GMAT 00 - Arithmetic Flashcards
Which quantity is bigger? A) square root 3 + cube root 3 + 4th root 3 B) 5.1
= 5.1 square root 3 = 1.7 (approx) cube root of 3 = smaller than 1.7 4th root of 3 = smaller than cube root of 3 = smaller than 1.7 1.7 + 1.7 + 1.7 = 5.1 since cube root and 4th root are smaller than 1.7, 5.1 is bigger.
108/9 (chunking technique)
= 12 108/9 = (99 + 9)/9 = 99/9 + 9/1 = 11 + 1 = 12
4x7 =
= 28
11x7 =
= 77
8x10 =
= 80
21x11 (arithmetic shortcut)
= 231 21x11 = (20+1) x 11 = (20x11) + 11 = 220 + 11 = 231
1,000 =
10^3
3^4 =
= 81
16x9 (arithmetic shortcut)
= 144 16x9 = 16x10 - 16 = 160 - 16 = 144
4x12 =
= 48
4/9 (decimal equivalent)
= .4444444
165/11
= 15 165/11 = (110 + 55) / 11 = 110/11 + 55/11 = 10 + 5 = 15 – Chunking –
0.0833333 (fraction conversion) =
= 1/12
9x4 =
= 36
4x8 =
= 32
is 405 divisible by 3, 5, and 9?
= Yes 4+0+5 = 9 9 = divisible by 3 9 = divisible by 9 405 = ends in 5 = divisible by 5
12x10 =
= 120
1/99 (decimal conversion) =
= 0.0101010101
17^2 =
= 289
2x11 =
= 22
10x8 =
= 80
Trick: Divide by 8
Cut your number in half three times
9x8 =
= 72
4^3 =
= 64
100 =
= 10^2
322/14 (chunking technique)
= 23 322/14 = (280 + 42)/14 = 280/14 + 42/14 = 20 + 3 = 23
9x6 =
= 54
10^2 =
= 100
Square Root of 5
= 2.2 (approx)
4x2 =
= 8
6x8 =
= 48
256 =
4^4 or 2^8
8x8 =
= 64
142/3 =
= 47 remainder 1 142/3 = (120 + 22)/3 = 120/3 + 22/3 = 40 + 7 remainder 1 = 47 remainder 1
196 =
= 14^2
4x11 =
= 44
is 108 divisible by 3, 4, 6, and 9? (Divisibility Rules)
= Yes 1 + 0 + 8 = 9 9 = divisible by 3 9 = divisible by 9 108 = 08/2 = 04/2 = 02 = Divisible by 4 108 = even = divisible by 2 108 = divisible by 2 and 3 = divisible by 6
Dimensions of a Right Isosceles Triangle
= X : X : X root 2
144 =
= 12^2
4^3 =
= 64
The Fast Fraction Shortcut
1/x + 1/y = (y+x)/xy also works for subtraction 1/x - 1/y = (y-x)/xy
225 =
= 15^2
13^2 =
= 169
6^3 =
= 216
An integer is divisible by 9 if… (Divisibility Rules)
If the sum of its digits is divisible by 9
400 =
= 20^2
Trick: 20% shortcut (or 30% or 40%)
To get 20% of a number, calculate 10% and multiply it by 2 (or 3 for 30% or 4 for 40%)
5x5 =
= 25
10x2 =
= 20
7x8 =
= 56
92/4 (arithmetic shortcut)
= 23 92/4 = (92/2) / 2 = 46/2 = 23
Dimensions of a 30-60-90 Right Triangle
= X : X root 3 : 2x
25^2 =
= 625
11x8 =
= 88
An integer is divisible by 10 if… (Divisibility Rules)
If it ends in a 0
25 =
= 5^2
125 =
5^3
16 =
2^4 or 4^2
20^2 =
= 400
32 =
2^5
11x3 =
= 33
9x11 =
= 99
1/12 (decimal conversion) =
= 0.083333333
30^2 =
= 900
10x6 =
= 60
Definition: A Perfect Cube
A Perfect Cube is a number that results from one number multiplied by itself twice (the square of itself) ex. 125 is a perfect cube (5x5x5 or 5x5^2)
9x12 =
= 108
3x9 =
= 27
289 =
= 17^2
3^2 =
= 9
1/4 (decimal conversion) =
= 0.25
11 x 14 (multiplication trick)
= 154 11x14 = 14 (10 + 1) = 140 + 14 = 154
The Perfect Square Minus 1 Trick
