1. It takes 3 parts of cement and 4 parts of sand to make a specific mixture. How many containers of cement is required to make a mixture of 28 containers?
find:
x = number of containers of cement
solution:
TOTAL = sum of parts
7/7 = 3/7 + 4/7 ---> denominator (7) = 3 + 4
mixture = cement + sand
7/7(28) = 3/7(28) + 4/7(28)
x = 3/7(28)
x = 12
2. How many liters of a 20% solution should be added to 40 liters of 80% solution to make a mixture containing 50% of concentration?
find:
v1 = number of liters of 20% solution
solution:
v1*concentration1 + v2*concentration2 = Vtotal*finalconcentration
v1(0.2) + 40(0.8) = (v1 + 40)(0.5)
0.2v1 + 32 = 0.5v1 + 20
0.3v1 = 12
v1 = 40
3. How many quarters, dimes, and nickels are there if their total is $3.60 and there are a total of 21 coins and the number of nickels is twice the number of dimes.
find:
n = number of nickels
d = number of dimes
q = number of quarters
given:
cents = 360
n = 2d ---> equation1
n + d + q = 21 ---> equation2
solution:
TOTAL = sum of parts
360 = 5n + 10d + 25q ---> equation3
substituting n = 2d in equation2
n + d + q = 21
2d + d + q = 21
3d + q = 21
q = 21 - 3d ---> equation4
substituting n = 2d in equation3
360 = 5n + 10d + 25q
360 = 5(2d) + 10d + 25q
360 = 10d + 10d + 25q
360 = 20d + 25q ---> equation5
substituting equation4 in equation5
360 = 20d + 25q
360 = 20d + 25(21 - 3d)
360 = 20d + 525 - 75d
55d = 525 - 360
55d = 165
d = 3
n = 2d = 2(3) = 6
q = 21 - 3d = 21 - 3(3) = 12
checking:
d cents = 3 * 10 = 30 cents
n cents = 6 * 5 = 30 cents
q cents = 12 * 25 = 300 cents
total = 30 + 30 + 300 = 360
Wednesday, October 24, 2012
Tuesday, October 23, 2012
IQ TEST: Math, Fractions, Series, Sequence, Military Time, Clock, Days of the week
1. What letter is three to the left of the letter that is immediately
to the right of the letter that is two to the left of the letter L?
A B C D E F G H I J K L M N O P Q R
solution:
start from the "inside" just like starting from the parentheses in math
two to the left of L ----> J
right of letter J ----> K
three to the left of K ----> H
2. What is the number that is one half of one quarter of one eighth of 448?
solution:
start from the "inside"
1/8 of 448 = 56
1/4 of 56 = 14
1/2 of 14 = 7
3. How many minutes is it before midnight if six tenth of an hour ago it
was twice as many minutes past 10 pm?
find:
x = minutes before midnight
given:
Total time from 10 pm to midnight ---> 120 minutes
t1 = 6/10 hr
unit analysis: hr * 60 min/hr
t1 = 6/10 * 60 = 36 minutes
solution:
TOTAL = sum of parts ---> working equation
substituting:
120 = x + 2x + 36
3x = 84
x = 84/3
x = 28 ---> answer
check:
28 minutes before midnight is 11:32 pm
36 minutes ago, the time is 10:56 pm
10:56 pm is 56 minutes after 10 pm
56 = twice of x
4. What time is it now if 2 hours later it would be half as long until 5 pm as it would be if it were an hour from now?
find:
x = time now
given:
5 pm ---> 17:00
solution:
17 - (x + 2) = 1/2 [17 - (x + 1)]
17 - x - 2 = 1/2 (17 - x - 1)
15 - x = 1/2 (16 - x)
15 - x = 8 - x/2
x/2 = 15 - 8
x = 14 ---> or 2 pm
5. a. What is the day today if the day before yesterday is 2 days after Friday? b. What is the day today if the day after tomorrow is 2 days before Saturday?
solution:
a.
put Friday first on the sequence, then start from the "inside"
FRIDAY sat SUNDAY mon TUESDAY wed THURSDAY fri SATURDAY
2 days after friday ---> sunday
sunday is the day before yesterday
monday is yesterday
TUESDAY = today ---> answer
b.
2 days before saturday ---> thursday
thursday is the day after tomorrow
wednesday is tomorrow
TUESDAY = today ---> answer
Sunday, October 21, 2012
Trigonometry - Law of sines, Bearing, Ship, Motorboat, Navy, Marines, Sailor
1. A ship takes a sighting of two motorboats. Motorboat A has a bearing of N 44 W and Motorboat B has a bearing of N 59 E. The distance between the two motorboats is 4 km and the bearing of Motorboat B from A is N 88 E. Compute the distance of the ship from each motorboat.
find:
SA = distance of the ship from Motorboat A
SB = distance of the ship from Motorboat B
given:
Bearing of Motorboat A = N 44 W
Bearing of Motorboat B = N 59 E
Bearing of Motorboat B from A = N 88 E
AB = 4 km, distance between Motorboat A and Motorboat B
solution:
A_B
S\/
angle ASB = 44 + 59 = 103
angle BAS = 180 - 88 - 44 = 48
angle ABS = 180 - 103 - 48 = 29
using Sine Law:
SA/sin ABS = SB/sin BAS = AB/sin ASB
SA/sin 29 = AB/sin 103
SA/0.485 = 4/0.974
SA = 0.485 * 4/0.974
SA = 1.99 km
SB/sin 48 = AB/sin 103
SB/0.743 = 4/0.974
SB = 0.743 * 4/0.974
SB = 3.05 km
find:
SA = distance of the ship from Motorboat A
SB = distance of the ship from Motorboat B
given:
Bearing of Motorboat A = N 44 W
Bearing of Motorboat B = N 59 E
Bearing of Motorboat B from A = N 88 E
AB = 4 km, distance between Motorboat A and Motorboat B
solution:
A_B
S\/
angle ASB = 44 + 59 = 103
angle BAS = 180 - 88 - 44 = 48
angle ABS = 180 - 103 - 48 = 29
using Sine Law:
SA/sin ABS = SB/sin BAS = AB/sin ASB
SA/sin 29 = AB/sin 103
SA/0.485 = 4/0.974
SA = 0.485 * 4/0.974
SA = 1.99 km
SB/sin 48 = AB/sin 103
SB/0.743 = 4/0.974
SB = 0.743 * 4/0.974
SB = 3.05 km
IQ TEST: Math, Probability, Permutations, Repetitions, Ordered Combinations, Exclusive events, Number of digits
1. If there are 10 people in a room and they would make handshakes with each one in the room, how many unique handshakes would be made?
