 3

2nd Column = 4 reduces to 3. 7 + 3 – 5 = 5

3rd Column = 3 reduces to 2 = 7 + 2 – 5 = 4

4th column = 3 reduces to 1. 7 + 1 – 5 = 3

5th column = 1 – 1 = 0

âˆ´ Answer = 3453_{7}