If you know your perfect squares, then you can figure out the total of any numbers multiplied around any perfect square. For example: 6x6 = 36 therefore 5x7 = (6x6 - 1) = 35
3x10 =
= 30
3x4 =
= 12
5x6 =
= 30
2^2 =
= 4
1/9 (decimal conversion) =
= 0.1111111
12x4 =
= 48
19^2 =
= 361
6x3 =
= 18
15^2 (numerical equivalent)
= 225
Trick: 10% shortcut
To get 10% of a number, shift the decimal one space to the left
243 =
3^5
0.09090909 (fraction conversion) =
= 1/11
6x9 =
= 54
12x5 =
= 60
7x5 =
= 35
8x2 =
= 16
1/7 (decimal conversion) =
= 0.14 (approx)
0.1111111 (fraction conversion) =
= 1/9
9x9 =
= 81
1/11 (decimal conversion) =
= 0.09090909
Square Root of 1
= 1
36 =
= 6^2
0.1 (fraction conversion) =
= 1/10
0.125 (fraction conversion) =
= 1/8
If n = 161/7, what is the value of n?
= 23 161/7 = (140 + 21) 7 = 140/7 + 21/7 = 20 + 3 = 23
10x5 =
= 50
is 480 diisible by 3, 6 and 9?
= 3 and 6 = Yes = 9 = No 480 = 4 + 8 = 12 = divisible by 3 = not divisible by 9 480 = even = divisible by 2 480 = divisible by 2 and 3 = divisible by 6
Which of these quantities are greater: A = Square root of 2 + square root of 6 + square root of 8 B = 4
Answer = A A = (approx) 1.4 + 2.4 + 2.8 = 6.6 B = 4 << This is a trap question: you can’t add roots directly >>
256 =
= 16^2
31 x 13 (multiplication trick)
= 403 31x13 = 31(10+3) = 310 + 93 = 403
Does 16x25 = 400?
Answer = Yes 16 = 4x4 (25)(4)(4) = (100)(4) = 400
2x5 =
= 10
3x6 =
= 18
5x1 =
= 5
7x99 (arithmetic shortcut)
= 693 7x99 = 7x100 - 7 = 700 - 7 = 693
11x6 =
= 66
24x5 (arithmetic shortcut)
= 120 24x5 = (20+4) x 5 = 20x5 + 4x5 = 100 + 20 = 120
Does 18 x 9 = 162?
Answer = yes 18 = (9)(2) (9)(2) x 9 = (9)(9)(2) = 81 x 2 = 162
0.33333 (fraction conversion) =
= 1/3
1/3 (decimal conversion) =
= 0.33333
6x6 =
= 36
11x9 =
= 99
5x2 =
= 10
19x21 =
= 399 19x21 = 20x20 - 1 19x21 = 400 - 1 = 399 – Perfect Square Minus 1 Trick –
3^3 =
= 27
Definition: A Perfect Square
A Perfect Square is a number that results from one number multiplied by itself Ex. 25 is a perfect square (5x5)
8 x 17 (multiplication trick)
= 136 8 x 17 = 8 (10 + 7) = 80 + 56 = 136
9x5 =
= 45
5%, 10%, and 15% of 40 (10% shortcut)
= 2, 4, and 6
2^3 =
= 8
361 =
= 19^2
64 =
2^6 or 4^3
2/7 (decimal equivalent)
= .28
2^6 =
= 64
90/5 (arithmetic shortcut)
= 18 90/5 = (18x5)/5 = 18
6x10 =
= 60
5x8 =
= 40
1/2 (decimal conversion) =
= 0.5
If a = 35x5x28 and b = 12x25x13, then what is the value of (a-b)/100?
Answer = 10 FACTOR a = 35 x 5 x 28 = (7x5) (5) (7x4) = (7x7) (5x5x4) = (49) (100) = 4,900 b = 12 x 25 x 13 = (3x4) (25) (13) = (3x13) (4x25) = (39) (100) = 3,900 a - b = 4,900 - 3,900 = 1,000 1,000 / 100 = 10
108/7 =
= 15 remainder 3 108/7 = (70 + 38) / 7 = 70/7 + 38/7 = 10 + 5 rem. 3 = 15 remainder 3
27 =
3^3
8x5 =
= 40
8^3 =
= 512
square root of two (numerical equivalent)
= 1.4 (approx)
2x10 =
= 20
Does 35 x 15 x 12 = 6300?