find:
H = number of unique handshakes
solution:
n = number of people = 10
H = (n - 1) + (n - 2) + (n - 3) .....until LAST ADDEND is 1
first term = PEOPLE - 1
H = 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1
H = 45
if there are only 5 people in the room
H = 4 + 3 + 2 + 1
H = 10
2. If a dice and a coin are tossed, what is the probability that you will get a 4 on the dice and heads on the coin?
solution:
P = Pdice * Pcoin
P = 1/6 * 1/2
P = 1/12
3. A circular spinner is marked W, X, Y, Z. what is the probability of NOT getting an X if spinned thrice?
solution:
if spinned once ---> P1 = 3/4
if spinned twice ---> P2 = 3/4 * 3/4 = 9/16
if spinned thrice ---> P1 = 3/4 * 3/4 * 3/4 = 27/64
4. There are 5 choices each question. What is the probability of correctly guessing 3 questions?
solution:
P = 1/5 * 1/5 * 1/5
P = 1/125
5. If you have 5 pants and 8 shirts, how many ways can you wear them?
solution:
W = 5 * 8
W = 40
6. In a restaurant, there are 3 choices of soup, 5 choices of entrees, and 2 choices of desert. How many ways can you order them if each order consists of one soup, one entree, and one desert?
solution:
W = 3 * 5 * 2
w = 30
7. If 4 warriors line up so that the order matters, how many ordered combinations (permutations) are possible?
P = 4 * 3 * 2 * 1
P = 24
8. If you count from 1 to 100, How many zeroes are there? How many sevens are there?
solution:
let
Zt = total number of 0's
St = total number of 7's
1 to 10 : z = 1, s = 1
11 to 20: z = 1, s = 1
21 to 30: z = 1, s = 1
31 to 40: z = 1, s = 1
41 to 50: z = 1, s = 1
51 to 60: z = 1, s = 1
61 to 70: z = 1, s = 2 (67, 70)
71 to 80: z = 1, s = 10 ( 71, 72, 73, 74, 75, 76, 77, 78, 79 ---> there are two 7's on 77 )
81 to 90: z = 1, s = 1
91 to 100: z = 2, s = 1 ( 100 ---> two zeroes on 100 )
Zt = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 2 = 11
St = 1 + 1 + 1 + 1 + 1 + 1 + 2 + 10 + 1 + 1 = 20
find:
H = number of unique handshakes
solution:
n = number of people = 10
H = (n - 1) + (n - 2) + (n - 3) .....until LAST ADDEND is 1
first term = PEOPLE - 1
H = 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1
H = 45
if there are only 5 people in the room
H = 4 + 3 + 2 + 1
H = 10
2. If a dice and a coin are tossed, what is the probability that you will get a 4 on the dice and heads on the coin?
solution:
P = Pdice * Pcoin
P = 1/6 * 1/2
P = 1/12
3. A circular spinner is marked W, X, Y, Z. what is the probability of NOT getting an X if spinned thrice?
solution:
if spinned once ---> P1 = 3/4
if spinned twice ---> P2 = 3/4 * 3/4 = 9/16
if spinned thrice ---> P1 = 3/4 * 3/4 * 3/4 = 27/64
4. There are 5 choices each question. What is the probability of correctly guessing 3 questions?
solution:
P = 1/5 * 1/5 * 1/5
P = 1/125
5. If you have 5 pants and 8 shirts, how many ways can you wear them?
solution:
W = 5 * 8
W = 40
6. In a restaurant, there are 3 choices of soup, 5 choices of entrees, and 2 choices of desert. How many ways can you order them if each order consists of one soup, one entree, and one desert?
solution:
