GMAT 00 - Arithmetic Flashcards
1
Q
Which quantity is bigger? A) square root 3 + cube root 3 + 4th root 3 B) 5.1
A
= 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.
2
Q
108/9 (chunking technique)
A
= 12 108/9 = (99 + 9)/9 = 99/9 + 9/1 = 11 + 1 = 12
3
Q
4x7 =
A
= 28
4
Q
11x7 =
A
= 77
5
Q
8x10 =
A
= 80
6
Q
21x11 (arithmetic shortcut)
A
= 231 21x11 = (20+1) x 11 = (20x11) + 11 = 220 + 11 = 231
7
Q
1,000 =
A
10^3
8
Q
3^4 =
A
= 81
9
Q
16x9 (arithmetic shortcut)
A
= 144 16x9 = 16x10 - 16 = 160 - 16 = 144
10
Q
4x12 =
A
= 48
11
Q
4/9 (decimal equivalent)
A
= .4444444
12
Q
165/11
A
= 15 165/11 = (110 + 55) / 11 = 110/11 + 55/11 = 10 + 5 = 15 – Chunking –
13
Q
0.0833333 (fraction conversion) =
A
= 1/12
14
Q
9x4 =
A
= 36
15
Q
4x8 =
A
= 32
16
Q
is 405 divisible by 3, 5, and 9?
A
= Yes 4+0+5 = 9 9 = divisible by 3 9 = divisible by 9 405 = ends in 5 = divisible by 5
17
Q
12x10 =
A
= 120
18
Q
1/99 (decimal conversion) =
A
= 0.0101010101
19
Q
17^2 =
A
= 289
20
Q
2x11 =
A
= 22
21
Q
10x8 =
A
= 80
22
Q
Trick: Divide by 8
A
Cut your number in half three times
23
Q
9x8 =
A
= 72
24
Q
4^3 =
A
= 64
25
100 =
= 10^2
26
322/14 (chunking technique)
= 23 322/14 = (280 + 42)/14 = 280/14 + 42/14 = 20 + 3 = 23
27
9x6 =
= 54
28
10^2 =
= 100
29
Square Root of 5
= 2.2 (approx)
30
4x2 =
= 8
31
6x8 =
= 48
32
256 =
4^4 or 2^8
33
8x8 =
= 64
34
142/3 =
= 47 remainder 1 142/3 = (120 + 22)/3 = 120/3 + 22/3 = 40 + 7 remainder 1 = 47 remainder 1
35
196 =
= 14^2
36
4x11 =
= 44
37
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
38
Dimensions of a Right Isosceles Triangle
= X : X : X root 2
39
144 =
= 12^2
40
4^3 =
= 64
41
The Fast Fraction Shortcut
1/x + 1/y = (y+x)/xy also works for subtraction 1/x - 1/y = (y-x)/xy
42
225 =
= 15^2
43
13^2 =
= 169
44
6^3 =
= 216
45
An integer is divisible by 9 if... (Divisibility Rules)
If the sum of its digits is divisible by 9
46
400 =
= 20^2
47
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%)
48
5x5 =
= 25
49
10x2 =
= 20
50
7x8 =
= 56
51
92/4 (arithmetic shortcut)
= 23 92/4 = (92/2) / 2 = 46/2 = 23
52
Dimensions of a 30-60-90 Right Triangle
= X : X root 3 : 2x
53
25^2 =
= 625
54
11x8 =
= 88
55
An integer is divisible by 10 if... (Divisibility Rules)
If it ends in a 0
56
25 =
= 5^2
57
125 =
5^3
58
16 =
2^4 or 4^2
59
20^2 =
= 400
60
32 =
2^5
61
11x3 =
= 33
62
9x11 =
= 99
63
1/12 (decimal conversion) =
= 0.083333333
64
30^2 =
= 900
65
10x6 =
= 60
66
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)
67
9x12 =
= 108
68
3x9 =
= 27
69
289 =
= 17^2
70
3^2 =
= 9
71
1/4 (decimal conversion) =
= 0.25
72
11 x 14 (multiplication trick)
= 154 11x14 = 14 (10 + 1) = 140 + 14 = 154
73
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
74
3x10 =
= 30
75
3x4 =
= 12
76
5x6 =
= 30
77
2^2 =
= 4
78
1/9 (decimal conversion) =
= 0.1111111
79
12x4 =
= 48
80
19^2 =
= 361
81
6x3 =
= 18
82
15^2 (numerical equivalent)
= 225
83
Trick: 10% shortcut
To get 10% of a number, shift the decimal one space to the left
84
243 =
3^5
85
0.09090909 (fraction conversion) =
= 1/11
86
6x9 =
= 54
87
12x5 =
= 60
88
7x5 =
= 35
89
8x2 =
= 16
90
1/7 (decimal conversion) =
= 0.14 (approx)
91
0.1111111 (fraction conversion) =
= 1/9
92
9x9 =
= 81
93
1/11 (decimal conversion) =
= 0.09090909
94
Square Root of 1
= 1
95
36 =
= 6^2
96
0.1 (fraction conversion) =
= 1/10
97
0.125 (fraction conversion) =
= 1/8
98
If n = 161/7, what is the value of n?
