1.3.7 Multiplying Two Numbers Ending in 5

This is helpful trick for multiplying two numbers ending in 5. Let’s look at its derivation, let $n_1 = a5 = 10a + 5$ and $n_2 = b5 = 10b + 5$ then:

$$n_1 \times n_2 = (10a + 5) \cdot (10b + 5)$$ $$= 100(ab) + 50(a + b) + 25$$ $$= 100(ab + \frac{a + b}{2}) + 25$$

So what does this mean:

1. If $a + b$ is even then the last two digits are 25.
2. If $a + b$ is odd then the last two digits are 75.
3. The remainder of the answer is just $a \cdot b + \lfloor \frac{a + b}{2} \rfloor$, where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.

Let’s look at an example in each case:

45 × 85 =

• Ones/Tens: Since $4 + 8$ is even -> 25
• Rest of Answer: $4 \times 8 + \frac{4 + 8}{2} = 32 + 6 = 38$

Answer: 3825

35 × 85 =

• Ones/Tens: Since $3 + 8$ is odd -> 75
• Rest of Answer: $3 \times 8 + \lfloor \frac{3 + 8}{2} \rfloor = 24 + 5 = 29$

Answer: 2975

Problem Set 1.3.7