W = 3 * 5 * 2
w = 30
7. If 4 warriors line up so that the order matters, how many ordered combinations (permutations) are possible?
P = 4 * 3 * 2 * 1
P = 24
8. If you count from 1 to 100, How many zeroes are there? How many sevens are there?
solution:
let
Zt = total number of 0's
St = total number of 7's
1 to 10 : z = 1, s = 1
11 to 20: z = 1, s = 1
21 to 30: z = 1, s = 1
31 to 40: z = 1, s = 1
41 to 50: z = 1, s = 1
51 to 60: z = 1, s = 1
61 to 70: z = 1, s = 2 (67, 70)
71 to 80: z = 1, s = 10 ( 71, 72, 73, 74, 75, 76, 77, 78, 79 ---> there are two 7's on 77 )
81 to 90: z = 1, s = 1
91 to 100: z = 2, s = 1 ( 100 ---> two zeroes on 100 )
Zt = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 2 = 11
St = 1 + 1 + 1 + 1 + 1 + 1 + 2 + 10 + 1 + 1 = 20
Greek and Latin Prefixes, Numerals used in English numerical system, counting, ordering, arrangement, measurement
SI, Metric Prefixes for Large and Small Numbers:
Prefix Symbol Value and their Meaning
Yotta--- Y --- 10^24 --- septillion
Zetta--- Z --- 10^21 --- sextillion
Exa----- E --- 10^18 --- quintillion
Peta---- P --- 10^15 --- quadrillion
Tera---- T --- 10^12 --- trillion
Giga---- G --- 10^9 ---- billion
Mega---- M --- 10^6 ---- million
myria--- my -- 10^4 ---- ten thousand
kilo---- k --- 10^3 ---- thousand
hecto--- h --- 10^2 ---- hundred
deka---- da--- 10 ------ ten
-------------------------------------
deci---- d --- 10^-1 ---- tenth
centi--- c --- 10^-2 ---- hundredth
milli--- m --- 10^-3 ---- thousandth
micro--- u --- 10^-6 ---- millionth
nano---- n --- 10^-9 ---- billionth
pico---- p --- 10^-12 --- trillionth
femto--- f --- 10^-15 --- quadrillionth
atto---- a --- 10^-18 --- quintillionth
zepto--- z --- 10^-21 --- sextillionth
yocto--- y --- 10^-24 --- septillionth
Latin Numbers and Prefix [Examples in brackets]:
1 = unus -------- uni [unary, unitary, unilateral]
2 = duo --------- bi, duo [binary, duple, biennial]
3 = tres, tria -- tri, ter [tertial, trinary, ternary]
4 = quattuor ---- quadri, quart [quartal, quaternary, quadriennial, quadrillion]
5 = quinque ----- quinque, quint [quintal, quinary, quinquenary, quinquennial]
6 = sex --------- sex(t), se [sextal, senary, sexenary]
7 = septem ------ sept [septimal, septenary, septet, septuple]
8 = octo -------- oct [octal, octaval, octonary, octet]
9 = novem ------- nonus, novem [nonal, nonary, nonet, nonuple]
10 = decem ------ dec(a), de [decimal, denary, dectet, decuple]
11 = undecim ---- undec, unde [undecimal, undenary, undecillion]
12 = duodecim --- duodec, duode [duodecimal, duodenary, duodecennial]
13 = tredecim -------- tredec, tridec [tridecimal, tridenary]
14 = quattuordecim --- quatuordec [quatuordecimal, quatuordenary]
15 = quindecim ------ quinde(c) [quindecimal, quindenary]
16 = sedecim --------- sede(c) [hexadecimal, sedenary]
17 = septendecim ----- septende(c) [septendecimal, septendenary]
18 = duodeviginti ---- decennoct [decennoctal, decenoctonary]
19 = undeviginti ----- decennov [decennoval, decennonary]
20 = viginti --------- vige, vice [vigesimal, vicesimal, vigenary, vicenary]
30 = triginta -------- trige, trice [trigesimal, tricenary]
40 = quadraginta ----- quadrage [quadragesimal, quadragenary]
50 = quinquaginta ---- quinquage [quinquagesimal, quinquagenary]
60 = sexaginta ------- sexage [sexagesimal, sexagenary]
70 = septuaginta ----- septuage [septuagesimal, septuagenary]
80 = octoginta ------- octage [octagesimal, octogenary]
90 = nonaginta ------- nonage [nonagesimal, nonagenary]
100 = centum --------- cente [centesimal, centenary, century, centennial]
1000 = mille --------- mille [millesimal, millenary, millennium, millennial]
Greek Numbers and their Prefixes [Examples in brackets]:
1 = en -------------------- eis, mia, ev, mono [monarch, monad, monogamy]
2 = dyo, duo, di ---------- duo, dis, di, dy [diarch, dyarch, dimeter, dyad, biathlon, bicycle]
3 = treis, tria ----------- treis, tria, tris, tri [triad, tripod, triangle]