Answer = yes 35 = (7x5) 15 = (5x3) 12 = (3x4) 35 x 15 x 12 = (7x5) x (5x3) x (3x4) = (5x5) x (4) x (3x3) x (7) = 25 x 4 x 9 x 7 = 100 x 63 = 6300
11^2 =
= 121
An integer is divisible by 5 if… (Divisibility Rules)
If it ends in 0 or 5
9x10 =
= 90
8 =
2^3
Trick: Multiply by 5
Multiply by 10, then divide by 2 (or vice versa)
7x2 =
= 14
64 (numerical equivalent)
= 4^3
Trick: 15% shortcut
To get 15% of a number, calculate 10% and 5% of the number, then add them together
7 x 123 (multiplication trick)
= 861 7x123 = 7 x (100 + 23) = 700 + 7(23) = 700 + 7(20 + 3) = 700 + 140 + 21 = 861
12x7 =
= 84
64 =
= 8^2
Square Root of 8
= 2.8 (approx)
8x12 =
= 96
2^4 =
= 16
2x6 =
= 12
3x11 =
= 33
4x6 =
= 24
10x3 =
= 30
9 =
3^2
0.25 (fraction conversion) =
= 1/4
Square Root of 6
= 2.4 (approx)
5x9 =
= 45
5x11 =
= 55
24^2 =
= 576
9x7 =
= 63
529 =
= 23^2
0.0101010101 (fraction conversion) =
= 1/99
4/2x4 = ?
4/2x4 = (4/2) x 4 = 2 x 4 = 8 Multiplication and Division are equal operations. Be sure to do them in order from left to right without hierarchy.
144/8 (arithmetic shortcut)
= 18 144/8 = (12x12)/8 = (4x3x6x2)/(4x2) = (4x2)(6x3)/4x2) = 18 or 8 x 20 = 160 - 8 = 152 - 8 = 144
625 =
= 25^2
Trick: Divide by 4
Cut your number in half twice
1/8 (decimal conversion) =
= 0.125
11x13 =
= 143 11x13 = 12x12 - 1 11x13 = 144-1 = 143 – Perfect Square Minus 1 Trick –
8x7 =
= 56
10x12 =
= 120
11x12 =
= 132
3x8 =
= 24
7x3 =
= 21
20%, 40%, and 60% of 120 (10% shortcut)
= 24, 48, and 72
1%, 10%, and 12% of 240 (10% shortcut)
= 2.4. 24, and 28.8
10x11 =
= 110
14^2 =
= 196
5x10 =
= 50
8x9 =
= 72
11x11 =
= 121
4x5 =
= 20
6x12 =
= 72
7x6 =
= 42
1/5 (decimal conversion) =
= 0.2
3x7 =
= 21
If x = 350/14 and y = 900/18, what is the value of x+y?