= 23 161/7 = (140 + 21) 7 = 140/7 + 21/7 = 20 + 3 = 23
99
10x5 =
= 50
100
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
101
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 \>\>
102
256 =
= 16^2
103
31 x 13 (multiplication trick)
= 403 31x13 = 31(10+3) = 310 + 93 = 403
104
Does 16x25 = 400?
Answer = Yes 16 = 4x4 (25)(4)(4) = (100)(4) = 400
105
2x5 =
= 10
106
3x6 =
= 18
107
5x1 =
= 5
108
7x99 (arithmetic shortcut)
= 693 7x99 = 7x100 - 7 = 700 - 7 = 693
109
11x6 =
= 66
110
24x5 (arithmetic shortcut)
= 120 24x5 = (20+4) x 5 = 20x5 + 4x5 = 100 + 20 = 120
111
Does 18 x 9 = 162?
Answer = yes 18 = (9)(2) (9)(2) x 9 = (9)(9)(2) = 81 x 2 = 162
112
0.33333 (fraction conversion) =
= 1/3
113
1/3 (decimal conversion) =
= 0.33333
114
6x6 =
= 36
115
11x9 =
= 99
116
5x2 =
= 10
117
19x21 =
= 399 19x21 = 20x20 - 1 19x21 = 400 - 1 = 399 -- Perfect Square Minus 1 Trick --
118
3^3 =
= 27
119
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)
120
8 x 17 (multiplication trick)
= 136 8 x 17 = 8 (10 + 7) = 80 + 56 = 136
121
9x5 =
= 45
122
5%, 10%, and 15% of 40 (10% shortcut)
= 2, 4, and 6
123
2^3 =
= 8
124
361 =
= 19^2
125
64 =
2^6 or 4^3
126
2/7 (decimal equivalent)
= .28
127
2^6 =
= 64
128
90/5 (arithmetic shortcut)
= 18 90/5 = (18x5)/5 = 18
129
6x10 =
= 60
130
5x8 =
= 40
131
1/2 (decimal conversion) =
= 0.5
132
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
133
108/7 =
= 15 remainder 3 108/7 = (70 + 38) / 7 = 70/7 + 38/7 = 10 + 5 rem. 3 = 15 remainder 3
134
27 =
3^3
135
8x5 =
= 40
136
8^3 =
= 512
137
square root of two (numerical equivalent)
= 1.4 (approx)
138
2x10 =
= 20
139
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
140
11^2 =
= 121
141
An integer is divisible by 5 if... (Divisibility Rules)
If it ends in 0 or 5
142
9x10 =
= 90
143
8 =
2^3
144
Trick: Multiply by 5
Multiply by 10, then divide by 2 (or vice versa)
145
7x2 =
= 14
146
64 (numerical equivalent)
= 4^3
147
Trick: 15% shortcut
To get 15% of a number, calculate 10% and 5% of the number, then add them together
148
7 x 123 (multiplication trick)
= 861 7x123 = 7 x (100 + 23) = 700 + 7(23) = 700 + 7(20 + 3) = 700 + 140 + 21 = 861
149
12x7 =
= 84
150
64 =
= 8^2
151
Square Root of 8
= 2.8 (approx)
152
8x12 =
= 96
153
2^4 =
= 16
154
2x6 =
= 12
155
3x11 =
= 33
156
4x6 =
= 24
157
10x3 =
= 30
158
9 =
3^2
159
0.25 (fraction conversion) =
= 1/4
160
Square Root of 6
= 2.4 (approx)
161
5x9 =
= 45
162
5x11 =
= 55
163
24^2 =
= 576
164
9x7 =
= 63
165
529 =
= 23^2
166
0.0101010101 (fraction conversion) =
= 1/99
167
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.