4 = tessera --------------- tettares, tettara, terakis, tetra, quad [tetrahedron, tetragramaton, tetrapack, quadrangle, quadrilateral]
5 = pente ----------------- pente, pentakis, penta [pentagon, pentangle, pentameter, pentecost]
6 = hexa ------------------ hex, hexa [hexagon, hexahedron, hexangle, hexameter, hex bolt]
7 = hepta ----------------- hepta, hepta [heptagon, heptameter, heptangle]
8 = okto ------------------ octô, octa [octagon, octave, octopus, octohedron]
9 = ennea ----------------- ennea, nona [nonagon, nano technology, ennead,enneahedron]
10 = deka ----------------- deca [decameter, decathlon, decagon, decade, decapolis]
11 = hendeka -------------- endeca, undeca, hendeca [hendecagon, undecagon, hendecarch, hendecahedron]
12 = dodeka --------------- dodeca [dodecagon, dodecahedron, dodecathlon]
13 = treiskaideka --------- triskaideca, trideca [triskaidecagon, trisdecagon, triskaidecahedron]
14 = tettares kai deka ---- tetrakaideca, tetradeca [tetradecagon, tetrakaidecagon, tetrakaidecahedron]
15 = pentekaideka --------- pendeca [pendecagon, pendecahedron]
16 = hekkaideka ----------- hexadeca [hexadecagon, hexadecahedron, hexadecimal]
17 = heptakaideka --------- heptadeca [heptadecagon, heptadecahedron]
18 = oktôkaideka ---------- octodeca [octodecagon, octodecahedron]
19 = enneakaideka --------- enneadeca [enneadecagon, enneadecahedron]
20 = eikosi (icosa-) ------ icos(a) [icosagon, icosahedron]
100 = hekaton ------------- hecato [hecatogon, hecatohedron]
1000 = chilioi, chiliai --- chilia [chiliagon, chiliahedron]
Prefix Symbol Value and their Meaning
Yotta--- Y --- 10^24 --- septillion
Zetta--- Z --- 10^21 --- sextillion
Exa----- E --- 10^18 --- quintillion
Peta---- P --- 10^15 --- quadrillion
Tera---- T --- 10^12 --- trillion
Giga---- G --- 10^9 ---- billion
Mega---- M --- 10^6 ---- million
myria--- my -- 10^4 ---- ten thousand
kilo---- k --- 10^3 ---- thousand
hecto--- h --- 10^2 ---- hundred
deka---- da--- 10 ------ ten
-------------------------------------
deci---- d --- 10^-1 ---- tenth
centi--- c --- 10^-2 ---- hundredth
milli--- m --- 10^-3 ---- thousandth
micro--- u --- 10^-6 ---- millionth
nano---- n --- 10^-9 ---- billionth
pico---- p --- 10^-12 --- trillionth
femto--- f --- 10^-15 --- quadrillionth
atto---- a --- 10^-18 --- quintillionth
zepto--- z --- 10^-21 --- sextillionth
yocto--- y --- 10^-24 --- septillionth
Latin Numbers and Prefix [Examples in brackets]:
1 = unus -------- uni [unary, unitary, unilateral]
2 = duo --------- bi, duo [binary, duple, biennial]
3 = tres, tria -- tri, ter [tertial, trinary, ternary]
4 = quattuor ---- quadri, quart [quartal, quaternary, quadriennial, quadrillion]
5 = quinque ----- quinque, quint [quintal, quinary, quinquenary, quinquennial]
6 = sex --------- sex(t), se [sextal, senary, sexenary]
7 = septem ------ sept [septimal, septenary, septet, septuple]
8 = octo -------- oct [octal, octaval, octonary, octet]
9 = novem ------- nonus, novem [nonal, nonary, nonet, nonuple]
10 = decem ------ dec(a), de [decimal, denary, dectet, decuple]
11 = undecim ---- undec, unde [undecimal, undenary, undecillion]
12 = duodecim --- duodec, duode [duodecimal, duodenary, duodecennial]
13 = tredecim -------- tredec, tridec [tridecimal, tridenary]
14 = quattuordecim --- quatuordec [quatuordecimal, quatuordenary]
15 = quindecim ------ quinde(c) [quindecimal, quindenary]
16 = sedecim --------- sede(c) [hexadecimal, sedenary]
17 = septendecim ----- septende(c) [septendecimal, septendenary]
18 = duodeviginti ---- decennoct [decennoctal, decenoctonary]
19 = undeviginti ----- decennov [decennoval, decennonary]
20 = viginti --------- vige, vice [vigesimal, vicesimal, vigenary, vicenary]
30 = triginta -------- trige, trice [trigesimal, tricenary]
40 = quadraginta ----- quadrage [quadragesimal, quadragenary]
50 = quinquaginta ---- quinquage [quinquagesimal, quinquagenary]
60 = sexaginta ------- sexage [sexagesimal, sexagenary]