x + y = 75 FACTOR X = 350/14 = (35)(10) / (2)(7) = (7)(5)(5)(2) / (2)(7) = (5)(5) = 25 X = 25 Y = 900/18 = (9)(100) / (9)(2) = 100/2 = 50 Y = 50 X + Y = 25 + 50 = 75
Which of the following is a factor of 5,427? A = 4 B = 5 C = 6 D = 7 E = 9
Answer = E = 9 > Not even, so not divisible by 4 = not A > Doesn’t end in 5, so not divisible by 5 = not B > Not even, so not divisible by 2 = not divisible by 2 & 3 = not divisible by 6 = not C > 5427 = 5400 + 27 = (100)(54) + 27 = not divisible by 7 = not D > 5+4+2+7 = 18 = divisible by 9 = E – Divisibility Rules –
81 =
9^2 or 3^4
7x4 =
= 28
7x10 =
= 70
0.2 (fraction conversion) =
= 1/5
The number 10^15 is divisible by all of the following EXCEPT: A) 500 B) 250 C) 64 D) 32 E) 6
Answer = E = 6 10^15 = 1,000,000,000,000,000 A & B = yes (end in 5) C = 64 = 8x8 = last 3 digits cut in half 3 times = (0 to 5 to 0 to 5) = still whole = yes D = 32 = 8x4 = last 3 digits cut in half 3 times (0 to 5 to 0 to 5) and last two digits cut in half twice (0 to 5 to 0) = yes E = 1,000,000,000,000,000,000 (sum of digits) = 1 = NOT divisible by 6 – Divisibility Rules –
2x2 =
= 4
11x5 =
= 55
1/100 (decimal conversion) =
= 0.01
Definition “Chunking”
A division trick in which you separate large numbers into smaller, more manageable numbers before performing division. Ex: 168 / 14 = (140 + 28) / 14 = 140/14 + 28/14 = 10 + 2 = 12
3x2 =
= 6
324 =
= 18^2
Which of these quantities is larger: A = 63x5x42 B = 49x9x30
Answer = they are equal A = 63x5x42 = (9x7) (5) (7x6) = (7x7) (9) (5x6) B = 49x9x30 = (7x7) (9) (3x5x2) = (7x7) (9) (5x6)
2x7 =
= 14
49 =
= 7^2
Which of the following quantities is greater: A = 241 X 11.6 B = 241(11) + 241(0.6)
Answer = They are Equal
Square Root of 10
= 3.2 (approx)
5^3 (numerical equivalent)
= 125
4^4 =
= 256
22^2 =
= 484
900 =
30^2
5^3 =
= 125
3^5 =
= 243
12 x 12.5 =
= 150 12 x 12.5 = 12 x (12 + 0.5) = (12x12) + (12 x 0.5) = 144 + 6 = 150
1%, 10%, and 98% of 110 (10% shortcut)
= 1.1, 11, and 107.8
23^2 =
= 529
1 + 4(5-3)^2 - 3x6 = ?
Answer = -1 PEMDAS 1 + 4(2)^2 - 3x6 = 1 + 4(4) - 3x6 = 1 + 16 - 18 = 17 - 18 = -1
14x16 =
= 224 14x16 = 15x15 - 1 14x15 = 225-1 = 224 – Perfect Square Minus 1 Trick –
9^3 =
= 729
3x5 =
= 15
5x7 =
= 35
484 =
= 22^2
5x4 =
= 20
5x3 =
= 15
255/11 (chunking technique)
= 23 2/11 255/11 = (22)(10)/11 + 35/11 = (2)(10) + 35/11 = 20 + 3 + 2/11 = 23 + 2/11
2^3 =
= 8
11x2 =
= 22
5x12 =
= 60
10x4 =
= 40
4^2 =
= 16
An integer is divisible by 2 if… (Divisibility Rules)
If its last digit can be cut in half
Trick: Divide by 5
Divide by 10, then multiply by 2 (or vice versa)
2x3 =
= 6
0.1666666 (fraction conversion) =
= 1/6
3x12 =
= 36
10x10 =
= 100
12x9 =
= 108
10^3 =
= 1,000
15^2 =
= 225
An integer is divisible by 6 if… (Divisibility Rules)
If it is divisible by 2 and by 3
21^2 =
= 441
18^2 =
= 324
Square Root of 9
= 3
12x6 =
= 72
576 =
= 24^2
Square Root of 4
= 2
4x4 =
= 16
is 189 divisible by 3 and 9? (Divisibility Rules)
= Yes 1 + 8 + 9 = 18 18 = divisible by 3 18 = divisible by 9
14x16 (arithmetic shortcut)
= 224 14x16 = 15x15 - 1 (perfect square trick) = 225 - 1 = 224
PEMDAS
Whenever you perform basic arithmetic, there is a specific order in which the arithmetic must be performed. This order is commonly known through the acronym PEMDAS P = Parentheses E = Exponents M = Multiplication D = Division A = Addition S = Subtraction Example: 1 + 4(5-3)^2 - 3x6 = ? Answer = -1 * Caveat: Multiplication and Division are equal operations, so instead of setting one before the other, be sure to do whichever comes first, going left to right across the problem.
