Rapid Maths Trick and Tips Flashcards
(107 cards)
1
Q
B1T1: When multiplying or dividing, …
A
…initially disregard any affixed zeroes or decimal points. Then reaffix/reinsert, if necessary, upon completing calculation.
2
Q
B1T3: To multiply a number by 4, …
A
…double the number, and then double once again.
3
Q
B1T4: To divide a number by 4, …
A
…halve the number, and then halve once again.
4
Q
B1T5: To multiply a number by 5, …
A
…divide the number by 2.
5
Q
B1T6: To divide a number by 5, …
A
…multiply the number by 2.
6
Q
B1T7: To square a number that ends in 5, …
A
…multiply the tens digit by the next whole number, and then affix the number 25.
7
Q
B1T8: To multiply a two-digit number by 11, …
A
…add the digits of the number, and insert the sum within the number itself. Carry if necessary.
8
Q
B1T9: To multiply a number by 25, …
A
…divide the number by 4.
9
Q
B1T10: To divide a number by 25, …
A
…multiply the number by 4.
10
Q
B1T11: To multiply a one- or two-digit number by 99, …
A
…subtract 1 from the number, and affix the difference between 100 and the number.
11
Q
B1T13: To multiply two numbers whose difference is 2, …
A
…square the number in the middle and subtract 1.
12
Q
B1T17: To multiply a number by 9, …
A
…multiply the number by 10 and then subtract the number.
13
Q
B1T18: To multiply a number by 12, …
A
…multiply the number by 10 and add twice the number itself.
14
Q
B1T19: To multiply a number by 15, …
A
…multiply the number by 10 and add half of the product.
15
Q
B1T20: To multiply two 2-digit numbers whose tens digits are the same and whose ones digits add up to ten, …
A
…multiply the tens digit by the next whole number, and affix the product of the ones digits.
16
Q
B1T21: To multiply a number by 1.5, 2.5, or the like, …
A
…first halve the number and double the 1.5 (or 2.5, or the like.) Then multiply.
17
Q
B1T22: To divide a number by 1.5, 2.5, or the like, …
A
…first double both the number and the 1.5 (or 2.5, or the like.) Then divide.
18
Q
B1T23: To square a two-digit number beginning in 5, …
A
…add 25 to the ones digit and affix the square of the ones digit.
19
Q
B1T24: To square a two-digit number ending in 1, …
A
…compute as in the following example: 31^2 = 30^2 + 30 + 31 = 961
20
Q
B1T25: To multiply two 2-digit numbers without showing work, …
A
…first multiply the ones digits together, then “cross-multiply,” and finally multiply the tens digits together. Carry if necessary.
21
Q
B1T26: To multiply two numbers whose difference is 4, …
A
…square the number exactly in the middle and subtract 4.
22
Q
B1T27: When a calculation seems slightly beyond your reach, …
A
…divide one number into two smaller ones. For example, view 8 × 14 as 8 × 7 × 2.
23
Q
B1T28: To multiply two numbers that are just over 100, …
A
…begin the answer with a 1. Then affix first the sum, and then the product, of the ones digits.
24
Q
B1T28: To multiply two numbers that are just over 100, …
A
…begin the answer with a 1. Then affix first the sum, and then the product, of the ones digits.
25
#B1T29: To subtract rapidly, ...
...view subtraction as addition, and work from left to right.
26
#B1T30: To subtract when numbers are on opposite sides of 100, 200, or the like, ...
...determine the two “distances” and add.
27
#B1T31: To make subtraction easier, ...
...alter the minuend and subtrahend in the same direction to make the subtrahend a multiple of 10, 100, or the like.
28
#B1T32: To make addition easier, ...
...round up an addend, add the addends, and then subtract the number that was added when rounding.
29
#B1T33: Group numbers to be added ...
...in combinations of 10, add numbers slightly out of order, and “see” two or three numbers as their sum.
30
#B1T34: When adding columns of numbers, ...
...enter the column totals without carrying, moving one column to the left each time.
31
#B1T35: To mentally add a column of numbers, add..
...one number at a time—first the tens digit, then the ones digit.
32
#B1T36: To mentally add a column of numbers, first...
...add all the tens digits, then add all the ones digits.
33
#B1T37: When adding long columns of numbers, lightly...
...cross out a digit every time you exceed 9, and proceed with just the ones digit.
34
#B1T38: When adding long columns of numbers, ... the column...
...divide the column into smaller, more manageable sections.
35
#B1T39: When adding just a few numbers, ...
...it is fastest to begin with the largest number and to end with the smallest.
36
#B1T40: To add 1 + 2 + 3 + ... + n, ...
...multiply n by (n + 1), and then divide by 2.
37
#B1T41: If you prefer not to “subtract by adding,” then ...
...you can subtract in two steps—first the tens digits, then the ones.
38
#B1T43: To multiply a number by 75, ....
...multiply the number by 3/4
39
#B1T44: To divide a number by 75, ...
...multiply the number by 1 1/3.
40
#B1T45: To divide a number by 8, ...
...multiply the number by 1 1/4.