70 = septuaginta ----- septuage [septuagesimal, septuagenary]
80 = octoginta ------- octage [octagesimal, octogenary]
90 = nonaginta ------- nonage [nonagesimal, nonagenary]
100 = centum --------- cente [centesimal, centenary, century, centennial]
1000 = mille --------- mille [millesimal, millenary, millennium, millennial]
Greek Numbers and their Prefixes [Examples in brackets]:
1 = en -------------------- eis, mia, ev, mono [monarch, monad, monogamy]
2 = dyo, duo, di ---------- duo, dis, di, dy [diarch, dyarch, dimeter, dyad, biathlon, bicycle]
3 = treis, tria ----------- treis, tria, tris, tri [triad, tripod, triangle]
4 = tessera --------------- tettares, tettara, terakis, tetra, quad [tetrahedron, tetragramaton, tetrapack, quadrangle, quadrilateral]
5 = pente ----------------- pente, pentakis, penta [pentagon, pentangle, pentameter, pentecost]
6 = hexa ------------------ hex, hexa [hexagon, hexahedron, hexangle, hexameter, hex bolt]
7 = hepta ----------------- hepta, hepta [heptagon, heptameter, heptangle]
8 = okto ------------------ octô, octa [octagon, octave, octopus, octohedron]
9 = ennea ----------------- ennea, nona [nonagon, nano technology, ennead,enneahedron]
10 = deka ----------------- deca [decameter, decathlon, decagon, decade, decapolis]
11 = hendeka -------------- endeca, undeca, hendeca [hendecagon, undecagon, hendecarch, hendecahedron]
12 = dodeka --------------- dodeca [dodecagon, dodecahedron, dodecathlon]
13 = treiskaideka --------- triskaideca, trideca [triskaidecagon, trisdecagon, triskaidecahedron]
14 = tettares kai deka ---- tetrakaideca, tetradeca [tetradecagon, tetrakaidecagon, tetrakaidecahedron]
15 = pentekaideka --------- pendeca [pendecagon, pendecahedron]
16 = hekkaideka ----------- hexadeca [hexadecagon, hexadecahedron, hexadecimal]
17 = heptakaideka --------- heptadeca [heptadecagon, heptadecahedron]
18 = oktôkaideka ---------- octodeca [octodecagon, octodecahedron]
19 = enneakaideka --------- enneadeca [enneadecagon, enneadecahedron]
20 = eikosi (icosa-) ------ icos(a) [icosagon, icosahedron]
100 = hekaton ------------- hecato [hecatogon, hecatohedron]
1000 = chilioi, chiliai --- chilia [chiliagon, chiliahedron]
Conversion of commonly used Fractions into Decimals and Percentage equivalents
Applications: These fractions and decimal equivalents are often used in (by)
-- Aptitude tests
-- IQ tests
-- College entrance examinations
-- Assessment tests
-- Achievement tests
-- Admission tests
-- Math wizard award tests
-- Military job selection tests
-- Psychometric entrance tests
-- Competitions & contests
-- Employment applications
-- Job promotion exams
-- Trade eligibility exams
-- Everyday accounting
-- Inventory listing
-- Machine shops
-- Car mechanic and repair shops
-- Sailors and Marine navigators
-- Engineers
-- Architects
-- Draftsmen
-- Physicists
-- Astronomers
-- Doctors
-- Economists
-- Merchandisers
-- Presidents in their speeches full of numbers and figures
-- Ordinary homes and offices
.. and other numerous applications in various trades, disciplines, crafts, commerce, businesses, professions & occupations
fifty percent (half):
1/2 = 0.5
1/2 = 2/4 = 3/6 = 4/8 = 0.5 (50%)
the third part:
1/3 = 0.333
1/3 = 2/6 = 3/9 = 4/12 = 5/15 = 0.333 (33.3%)
"the devil's number", "number of the beast":
2/3 = 0.666
2/3 = 4/6 = 6/9 = 8/12 = 10/15 = 0.666 (66.6%)
twenty-five percent:
1/4 = 0.25
1/4 = 2/8 = 3/12 = 5/20 = 0.25 (25%)
seventy-five percent:
3/4 = 0.75
3/4 = 6/8 = 9/12 = 12/16 = 15/20 = 0.75 (75%)
twenty-forty-sixty-eighty percent: (magic of the fifth part)
1/5 = 0.2 (20%) = 3/15 = 4/20 = 5/25
2/5 = 0.4 (40%) = 6/15 = 8/20 = 10/25
3/5 = 0.6 (60%) = 9/15 = 12/20 = 15/25
4/5 = 0.8 (80%) = 12/15 = 16/20 = 20/25
common divisions of six:
1/6 = 0.166
1/6 = 2/12 = 0.166 (16.6%)
5/6 = 0.833
5/6 = 10/12 = 0.833
divisions of seven:
1/7 = 0.143
2/7 = 0.286
3/7 = 0.429
4/7 = 0.571
5/7 = 0.714
6/7 = 0.857
