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Multiplication by 9

Many young school children find it hard to multiply by 9, using the conventional right to left method, whether or not they mastered the 9 times table.

But as we have discussed in our books 25 Math Short Cuts (pp. 25 – 27) and Algebra Made Easy as Arithmetic (pp. 14 – 15), multiplying by 9 is easy if we consider it as (10 -1).

Using this method we could easily get 4 x 9 = 40 – 4 = 36 and 7 x 9 = 70 -7 = 63.

Also, we have,  13 x 9 = 130 – 13 = 117 and  257 x 9 = 2570 – 257 = 2313.

But it would not be easy for the following: 74 x 9 = 740 – 74 or 864 x 9 = 8640 – 864.

However, we can develop a simple way to multiply by 9 and mastering this would be very valuable in higher mathematics.

Let us consider one of our examples, 257 x 9. If we perform the subtraction using the left to right method,

               2 5 7 0

  •  2 5 7

   2 3 2 (7)

We can see that after bringing down 2, we subtracted 2 from 5, which is the next digit to it in the multiplicand to get 3. Then we also deducted 5 from 7 which again is the next digit to it  to get 2 and finally we are subtracted the last digit 7 from 0 to get negative or “bar” 7 which we represented as (7).  

We can then evaluate 232(7) as 2320 – 7 or 2313. (Please see https://www.math-inic.com/blog/msc-4-using-bar-numbers/ for a discussion on bar numbers.)

We can now multiply long numbers by 9 using our technique.

1 4 5 8 x 9 = 1313(8) = 13,122

Here we used the following steps:

  1. Copy the first digit of the multiplicand as the 1st digit of the answer – 1
  2. Subtract the first from the next digit in the multiplicand – (4 – 1 = 3)
  3. Repeat step 2 starting from the second digit up to the penultimate digit of the multiplicand (5 – 4 = 1) and (8 – 5 = 3)
  4. Write down the last digit as a negative number which is the same as subtracting it from 0.
  5. Express the answer in the conventional format.

Let have some more examples:

789 x 9 = 711(9) = 7101

3584 x 9 = 323(4)(4) = 32,256

35174 x 9 = 32(4)6(3)(4) = 316,566

Try multiplying these numbers by 9 using our technique.

  1. 35
  2. 53
  3. 258
  4. 285
  5. 3478
  6. 3749
  7. 25688
  8. 52368
  9. 364869
  10. 754868

Answers to previous exercises:

  1. 542 ÷ 5 = 108.4
  2. 743 ÷ 50 = 14.86
  3. 867 ÷ 500 = 1.734
  4. 927 ÷ 0.5 = 1,854
  5. 1024 ÷ 5 = 204.8
  6. 7526 ÷ 50 = 150.52
  7. 13856 ÷ 500 = 27.712
  8. 23254 ÷ 0.5 = 46,508
  9. 78946 ÷ 5000 =15.7892
  10. 35795 ÷ 0.05 = 715,900
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