168
144/8 (arithmetic shortcut)
= 18 144/8 = (12x12)/8 = (4x3x6x2)/(4x2) = (4x2)(6x3)/4x2) = 18 or 8 x 20 = 160 - 8 = 152 - 8 = 144
169
625 =
= 25^2
170
Trick: Divide by 4
Cut your number in half twice
171
1/8 (decimal conversion) =
= 0.125
172
11x13 =
= 143 11x13 = 12x12 - 1 11x13 = 144-1 = 143 -- Perfect Square Minus 1 Trick --
173
8x7 =
= 56
174
10x12 =
= 120
175
11x12 =
= 132
176
3x8 =
= 24
177
7x3 =
= 21
178
20%, 40%, and 60% of 120 (10% shortcut)
= 24, 48, and 72
179
1%, 10%, and 12% of 240 (10% shortcut)
= 2.4. 24, and 28.8
180
10x11 =
= 110
181
14^2 =
= 196
182
5x10 =
= 50
183
8x9 =
= 72
184
11x11 =
= 121
185
4x5 =
= 20
186
6x12 =
= 72
187
7x6 =
= 42
188
1/5 (decimal conversion) =
= 0.2
189
3x7 =
= 21
190
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
191
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 --
192
81 =
9^2 or 3^4
193
7x4 =
= 28
194
7x10 =
= 70
195
0.2 (fraction conversion) =
= 1/5
196
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 --
197
2x2 =
= 4
198
11x5 =
= 55
199
1/100 (decimal conversion) =
= 0.01
200
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
201
3x2 =
= 6
202
324 =
= 18^2
203
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)
204
2x7 =
= 14
205
49 =
= 7^2
206
Which of the following quantities is greater: A = 241 X 11.6 B = 241(11) + 241(0.6)
Answer = They are Equal
207
Square Root of 10
= 3.2 (approx)
208
5^3 (numerical equivalent)
= 125
209
4^4 =
= 256
210
22^2 =
= 484
211
900 =
30^2
212
5^3 =
= 125
213
3^5 =
= 243
214
12 x 12.5 =
= 150 12 x 12.5 = 12 x (12 + 0.5) = (12x12) + (12 x 0.5) = 144 + 6 = 150
215
1%, 10%, and 98% of 110 (10% shortcut)
= 1.1, 11, and 107.8
216
23^2 =
= 529
217
1 + 4(5-3)^2 - 3x6 = ?
Answer = -1 PEMDAS 1 + 4(2)^2 - 3x6 = 1 + 4(4) - 3x6 = 1 + 16 - 18 = 17 - 18 = -1
218
14x16 =
= 224 14x16 = 15x15 - 1 14x15 = 225-1 = 224 -- Perfect Square Minus 1 Trick --
219
9^3 =
= 729
220
3x5 =
= 15
221
5x7 =
= 35
222
484 =
= 22^2
223
5x4 =
= 20
224
5x3 =
= 15
225
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
226
2^3 =
= 8
227
11x2 =
= 22
228
5x12 =
= 60
229
10x4 =
= 40
230
4^2 =
= 16
231
An integer is divisible by 2 if... (Divisibility Rules)
If its last digit can be cut in half
232
Trick: Divide by 5
Divide by 10, then multiply by 2 (or vice versa)
233
2x3 =
= 6
234
0.1666666 (fraction conversion) =
= 1/6
235
3x12 =
= 36
236
10x10 =
= 100
237
12x9 =
= 108
238
10^3 =
= 1,000
239
15^2 =
= 225
240
An integer is divisible by 6 if... (Divisibility Rules)
If it is divisible by 2 and by 3
241
21^2 =
= 441
242
18^2 =
= 324
243
Square Root of 9
= 3
244
12x6 =
= 72
245
576 =
= 24^2
246
Square Root of 4
= 2
247
4x4 =
= 16
248
is 189 divisible by 3 and 9? (Divisibility Rules)
= Yes 1 + 8 + 9 = 18 18 = divisible by 3 18 = divisible by 9
249
14x16 (arithmetic shortcut)
= 224 14x16 = 15x15 - 1 (perfect square trick) = 225 - 1 = 224
250
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.