common divisions of eight: (x/8 * 12.5) sangngapulo + "benteng"
1/8 = 0.125
1/8 = 2/16 = 0.125 (12.5%)
3/8 = 0.375
3/8 = 6/16 = 0.375 (37.5%)
5/8 = 0.625
5/8 = 10/16 = 0.625 (62.5%)
7/8 = 0.875
7/8 = 14/16 = 0.875 (87.5%)
repeating sequence:
1/9 = 0.111
2/9 = 0.222
4/9 = 0.444
5/9 = 0.555
7/9 = 0.777
8/9 = 0.888
ten percent to ninety percent:
1/10 = 0.1 (10%)
2/10 = 0.2 (20%) = 1/5
3/10 = 0.3 (30%)
4/10 = 0.4 (40%) = 2/5
5/10 = 0.5 (50%)
6/10 = 0.6 (60%) = 3/5
7/10 = 0.7 (70%)
8/10 = 0.8 (80%) = 4/5
9/10 = 0.9 (90%)
first two digits (repeating sequence):
1/11 = 0.0909
2/11 = 0.1818
3/11 = 0.2727
4/11 = 0.3636
5/11 = 0.4545
6/11 = 0.5454
7/11 = 0.6363
8/11 = 0.7272
9/11 = 0.8181
10/11 = 0.9090
divisions of the dozen:
1/12 = 0.083
5/12 = 0.416
7/12 = 0.583
11/12 = 0.916
common divisions of sixteen: (diff = 0.125); (+12 +13, 25 75 alt)
1/16 = 0.0625
3/16 = 0.1875
5/16 = 0.3125
7/16 = 0.4375
9/16 = 0.5625
11/16 = 0.6875
13/16 = 0.8125
15/16 = 0.9375
common division of twenty: (magic 5 --> multiply by 5 and get percent)
1/20 = 0.05 (5%)
2/20 = 0.1 (10%)
3/20 = 0.15 (15%)
4/20 = 0.2 ( 20%)
5/20 = 0.25 (25%)
6/20 = 0.3 (30%)
7/20 = 0.35 (35%)
8/20 = 0.4 (40%)
9/20 = 0.45 (45%)
10/20 = 0.5 (50%)
11/20 = 0.55 (55%)
12/20 = 0.6 (60%)
13/20 = 0.65 (65%)
14/20 = 0.7 (70%)
15/20 = 0.75 (75%)
16/20 = 0.8 (80%)
17/20 = 0.85 (85%)
18/20 = 0.9 (90%)
19/20 = 0.95 (95%)
common division of twenty five: (magic 4 --> multiply by 4 to get percent)
1/25 = 0.04 = 4%
2/25 = 0.08 = 8%
3/25 = 0.12 = 12%
4/25 = 0.16 = 16%
5/25 = 0.2 = 20%
6/25 = 0.24 = 24%
7/25 = 0.28 = 28%
8/25 = 0.32 = 32%
9/25 = 0.36 = 36%
10/25 = 0.4 = 40%
11/25 = 0.44 = 44%
12/25 = 0.48 = 48%
13/25 = 0.52 = 52%
14/25 = 0.56 = 56%
15/25 = 0.6 = 60%
16/25 = 0.64 = 64%
17/25 = 0.68 = 68%
18/25 = 0.72 = 72%
19/25 = 0.76 = 76%
20/25 = 0.8 = 80%
21/25 = 0.84 = 84%
22/25 = 0.88 = 88%
23/25 = 0.92 = 92%
24/25 = 0.96 = 96%
common divisions of thirty-two: (sequence difference = 1/16 = 0.0625)
1/32 = 0.03125
3/32 = 0.09375
5/32 = 0.15625
7/32 = 0.21875
9/32 = 0.28125 (*)
11/32 = 0.34375
13/32 = 0.40625
15/32 = 0.46875
17/32 = 0.53125 (*)
19/32 = 0.59375
21/32 = 0.65625
23/32 = 0.71875
25/32 = 0.78125 (*)
27/32 = 0.84375
29/32 = 0.90625
31/32 = 0.96875
one percent to twenty percent: (x/25 * 4); (x/50 * 2)
1/100 == 0.01 = 1%
1/50 === 0.02 = 2%
3/100 == 0.03 = 3%
1/25 === 0.04 = 4%
1/20 === 0.05 = 5%
3/50 === 0.06 = 6%
7/100 == 0.07 = 7%
2/25 === 0.08 = 8%
9/100 == 0.09 = 9%
1/10 === 0.1 == 10%
11/100 = 0.11 = 11%
3/25 === 0.12 = 12%
13/100 = 0.13 = 13%
7/50 === 0.14 = 14%
3/20 === 0.15 = 15%
4/25 === 0.16 = 16%
17/100 = 0.17 = 17%
9/50 === 0.18 = 18%
19/100 = 0.19 = 19%
1/5 ==== 0.2 == 20%
divisions of forty: (increments --> quarter of ten percent) (x/40 * 2.5)
1/40 = 0.025 == 2.5% == 1/40
2/40 = 0.05 === 5% ==== 1/20
3/40 = 0.75 === 7.5% == 3/40
4/40 = 0.1 ==== 10% === 1/10
5/40 = 0.125 == 12.5% = 1/8
6/40 = 0.15 === 15% === 3/20
7/40 = 0.175 == 17.5% = 7/40
8/40 = 0.2 ==== 20% === 1/5
9/40 = 0.225 == 22.5% = 9/40
10/40 = 0.25 == 25% === 1/4
11/40 = 0.275 = 27.5% = 11/40
12/40 = 0.3 === 30% === 3/10
13/40 = 0.325 = 32.5% = 13/40
14/40 = 0.35 == 35% === 7/20
15/40 = 0.375 = 37.5% = 3/8
16/40 = 0.4 === 40% === 2/5
17/40 = 0.425 = 42.5% = 17/40
18/40 = 0.45 == 45% === 9/20
19/40 = 0.475 = 47.5% = 19/40
20/40 = 0.5 === 50% === 1/2
21/40 = 0.525 = 52.5% = 21/40
22/40 = 0.55 == 55% === 11/20
23/40 = 0.575 = 57.5% = 23/40
24/40 = 0.6 === 60% === 3/5
25/40 = 0.625 = 62.5% = 5/8
26/40 = 0.65 == 65% === 13/20
27/40 = 0.675 = 67.5% = 27/40
28/40 = 0.7 === 70% === 7/10
29/40 = 0.725 = 72.5% = 29/40
30/40 = 0.75 == 75% === 3/4
31/40 = 0.775 = 77.5% = 31/40
32/40 = 0.8 === 80% === 4/5
33/40 = 0.825 = 82.5% = 33/40
34/40 = 0.85 == 85% === 17/20
35/40 = 0.875 = 87.5% = 7/8
36/40 = 0.9 === 90% === 9/10
37/40 = 0.925 = 92.5% = 37/40
38/40 = 0.95 == 95% === 19/20
39/40 = 0.975 = 97.5% = 39/40
increments of quarter percent:
1/400 = 0.0025 = 0.25% = quarter of a percent ====== 1/4%