41
#B1T46: To divide a number by 15, ...
...multiply the number by 2/3
42
#B1T47: To estimate multiplication by 33 or 34, ...
...divide by 3.
43
#B1T48: To estimate division by 33 or 34, ...
...multiply by 3.
44
#B1T49: To estimate multiplication by 49 or 51, ...
...divide by 2.
45
#B1T50: To estimate division by 49 or 51, ...
...multiply by 2.
46
#B1T51: To estimate multiplication by 66 or 67, ...
...multiply by 2/3.
47
#B1T52: To estimate division by 66 or 67, ...
...multiply by 1.5.
48
#B1T53: To estimate division by 9, ...
...multiply by 11.
49
#B1T54: To estimate division by 11, ...
...multiply by 9.
50
#B1T55: To estimate division by 14, ...
...multiply by 7.
51
#B1T56: To estimate division by 17, ...
...multiply by 6.
52
#B1T57: When a multiplication seems slightly beyond your grasp, ...
...regroup, as in the following example: 43 × 6 = (40 × 6) + (3 × 6) = 258.
53
#B1T58: When a multiplicand or multiplier is just shy of a multiple of 10 or 100, ...
...round up and subtract, as in the following example: 15 × 29 = (15 × 30) - (15 × 1) = 435.
54
#B1T59: To multiply a three-digit or larger number by 11, ...
...first carry down the ones digit of the number. Then add the ones and tens digits, the tens and hundreds digit, and so forth. Carry when necessary.
55
#B1T60: Dividing by all nines ...
...produces a repeating pattern. For example, 2/9 = 0.222..., 37/99 = 0.3737..., and 486/999 = 0.486486...
56
#B2T1: When multiplying by a one-digit number, ...
...enter each product without carrying, moving one column to the left each time.
57
#B2T2: When adding a column of numbers, ...
...“bunch up” the partial sums without carrying, moving one column to the left each time.
58
#B2T3: Where possible, subtract without...
...borrowing by handling two columns at once.
59
#B2T4: When dividing with even numbers, ...
...cut each number in half to simplify the computation.
60
#B2T5: When adding pluses and minuses, you may ...
...(a) proceed one number at a time, (b) add the pluses, then the minuses, and combine, or (c) add, netting out the minuses against the pluses.
61
#B2T6: When adding numbers that are close to each other, ...
...guess at the midpoint, multiply, then compute with the distances from the midpoint.
62
#B2T7: Where possible, ... a two-digit multiplier ...
...convert a two-digit multiplier into two one-digit multipliers. For example, 384 × 42 could be computed as 384 × 7 × 6.
63
#B2T8: When multiplying three or more numbers, ...
...proceed in a different order if it will simplify the computation.
64
#B2T9: Multiply by 6 by ...
...multiplying by 3, then 2 (or by 2, then 3). For example, 18 × 6 could be computed as 18 × 3 × 2.
65
#B2T10: Divide by 6 by ...
...dividing by 3, then 2 (or by 2, then 3). For example, 102 / 6 could be computed as 102 / 2 / 3.
66
#B2T11: When adding the same digit several times in a column, ...
...multiply to speed up the calculation.
67
#B2T12: When adding numbers just under multiples of $1, ...
...round up, add, then deduct the amounts that were “added on.”
68
#B2T13: When multiplying, look for ...
...“digit-multiples” to speed up the computation.
69
#B2T14: When dividing, ...
...multiply and subtract in your head to conserve both space and time.
70
#B2T15: Where possible, ... a two-digit divisor into ...
...convert a two-digit divisor into two one-digit divisors. For example, 4,312 + 56 could be computed as 4,312 / 8 / 7.
71
#B2T16: Multiply a two-digit number by ...
...a one-digit number by working from left to right, focusing on the “place value” of each digit.
72
#B2T17: Multiply a three-digit number by ...
...a one-digit number by working from left to right, focusing on the “place value” of each digit.
73
#B2T18: Where possible, treat a number being divided as ...
...100 (or a multiple of 100) plus something. For example, 115 / 5 could be computed as (100 + 15) / 5, or (100 / 5) + (15 / 5).
74
#B2T19: Where possible, ... a number being divided ...
...split a number being divided into two parts, each of which can be divided evenly by the divisor. For example, 105 / 3 could be computed as (99 + 6) / 3, or (99 / 3) + (6 / 3).
75
#B2T20: To multiply a number by 21, 31, or the like, ...
...multiply by 1 less (than the 21, 31, etc.), then add the number. For example, 45 × 21 could be computed as (45 × 20) + 45.
76
#B2T21: To multiply a number by 19, 29, or the like, ...
...multiply by 1 more (than the 19, 29, etc.), then subtract the number. For example, 7 × 29 could be computed as (7 × 30) - 7.
77
#B2T22: Multiply by 12 by ...
...breaking the 12 into smaller parts. For example, 16 × 12 could be computed as 16 × 6 × 2.
78
#B2T23: Divide by 12 by ...
...breaking the 12 into smaller parts. For example, 156 / 12 could be computed as 156 / 2 / 2 / 3.
