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(**) Multiplying Two Numbers That End In 5:

   A.  When multiplying two numbers that both end in 5, you can do the following from algebra:

(10a + 5) (10b + 5) = 100 (ab + ^{(a + b)}/₂) + 25, if (a + b) is even

                                100 (ab + ^{(a + b – 1)}/₂) + 75, if (a + b) is odd            

   B.  Using numbers instead of variables we get the following rules:

If (a + b) is even:  1.  Write down 25.

2.      Multiply the ten’s digits together.

3.      Add the ten’s digits together and divide by 2.

4.      Add step 2 and step 3.  Write this result.

If (a + b) is odd:    1.  Write down 75.

2.      Multiply the ten’s digits together.

3.      Add the ten’s digits together, subtract 1, then divide by 2.

4.      Add step 2 and step 3.  Write this result.

Ex [1]  35 x 55 =_________.

a)      Since 3 + 5 is even, write 25.

b)      3 x 5 = 15.

c)      ^{(3 + 5)}/₂ = 4.

d)      15 + 4 = 19.  Write 19.

e)      The answer is 1925.

Ex [2]  85 x 75 =_________.

a)      Since 8 + 7 is odd, write 75.

b)      8 x 7 = 56.

c)      ^{(8 + 7 – 1)}/₂ = 7.

d)      56 + 7 = 63.  Write 63.

e)      The answer is 6375.

Ex [3]  125 x 155 =_________.

a)      Since 12 + 15 is odd, write 75.

b)      12 x 15 = 180.  See Multiplying by 15 or Double and Half.

c)      ^{(12 + 15 – 1)}/₂ = 13.

d)      180 + 13 = 193.  Write 193.

e)      The answer is 19375.

   C.  Note:  If you encounter a problem, as in Ex [2], where the difference of the ten’s digit is 1, then you can add one to the largest ten’s digit and multiply by the remaining ten’s digit. 

*Ex [2]  85 x 75 =_________.

a)  Since 8 + 7 is odd, write 75.

b)  8 + 1 = 9.

c)  9 x 7 = 63.  Write 63.

d)  The answer is 6375.

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