1/200 = 0.005 == 0.5% == half of a percent ========= 1/2%
3/400 = 0.0075 = 0.75% = three quarters of a percent = 3/4%
1/100 = 0.01 === 1%
1/80 == 0.0125 = 1.25%
3/200 = 0.015 == 1.5%
7/400 = 0.0175 = 1.75%
1/50 === 0.02 === 2%
9/400 == 0.0225 = 2.25%
1/40 === 0.025 == 2.5%
11/400 = 0.0275 = 2.75%
3/100 == 0.03 === 3%
13/400 = 0.0325 = 3.25%
7/200 == 0.035 == 3.5%
3/80 === 0.0375 = 3.75%
1/25 === 0.04 === 4%
17/400 = 0.0425 = 4.25%
9/200 == 0.045 == 4.5%
19/400 = 0.0475 = 4.75%
1/20 === 0.05 === 5%
21/400 = 0.0525 = 5.25%
11/200 = 0.055 == 5.5%
23/400 = 0.0575 = 5.75%
3/50 === 0.06 === 6%
1/16 === 0.0625 = 6.25% --> 100/16 = 6.25
13/200 = 0.065 == 6.5%
27/400 = 0.0675 = 6.75%
7/100 == 0.07 === 7%
29/400 = 0.0725 = 7.25%
3/40 === 0.075 == 7.5%
31/400 = 0.0775 = 7.75%
2/25 === 0.08 === 8%
33/400 = 0.0825 = 8.25%
17/200 = 0.085 == 8.5%
7/80 === 0.0875 = 8.75%
9/100 == 0.09 === 9%
37/400 = 0.0925 = 9.25%
19/200 = 0.095 == 9.5%
39/400 = 0.0975 = 9.75%
1/10 === 0.1 ==== 10%
41/400 = 0.1025 = 10.25%
21/200 = 0.105 == 10.5%
43/400 = 0.1075 = 10.75%
11/100 = 0.11 === 11%
9/80 === 0.1125 = 11.25%
23/200 = 0.115 == 11.5%
47/400 = 0.1175 = 11.75%
3/25 === 0.12 === 12%
49/400 = 0.1225 = 12.25%
1/8 ==== 0.125 == 12.5% --> 100/8 = 12.5
51/400 = 0.1275 = 12.75%
other interesting fractions with definite patterns:
1/14 = 0.0714
1/15 = 0.0666 (6.66%)
1/20 = 0.05 (5%)
1/22 = 0.04545
1/25 = 0.04 (4%)
10/25 = 0.4 (40%)
24/25 = 0.96 (96%)
1/30 = 0.03333 (3.33%)
1/40 = 0.025 (2.5%)
1/45 = 0.02222
1/49 = 0.020408
1/50 = 0.02 (2%)
49/50 = 0.98 (98%)
1/60 = 0.016666
1/64 = 0.015625
1/66 = 0.0151515 (*)
1/75 = 0.013333
1/90 = 0.011111
1/99 = 0.010101
75/99 = 0.7575
1/100 = 0.01 (1%)
1/120 = 0.0083333
1/128 = 0.0078125
1/256 = 0.004 (0.4%)
1/512 = 0.002 (0.2%)
123/999 = 0.123123
1/1000 = 0.001 (0.1%)
-- Aptitude tests
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-- Psychometric entrance tests
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-- Employment applications
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-- Sailors and Marine navigators
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-- Merchandisers
-- Presidents in their speeches full of numbers and figures
-- Ordinary homes and offices
.. and other numerous applications in various trades, disciplines, crafts, commerce, businesses, professions & occupations
fifty percent (half):
1/2 = 0.5
1/2 = 2/4 = 3/6 = 4/8 = 0.5 (50%)
the third part:
1/3 = 0.333
1/3 = 2/6 = 3/9 = 4/12 = 5/15 = 0.333 (33.3%)
"the devil's number", "number of the beast":
2/3 = 0.666
2/3 = 4/6 = 6/9 = 8/12 = 10/15 = 0.666 (66.6%)
twenty-five percent:
1/4 = 0.25
1/4 = 2/8 = 3/12 = 5/20 = 0.25 (25%)
seventy-five percent:
3/4 = 0.75
3/4 = 6/8 = 9/12 = 12/16 = 15/20 = 0.75 (75%)
twenty-forty-sixty-eighty percent: (magic of the fifth part)
1/5 = 0.2 (20%) = 3/15 = 4/20 = 5/25
2/5 = 0.4 (40%) = 6/15 = 8/20 = 10/25
3/5 = 0.6 (60%) = 9/15 = 12/20 = 15/25
4/5 = 0.8 (80%) = 12/15 = 16/20 = 20/25
common divisions of six:
1/6 = 0.166
1/6 = 2/12 = 0.166 (16.6%)
5/6 = 0.833
5/6 = 10/12 = 0.833
divisions of seven:
1/7 = 0.143
2/7 = 0.286
3/7 = 0.429
4/7 = 0.571
5/7 = 0.714
6/7 = 0.857
common divisions of eight: (x/8 * 12.5) sangngapulo + "benteng"
1/8 = 0.125
1/8 = 2/16 = 0.125 (12.5%)
3/8 = 0.375
3/8 = 6/16 = 0.375 (37.5%)
5/8 = 0.625
5/8 = 10/16 = 0.625 (62.5%)
7/8 = 0.875
7/8 = 14/16 = 0.875 (87.5%)
repeating sequence:
1/9 = 0.111
2/9 = 0.222
4/9 = 0.444
5/9 = 0.555
7/9 = 0.777
8/9 = 0.888
ten percent to ninety percent:
1/10 = 0.1 (10%)
2/10 = 0.2 (20%) = 1/5
3/10 = 0.3 (30%)
4/10 = 0.4 (40%) = 2/5
5/10 = 0.5 (50%)
6/10 = 0.6 (60%) = 3/5
7/10 = 0.7 (70%)
8/10 = 0.8 (80%) = 4/5
9/10 = 0.9 (90%)
first two digits (repeating sequence):
1/11 = 0.0909
2/11 = 0.1818
3/11 = 0.2727
4/11 = 0.3636
5/11 = 0.4545
6/11 = 0.5454
7/11 = 0.6363
8/11 = 0.7272
9/11 = 0.8181
10/11 = 0.9090
divisions of the dozen:
1/12 = 0.083
5/12 = 0.416
