Intermediate Multiplication

Squaring numbers
Numbers that are close-together
Distributing products
Distributed subtraction
Distributed addition

The best method to use for mentally multiplying 2-digit numbers depends on the numbers you're given. If both numbers are the same, use the squaring method. If they're close to each other, use the close-together method. If one of the numbers can be factored into small numbers, use the factoring method. If one of the numbers is near 100 or it ends in 7, 8, or 9, try the subtraction method. If one of the numbers ends in a small digit, such as 1, 2, 3, or 4, or when all else fails, use the addition method.

Addition Method

Applied to any multiplication problem
Best to use when one of the numbers being multiplied ends in a small digit

Round that number down to the nearest easy number:

41 x 17 =
(40 x 17) + (1 x 17) =
680 + 17 = 697

32 x 71 =
(30 x 71) + (2 x 71) =
2130 + 142 = 2272

34 x 14 =
(34 x 10) + (34 x 4) =
340 + 126 = 466

Subtraction Method

When a 2-digit numbers ends with a large digit, such as 7, 8, or 9

53 x 89 is easier to figure by rounding up to an easy 2-digit number and then subtracting a distributed product...

Treat the 89 as 90 - 1, distribute the 53 and then calculate...

53 x 89 =
53 x (90 - 1) =
(53 x 90) - (53 x 1) =
4770 - 53 = 4717

9 x 79 =
(9 x 80) – (9 x 1) =
720 - 9 = 711

19 x 37 =
(20 x 37) – (1 x 37) =
740 - 37 = 703

97 x 22 =
(100 x 22) - (3 x 22) =
2200 - 66 = 2134

Factoring Method

This third strategy is easier because you compute a number and then immediately put it to use.
You often have several choices when using the factoring method, and you may wonder in what order you should multiply the factors.
Practice math of least resistance by looking at possible factors and choosing the easiest path.

For the problem like 97 x 22 above, instead of rounding either number up or down, factor the 22 as (11 x 2) obtaining 97 x 11 x 2.

Now perform (97 x 10) + (97 x 1) = 1067 first, then multiply that result by 2 to get 2134 as shown...

97 x 22 =
97[11 x 2] =
97(10 + 1) x 2 =
(97 x 10) + (97 x 1) x 2 =
(970 + 97) x 2 =
1067 x 2 = 2134

Squaring

A strategy for any two digit number

In a problem like 13 x 13, lower one of the 13s to the easier number 10 by going down 3, and then you must go up with the other 13 adding 3 to get 16.

The first part of the calculation will now be 10 x 16, and to that add the square of the up/down number, which in this case was 3...

(10 x 16) + 3² = 169

Close-Together Method

This strategy is for 2-digit numbers close to each other

For 26 x 23 find a easy number that is close to both numbers: here use 20.

Next, note how far away each is from the easier number 20: 26 is 6 away and 23 is 3 away.

Add the original numbers together (26 + 23 = 49), subtract the easier number 20 and obtain 29.

After we multiply 29 x 20 = 580, then add the product of the distances (6 x 3 = 18).

26 x 23 =
((20x((26+23)-20) + (6x3) =
(20x29) + (6x3) =
580 + 18 = 598

If you have numbers like 17 x 76 where the larger number is even, divide that larger number by two and double the smaller number.

17 x 76 =
(17x2) x (76/2) =
(34) x (38) =

That will leave a close-together problem of 34 x 38.

34 x 38 =
(30 x ((34+38)-30) + (4x8) =
(30 x 42) + (4x8) =
1260 + 32 = 1292

Calculate the following:

14² =
(10 x 18) + 4² =
180 + 16 =
196
18² =
(20 x 16) + 2² =
320 + 4 =
324
22² =
(20 x 24) + 2² =
480 + 4 =
484
23² =
(20 x 26) + 3² =
520 + 9 =
529
24² =
(20 x 28) + 4² =
560 + 16 =
576
25² =
(20 x 30) + 5² =
600 + 25 =
625
29² =
(30 x 28) + 1² =
840 + 1 =
841
31² =
(30 x 32) + 1² =
960 + 1 =
961
35² =
(30 x 40) + 5² =
1200 + 25 =
1225
36² =
(40 x 32) + 4² =
1280 + 16 =
1296
41² =
(40 x 42) + 1² =
1680 + 1 =
1681
44² =
(40 x 48) + 4² =
1920 + 16 =
1936
45² =
(40 x 50) + 5² =
2000 + 25 =
2025
47² =
(50 x 44) + 3² =
2200 + 9 =
2209
56² =
(60 x 52) + 4² =
3120 + 16 =
3136
64² =
(60 x 68) + 4² =
4080 + 16 =
4096
71² =
(70 x 72) + 1² =
5040 + 1 =
5041
82² =
(80 x 84) + 2² =
6720 + 4 =
6724
86² =
(90 x 82) + 4² =
7380 + 16 =
7396
93² =
(90 x 96) + 3² =
8640 + 9 =
8649
99² =
(100 x 98) + 1² =
9800 + 1 =
9801
Using the addition method...
31 x 23 =
(30 + 1) x 23 =
(30 x 23) + (1 x 23) =
690 + 23 =
713
61 x 13 =
(60 + 1) x 13 =
(60 x 13) + (1 x 13) =
780 + 13 =
793
52 x 68 =
(50 + 2) x 68 =
(50 x 68) + (2 x 68) =
3400 + 136 =
3536
94 x 26 =
(90 + 4) x 26 =
(90 x 26) + (4 x 26) =
2340 + 104 =
2444
47 x 91 =
47 x (90 + 1) =
(47 x 90) + (47 x 1) =
4230 + 47 =
4277
Using the subtraction method...
39 x 12 =
(40 - 1) x 12 =
480 - 12 =
468
79 x 41 =
(80 - 1) x 41 =
3280 - 41 =
3239
98 x 54 =
(100 - 2) x 54 =
5400 - 108 =
5292
87 x 66 =
(90 - 3) x 66 =
(90 x 66) - (3 x 66) =
5940 - 198 =
5742
38 x 73 =
(40 - 2) x 73 =
(40 x 73) - (2 x 73) =
2920 - 146 =
2774
Using the factoring method...
75 x 56 =
75 x 8 x 7 =
600 x 7 =
4200
67 x 12 =
67 x 6 x 2 =
402 x 2 =
804
83 x 14 =
83 x 7 x 2 =
581 x 2 =
1162
79 x 54 =
79 x 9 x 6 =
711 x 6 =
4266
45 X 56 =
45 x 8 x 7 =
360 x 7 =
2520
68 x 28 =
68 x 7 x 4 =
476 x 4 =
1904
Using the close-together method...
13 x 19 =
(10 x 22) + (3 x 9) =
220 + 27 =
247
86 x 84 =
(80 x 90) + (6 x 4) =
7200 + 24 =
7224
77 x 71 =
(70 x 78) + (7 x 1) =
5460 + 7 =
5467
81 x 86 =
(80 x 87) + (1 x 6) =
6960 + 6 =
6966
98 x 93 =
(100 x 91) + (-2 x –7) =
9100 + 14 =
9114
67 x 73 =
(70 x 70) + (-3 x 3) =
4900 – 9 =
4891
Using more than one method...
14 x 23 =
23 x 7 x 2 =
161 x 2 =
322, or

14 x 23 =
23 x 2 x 7 =
46 x 7 =
322, or

14 x 23 =
(14 x 20) + (14 x 3) =
280 + 42 =
322
35 x 97 =
35 x (100 – 3) =
3500 – 35 x 3 =
3500 – 105 =
3395, or

35 x 97 =
97 x 7 x 5 =
679 x 5 =
3395
22 x 53 =
53 x 11 x 2= 583 x 2 =
1166, or

53 x 22 =
(50 + 3) x 22 =
50 x 22 + 3 x 22 =
1100 + 66 =
1166
49 x 88 =
(50 – 1) x 88 =
(50 x 88) – (1 x 88) =
4400 - 88 =
4312, or

88 x 49 =
88 x 7 x 7 =
616 x 7 =
4312, or

49 x 88 =
49 x 11 x 8 =
539 x 8 =
4312
42 x 65 =
(40 x 65) + (2 x 65) =
2600 + 130 =
2730, or

65 x 42 =
65 x 6 x 7 =
390 x 7 =
2730