0.5 (fraction conversion) =
= 1/2
9x3 =
= 27
An integer is divisible by 7 if… (Divisibility Rules)
No rule - chunk the number (factor it down, look for a 7)
Trick: Multiply by 99
Multiply by 100, then subtract the original number
12x12 =
= 144
16 =
= 4^2
0.01 (fraction conversion) =
= 1/100
7x7 =
= 49
What is the 16th digit to the right of the decimal point in the decimal equivalent of 3/11?
= 7 1/11 = 0.0909090909 (repeating) therefore, 3 x 0.0909090909 (repeating) = 0.27272727 (repeating) 16 is an even number, so the digit will be the even result = 7
Square Root of 7
= 2.6 (approx)
6x11 =
= 66
0.55555555 (fraction equivalent)
= 5/9
7x11 =
= 77
6x7 =
= 42
16^2 =
= 256
Square Root of 2
= 1.4 (approx)
Trick: 1% shortcut
To get 1% of a number, simply shift its decimal spot two spaces to the left
2^5 =
= 32
Square Root of 3
= 1.7 (approx)
Trick: 5 % shortcut
To get 5% of a number, shift the decimal one space to the left and cut the number in half
9x2 =
= 18
square root of 7 (numerical equivalent)
= 2.6 (approx)
12x3 =
= 36
2x12 =
= 24
6x5 =
= 30
20% of 180
= 36 20% of 180 = 2 x 10% of 180 = 2 x 18 = 36
165/3 (chunking technique)
= 55 165/3 = (150 + 15)/3 = 150/3 + 15/3 = 50 + 5 = 55
7^3 =
= 343
An integer is divisible by 4 if… (Divisibility Rules)
If its last TWO digits can be cut in half TWICE
8x4 =
= 32
12^2 =
= 144
7x12 =
= 84
0.14 (approx. fraction conversion) =
= 1/7
121 =
= 11^2
8x3 =
= 24
3^3 =
= 27
4 =
= 2^2
Trick: Multiply by 9
Multiply by 10 and subtract the original number
10x7 =
= 70
An integer is divisible by 3 if… (Divisibility Rules)
If the sum of its digits is divisible by 3
Which quantity is bigger? A = 233.104(5.005) B) 5(233.104) + 0.005(233.104)
C = they are equal 233.104(5.005) = 5(233.104) + 0.005(233.104) 233.104(5.005) = (5.005)(233.104)
4x9 =
= 36
12x8 =
= 96
Does 14 x 35 = 500?
Answer = no 14 = (7x2) 35 = (7x5) (7x2) x (7x5) = (7x7) (5x2) = (49) (10) = 490
The Multiplication Trick
When multiplying by a big, complicated number, see if you can split that number into numbers less complicated ex: 9 x 21 = 9 x (20+1) = (9 x 20) + (9 x 1) = 180 + 9 = 189
11x4 =
= 44
81/27
= 3 FACTOR Division is a fraction. You can always factor in a fraction. 81/27 = (9)(9) / (9)(3) = 9/3 = 3
7x9 =
= 63
8x6 =
= 48
3x3 =
= 9
81 =
= 9^2
8x11 =
= 88
16 (numerical equivalent)
= 4^2
The Square Root Trick
Values larger than the square root of 4 increase in increments of 0.2 and values smaller than the square root of 4 decrease in increments of 0.3 << This only works up to the square root of 10 >>
10x9 =
= 90
12x11 =
= 132
1/6 (decimal conversion) =
= 0.16666666
Which quantity is bigger? A) the 30th digit to the right of the decimal point in the decimal equivalent of 18/99? B) the 41st digit to the right of the decimal point in the decimal equivalent of 18/99?
= A 1/99 = .01010101010101 therefore 18/99 = .181818181818181818 30 = even = 8 41 = odd = 1 8 > 1 = A is bigger
0.375 (fraction equivalent)
= 3/8
9 =
= 3^2
441 =
= 21^2
4x10 =
= 40
6x2 =
= 12
4x3 =
= 12
1/10 (decimal conversion) =
= 0.1
2x4 =
= 8
An integer is divisible by 8 if… (Divisibility Rules)
If its last THREE digits can be cut in half THREE times
216 =
6^3
169 =
= 13^2
11x10 =
= 110
6x4 =
= 24
2x8 =
= 16
Trick: Multiply by 11
Multiply by 10, then add the original number
2x9 =
= 18
12x2 =
= 24
