Chapter

This is a specific technique for multiplication of any number with 9s (9, 99, 999, 9999 etc..).  This method involves 3 distinct types:

• Type A:  When the number of 9s in the multiplier are equal to the number of digits in the multiplicand (i.e the number which has to be multiplied by 9s)
• Type B:  When the number of 9s in the multiplier are more than the number of digits in the multiplicand (i.e the number which has to be multiplied by 9s)
• Type C:  When the number of 9s in the multiplier are less than the number of digits in the multiplicand (i.e the number which has to be multiplied by 9s)

Type A:

When the number of 9s in the multiplier are equal to the number of digits in the multiplicand (i.e the number which has to be multiplied by 9s)

For example:  43 x 99

Step 1:  Subtract 1 from the multiplicand and write the result on the Left Hand Side (LHS).  43 – 1 = 42

Step2:  Now subtract the LHS number 42 from 99 on the RHS. 

99 – 42 = 57

Step 3:  Combine both the LHS and RHS to obtain the final answer 4257.

Example 2:  376  x 999 = 375624

Example 3:  Calculation involving decimals:

68.5 x 9.99

Step 1:  Remove the decimal from both the numbers with the result 685 x 999

Step 2:  Multiply 685 x 999 = 684315

Step 3:  Insert the decimal back in the final answer at the appropriate place by calculating the number of decimal positions taken off in step 1 (2 + 1 = 3)

So place the decimal after 3 digits from the right giving the final answer as 684.315

Exercises:

37 x 99 = 3663

46 x 99 = 4554

358 x 999 = 357642

529 x 999 = 528471

340 x 999 = 339660

41.5 x 9.99 = 414.585

98.1 x 99.9 = 9800.19

86.48 x 9.999 = 864.71352

438.5 x 99.99 = 43845.615

51.39 x 0.9999 = 51.384861

Type B:

When the number of 9s in the multiplier are more than the number of digits in the multiplicand (i.e the number which has to be multiplied by 9s)

Example 1:  487 x 9999

Step 1:  Make the number of digits equal to the number of 9s by appending zeros to the left of the number.  So, the number above will become 0487.

Step 2:  Follow the steps in Type A above.  (0487 x 9999 = 04869513)

Step 3:  Derive the final answer by removing the zeros in front of the number giving you the final answer as 4869513

Example 2:  73.5 x 9.9999

Step 1:  Re-write the number as 00735 x 99999

Step 2:  Derive the answer as 0073499265

Step 3:  Remove the zeros in front of the number and put back the decimal at the usual place to get the final answer:  734.99265

Exercises: 

45 x 999 = 44955

30 x 999 = 29970

64 x 9999 = 639936

481 x 9999 = 4809519

759 x 99999 = 75899241

38.4 x 99.99 = 3839.616

41.73 x 99.999 = 4172.95827

91.2 x 99.999 = 9119.9088

88.75 x 999.999 = 88749.91125

51.03 x 0.99999 = 51.0294897

Type C:

When the number of 9s in the multiplier are less than the number of digits in the multiplicand (i.e the number which has to be multiplied by 9s)

Example 1:  32 x 9

Step 1:  Put as many trailing zeros after the number equal to the number of 9s in the multiplier making the derived number 320

Step 2:  Now subtract the original number 32 from the derived number in step 1 (320 -32) to obtain the final answer as 288.

Observe that Type C method is different from both Type A and Type B.

Example 2:  756 x 99 = 75600 - 756 = 74844

Example 3:  6006 x 99 = 600600 – 6006 = 594594

Exercises: 

56 x 9 = 504

156 x 99 = 15444

459 x 99 = 45441

786 x 99 = 77814

1008 x 999 = 1006992

3003 x 99 = 297297

123.4 x 9.9 = 1221.66

364 x 99 = 36036

5891 x 999 = 5885109

48731 x 9999 = 487261269