251
0.5 (fraction conversion) =
= 1/2
252
9x3 =
= 27
253
An integer is divisible by 7 if... (Divisibility Rules)
No rule - chunk the number (factor it down, look for a 7)
254
Trick: Multiply by 99
Multiply by 100, then subtract the original number
255
12x12 =
= 144
256
16 =
= 4^2
257
0.01 (fraction conversion) =
= 1/100
258
7x7 =
= 49
259
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
260
Square Root of 7
= 2.6 (approx)
261
6x11 =
= 66
262
0.55555555 (fraction equivalent)
= 5/9
263
7x11 =
= 77
264
6x7 =
= 42
265
16^2 =
= 256
266
Square Root of 2
= 1.4 (approx)
267
Trick: 1% shortcut
To get 1% of a number, simply shift its decimal spot two spaces to the left
268
2^5 =
= 32
269
Square Root of 3
= 1.7 (approx)
270
Trick: 5 % shortcut
To get 5% of a number, shift the decimal one space to the left and cut the number in half
271
9x2 =
= 18
272
square root of 7 (numerical equivalent)
= 2.6 (approx)
273
12x3 =
= 36
274
2x12 =
= 24
275
6x5 =
= 30
276
20% of 180
= 36 20% of 180 = 2 x 10% of 180 = 2 x 18 = 36
277
165/3 (chunking technique)
= 55 165/3 = (150 + 15)/3 = 150/3 + 15/3 = 50 + 5 = 55
278
7^3 =
= 343
279
An integer is divisible by 4 if... (Divisibility Rules)
If its last TWO digits can be cut in half TWICE
280
8x4 =
= 32
281
12^2 =
= 144
282
7x12 =
= 84
283
0.14 (approx. fraction conversion) =
= 1/7
284
121 =
= 11^2
285
8x3 =
= 24
286
3^3 =
= 27
287
4 =
= 2^2
288
Trick: Multiply by 9
Multiply by 10 and subtract the original number
289
10x7 =
= 70
290
An integer is divisible by 3 if... (Divisibility Rules)
If the sum of its digits is divisible by 3
291
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)
292
4x9 =
= 36
293
12x8 =
= 96
294
Does 14 x 35 = 500?
Answer = no 14 = (7x2) 35 = (7x5) (7x2) x (7x5) = (7x7) (5x2) = (49) (10) = 490
295
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
296
11x4 =
= 44
297
81/27
= 3 FACTOR Division is a fraction. You can always factor in a fraction. 81/27 = (9)(9) / (9)(3) = 9/3 = 3
298
7x9 =
= 63
299
8x6 =
= 48
300
3x3 =
= 9
301
81 =
= 9^2
302
8x11 =
= 88
303
16 (numerical equivalent)
= 4^2
304
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 \>\>
305
10x9 =
= 90
306
12x11 =
= 132
307
1/6 (decimal conversion) =
= 0.16666666
308
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
309
0.375 (fraction equivalent)
= 3/8
310
9 =
= 3^2
311
441 =
= 21^2
312
4x10 =
= 40
313
6x2 =
= 12
314
4x3 =
= 12
315
1/10 (decimal conversion) =
= 0.1
316
2x4 =
= 8
317
An integer is divisible by 8 if... (Divisibility Rules)
If its last THREE digits can be cut in half THREE times
318
216 =
6^3
319
169 =
= 13^2
320
11x10 =
= 110
321
6x4 =
= 24
322
2x8 =
= 16
323
Trick: Multiply by 11
Multiply by 10, then add the original number
324
2x9 =
= 18
325
12x2 =
= 24
