### Example 1

11.44_{eight} – 3.65_{eight} + 2.61_{eight}

**Apply BODMAS**

Rewrite 11.44_{8} – 3.65_{8} + 2.61_{8} as:

⇒ 11.44_{8} + 2.61_{8 }– 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 …….

**Let’s 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 ÷ 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 ÷ 6 = 1 r 0….. 1 + x + 3 = 6. (remember we are carrying 1 from the 2nd column).

∴ 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 ÷ 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

∴ 3242_{6} + 2351_{6} = 10033_{6}

**Class Activity**

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

### Example 3

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

**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 the 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 ∴ 6 + 1 – 3 = 4.

So we can keep trying 6. Remember we borrowed 1 from the 2nd column.

2nd Column = 0 – 1 = -1.

– 1 is less than the result 3 so we have to borrow from the 3rd column∴ 6 + (- 1) – x = 3.

We now have 5 – x = 3. The missing number in this column is 2.

3rd Column = 2 – 1 = 1.

1 is less than the result 4 so we have to borrow from the 4th column ∴ 6 + 1 – x = 4.

We now have 7 – x = 4. The missing number in this column is 3.

4th column = 4 – 1 = 3, ∴ 3 – 3 = 0

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

∴ 4201_{6} – 3323_{6} = 434_{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 quickly.

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 1^{st} step, from the final result to get the missing numbers.

**Explanation**

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 = 2 reduces to 1. (7 + 1) – 5 = 3

5th column = 1 – 1 = 0

∴ Answer = 3453_{7}

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