9
x
=
10
x
−
x
{\displaystyle 9x=10x-x}
. Using this formula let’s you use the easy calculation of
10
x
{\displaystyle 10x}
to find the more difficult calculation of
9
x
{\displaystyle 9x}
. This method works for any number multiplied by 9, but there are also many tricks and patterns in the nines times table, which will allow you to easily multiply a single-digit number by 9.
Multiplying by Ten
Learn the formula. The formula is 9 x = 10 x − x {\displaystyle 9x=10x-x} 9x=10x-x. This method works for 9 times any number. In this instance, identify the factor you are multiplying by 9 as x {\displaystyle x} x.
Multiply x {\displaystyle x} x by 10. Multiplying a number by 10 is easy to do in your head. All you need to do is add a zero to the end of the number. For example: 9 ( 115 ) = 10 ( 115 ) − 115 {\displaystyle 9(115)=10(115)-115} 9(115)=10(115)-115 9 ( 115 ) = 1150 − 115 {\displaystyle 9(115)=1150-115} 9(115)=1150-115
Subtract x {\displaystyle x} x from 10 x {\displaystyle 10x} 10x. This will give you the product of 9 x {\displaystyle 9x} 9x. For example, 9 ( 115 ) = 1150 − 115 = 1035 {\displaystyle 9(115)=1150-115=1035} 9(115)=1150-115=1035.
Using Your Fingers
Hold up your hands. Face your palms away from you, and spread out all ten of your fingers. Note that this method only works when you are multiplying 9 by a number between 1 and 10.
Identify the factor you are multiplying by 9. Beginning on the left, count your fingers until you reach that digit. Put down that finger. For example, if you are multiplying 9 × 6 {\displaystyle 9\times 6} 9\times 6, you would put down the thumb of your right hand.
Look at the number of fingers on the left. You are looking at all the fingers to the left of the finger you put down. This number of fingers tells you the digit in the tens place of your product. For example, you have 5 fingers to the left of the thumb you put down. That means the tens digit of your product is 5.
Look at the number of fingers on the right. You are looking at all the fingers to the right of the finger you put down. This number of fingers tells you the digit in the ones place of your product. For example, you have 4 fingers to the right of the thumb you put down. That means the ones digit of your product is 4. So, 9 × 6 = 54 {\displaystyle 9\times 6=54} 9\times 6=54.
Memorizing Patterns
Notice that the product’s digits add up to 9. This is true of the product of any number and 9. Knowing this can help you check your work as you multiply in your head. For example: 9 × 2 = 81 {\displaystyle 9\times 2=81} 9\times 2=81; 8 + 1 = 9 {\displaystyle 8+1=9} 8+1=9 9 × 5 = 45 {\displaystyle 9\times 5=45} 9\times 5=45; 4 + 5 = 9 {\displaystyle 4+5=9} 4+5=9 9 × 12 = 108 {\displaystyle 9\times 12=108} 9\times 12=108; 1 + 0 + 8 = 9 {\displaystyle 1+0+8=9} 1+0+8=9
Identify the pattern in the 9 x 10 times table. Notice that the tens digit starts at 0 and goes up to 9. The ones digit works in reverse, starting with 9 and moving down to 0. For example: 9 × 1 = 09 {\displaystyle 9\times 1=09} 9\times 1=09 9 × 2 = 18 {\displaystyle 9\times 2=18} 9\times 2=18 9 × 3 = 27 {\displaystyle 9\times 3=27} 9\times 3=27 9 × 4 = 36 {\displaystyle 9\times 4=36} 9\times 4=36 9 × 5 = 45 {\displaystyle 9\times 5=45} 9\times 5=45 9 × 6 = 54 {\displaystyle 9\times 6=54} 9\times 6=54 9 × 7 = 63 {\displaystyle 9\times 7=63} 9\times 7=63 9 × 8 = 72 {\displaystyle 9\times 8=72} 9\times 8=72 9 × 9 = 81 {\displaystyle 9\times 9=81} 9\times 9=81 9 × 10 = 90 {\displaystyle 9\times 10=90} 9\times 10=90
Use the formula 9 × a = b c {\displaystyle 9\times a=bc} 9\times a=bc. In this formula b = a − 1 {\displaystyle b=a-1} b=a-1 and c = 9 − b {\displaystyle c=9-b} c=9-b. The expression b c {\displaystyle bc} bc does not indicate multiplication in this formula; rather, b {\displaystyle b} b equals the tens digit in the product, and c {\displaystyle c} c equals the ones digit in the product In other words, to find the tens digit of the product, subtract 1 from the number you are multiplying by 9. Then, subtract that difference from 9 to find the ones digit. Note that this method only works when multiplying a single-digit number by 9. For example, to calculate 9 × 7 {\displaystyle 9\times 7} 9\times 7, you would first find: b = a − 1 {\displaystyle b=a-1} b=a-1 b = 7 − 1 = 6 {\displaystyle b=7-1=6} b=7-1=6Thus, 6 is the product’s tens digit.Second, you would find: c = 9 − b {\displaystyle c=9-b} c=9-b c = 9 − 6 = 3 {\displaystyle c=9-6=3} c=9-6=3Thus, 3 is the product’s ones digit. So, 9 × 7 = 63 {\displaystyle 9\times 7=63} 9\times 7=63
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