79
#B2T24: Where possible, treat a number being divided as ...
...100 (or a multiple of 100) minus something. For example, 92 / 4 could be computed as (100 - 8) / 4, or (100 / 4) - (8 / 4).
80
#B2T25: Where possible, treat a number being divided as something...
...minus something, where both numbers can be divided evenly by the divisor. For example, 144 / 3 could be computed as (150 - 6) / 3, or (150 / 3) - (6 / 3).
81
#B2T26: When subtracting numbers just under 100 (or multiple of 100), ...
...subtract the next higher multiple of 100, then add back the amount oversubtracted.
82
#B2T27: To multiply with the numbers 11 through 19, ...
...compute as in the following example: 17 × 12 = [(17 + 2) × 10] + (7 × 2).
83
#B2T28: To square a number ending in 1, ...
...compute as in the following example: 31^2 = (32 × 30) + 1. To square a number ending in 9, compute as in the following example: 19^2 = (18 × 20) + 1.
84
#B2T29: To square a number ending in 2, ...
...compute as in the following example: 32^2 = (34 × 30) + 4. To square a number ending in 8, compute as in the following example: 28^2 = (26 × 30) + 4.
85
#B2T30: To square a number ending in 3, ...
...compute as in the following example: 23^2 = (26 × 20) + 9. To square a number ending in 7, compute as in the following example: 17^2 = (14 × 20) + 9.
86
#B2T31: To multiply by a multiple of 11, ...
...multiply by the previous multiple of 10, then add 10%.
87
#B2T32: To multiply by a multiple of 9, ...
...multiply by the next multiple of 10, then subtract 10%.
88
#B2T33: To square a number ending in 6, ...
...compute as in the following example: 16^2 = 15^2 + 15 + 16.
89
#B2T34: To square a number ending in 4, ...
...compute as in the following example: 24^2 = 25^2 - 25 - 24.
90
#B2T35: Multiply a four-digit number by a one-digit number by ...
...working from left to right, focusing on the “place value” of each digit.
91
#B2T36: Multiply a two-digit number by a two-digit number by ...
...multiplying tens digits, cross-multiplying twice, then multiplying ones digits. Then add the products.
92
#B2T37: To square a number between 90 and 100, ...
...subtract from the number its distance from 100 for the left half of the answer. Then square the distance for the right half.
93
#B2T38: To square a number between 100 and 110, ...
...add to the number its distance from 100 for the left portion of the answer. Then square the distance for the right portion.
94
#B2T39: To multiply two numbers that are a little under 100, ...
...subtract from either number the distance from 100 of the other number for the left half of the answer. Then multiply the two distances together for the right half.
95
#B2T40: To multiply two consecutive numbers ending in 5, ...
...square the number exactly in the middle and subtract 25. For example, 15 × 25 = 20^2 - 25.
96
#B2T41: To multiply two alternate numbers ending in 5, ...
...square the number exactly in the middle and subtract 100. For example, 35 × 55 = 45^2 - 100.
97
#B2T42: To multiply two three-digit numbers whose middle digits are zero (e.g., 409 × 703), ...
...multiply hundreds digits, cross-multiply twice, add the products, then multiply the ones digits. Then combine the amounts from Steps 1, 3, and 4.
98
#B2T43: To multiply with one number just under a whole, ...
...round up the mixed number, multiply, then subtract the overstated amount.
99
#B2T44: When adding a column of numbers, ...
...place at the bottom of the next column each carried amount. A new digit of the answer will appear as each column is added.
100
#B2T45: When adding two numbers, ... one number so that...
...break apart one number so that one of its components plus the other number will total 100 (or a multiple of 100).
101
#B2T46: To multiply two two-digit numbers consisting of a number with a repeated digit (as 66) and another number whose digits add to 10 (as 37), ...
...first pretend that the latter number is 10 greater than it is (47, using our example above). Then multiply the tens digits together, then the ones digits together, writing the answer from left to right.
102
#B2T47: To multiply two two-digit numbers whose tens digits add to 10 and whose ones digits are identical (e.g., 76 × 36), ...
...multiply the tens digits together and add the ones digit for the left half of the answer. Then square the ones digit for the right half.
103
#B2T48: When multiplying a whole number by a fraction (e.g., 28 × 3/4), ...
...move the denominator of the fraction underneath the whole number if it will simplify the calculation.
104
#B2T49: Where manageable, multiply by two or three digits ...
...at once. For example, when computing 812 × 5, multiply 12 by 5 in one fell swoop.
105
#B2T50: To multiply a two-digit number by 111, ...
...add the digits of the number, and insert the sum within the number itself twice. This trick will only work for digit-sums of 9 or less.
106
#B2T51: To multiply a three-digit or larger number by 111, ...
...first carry down the ones digit of the number. Then add the ones and tens digits; the ones, tens, and hundreds digits; and so forth. Finally, carry down the first digit of the number. Carry when necessary.
107
#B2T52: To multiply a one-, two-, or three-digit number by 999, ...
...subtract 1 from the number, and affix the difference between 1,000 and the number.