7/12 = 0.583
11/12 = 0.916
common divisions of sixteen: (diff = 0.125); (+12 +13, 25 75 alt)
1/16 = 0.0625
3/16 = 0.1875
5/16 = 0.3125
7/16 = 0.4375
9/16 = 0.5625
11/16 = 0.6875
13/16 = 0.8125
15/16 = 0.9375
common division of twenty: (magic 5 --> multiply by 5 and get percent)
1/20 = 0.05 (5%)
2/20 = 0.1 (10%)
3/20 = 0.15 (15%)
4/20 = 0.2 ( 20%)
5/20 = 0.25 (25%)
6/20 = 0.3 (30%)
7/20 = 0.35 (35%)
8/20 = 0.4 (40%)
9/20 = 0.45 (45%)
10/20 = 0.5 (50%)
11/20 = 0.55 (55%)
12/20 = 0.6 (60%)
13/20 = 0.65 (65%)
14/20 = 0.7 (70%)
15/20 = 0.75 (75%)
16/20 = 0.8 (80%)
17/20 = 0.85 (85%)
18/20 = 0.9 (90%)
19/20 = 0.95 (95%)
common division of twenty five: (magic 4 --> multiply by 4 to get percent)
1/25 = 0.04 = 4%
2/25 = 0.08 = 8%
3/25 = 0.12 = 12%
4/25 = 0.16 = 16%
5/25 = 0.2 = 20%
6/25 = 0.24 = 24%
7/25 = 0.28 = 28%
8/25 = 0.32 = 32%
9/25 = 0.36 = 36%
10/25 = 0.4 = 40%
11/25 = 0.44 = 44%
12/25 = 0.48 = 48%
13/25 = 0.52 = 52%
14/25 = 0.56 = 56%
15/25 = 0.6 = 60%
16/25 = 0.64 = 64%
17/25 = 0.68 = 68%
18/25 = 0.72 = 72%
19/25 = 0.76 = 76%
20/25 = 0.8 = 80%
21/25 = 0.84 = 84%
22/25 = 0.88 = 88%
23/25 = 0.92 = 92%
24/25 = 0.96 = 96%
common divisions of thirty-two: (sequence difference = 1/16 = 0.0625)
1/32 = 0.03125
3/32 = 0.09375
5/32 = 0.15625
7/32 = 0.21875
9/32 = 0.28125 (*)
11/32 = 0.34375
13/32 = 0.40625
15/32 = 0.46875
17/32 = 0.53125 (*)
19/32 = 0.59375
21/32 = 0.65625
23/32 = 0.71875
25/32 = 0.78125 (*)
27/32 = 0.84375
29/32 = 0.90625
31/32 = 0.96875
one percent to twenty percent: (x/25 * 4); (x/50 * 2)
1/100 == 0.01 = 1%
1/50 === 0.02 = 2%
3/100 == 0.03 = 3%
1/25 === 0.04 = 4%
1/20 === 0.05 = 5%
3/50 === 0.06 = 6%
7/100 == 0.07 = 7%
2/25 === 0.08 = 8%
9/100 == 0.09 = 9%
1/10 === 0.1 == 10%
11/100 = 0.11 = 11%
3/25 === 0.12 = 12%
13/100 = 0.13 = 13%
7/50 === 0.14 = 14%
3/20 === 0.15 = 15%
4/25 === 0.16 = 16%
17/100 = 0.17 = 17%
9/50 === 0.18 = 18%
19/100 = 0.19 = 19%
1/5 ==== 0.2 == 20%
divisions of forty: (increments --> quarter of ten percent) (x/40 * 2.5)
1/40 = 0.025 == 2.5% == 1/40
2/40 = 0.05 === 5% ==== 1/20
3/40 = 0.75 === 7.5% == 3/40
4/40 = 0.1 ==== 10% === 1/10
5/40 = 0.125 == 12.5% = 1/8
6/40 = 0.15 === 15% === 3/20
7/40 = 0.175 == 17.5% = 7/40
8/40 = 0.2 ==== 20% === 1/5
9/40 = 0.225 == 22.5% = 9/40
10/40 = 0.25 == 25% === 1/4
11/40 = 0.275 = 27.5% = 11/40
12/40 = 0.3 === 30% === 3/10
13/40 = 0.325 = 32.5% = 13/40
14/40 = 0.35 == 35% === 7/20
15/40 = 0.375 = 37.5% = 3/8
16/40 = 0.4 === 40% === 2/5
17/40 = 0.425 = 42.5% = 17/40
18/40 = 0.45 == 45% === 9/20
19/40 = 0.475 = 47.5% = 19/40
20/40 = 0.5 === 50% === 1/2
21/40 = 0.525 = 52.5% = 21/40
22/40 = 0.55 == 55% === 11/20
23/40 = 0.575 = 57.5% = 23/40
24/40 = 0.6 === 60% === 3/5
25/40 = 0.625 = 62.5% = 5/8
26/40 = 0.65 == 65% === 13/20
27/40 = 0.675 = 67.5% = 27/40
28/40 = 0.7 === 70% === 7/10
29/40 = 0.725 = 72.5% = 29/40
30/40 = 0.75 == 75% === 3/4
31/40 = 0.775 = 77.5% = 31/40
32/40 = 0.8 === 80% === 4/5
33/40 = 0.825 = 82.5% = 33/40
34/40 = 0.85 == 85% === 17/20
35/40 = 0.875 = 87.5% = 7/8
36/40 = 0.9 === 90% === 9/10
37/40 = 0.925 = 92.5% = 37/40
38/40 = 0.95 == 95% === 19/20
39/40 = 0.975 = 97.5% = 39/40
increments of quarter percent:
1/400 = 0.0025 = 0.25% = quarter of a percent ====== 1/4%
1/200 = 0.005 == 0.5% == half of a percent ========= 1/2%
3/400 = 0.0075 = 0.75% = three quarters of a percent = 3/4%
1/100 = 0.01 === 1%
1/80 == 0.0125 = 1.25%
3/200 = 0.015 == 1.5%
7/400 = 0.0175 = 1.75%
1/50 === 0.02 === 2%
9/400 == 0.0225 = 2.25%
1/40 === 0.025 == 2.5%
11/400 = 0.0275 = 2.75%
3/100 == 0.03 === 3%
13/400 = 0.0325 = 3.25%
7/200 == 0.035 == 3.5%
3/80 === 0.0375 = 3.75%
1/25 === 0.04 === 4%
17/400 = 0.0425 = 4.25%
9/200 == 0.045 == 4.5%
19/400 = 0.0475 = 4.75%
1/20 === 0.05 === 5%
21/400 = 0.0525 = 5.25%
11/200 = 0.055 == 5.5%
23/400 = 0.0575 = 5.75%
3/50 === 0.06 === 6%
1/16 === 0.0625 = 6.25% --> 100/16 = 6.25
13/200 = 0.065 == 6.5%
27/400 = 0.0675 = 6.75%
7/100 == 0.07 === 7%
29/400 = 0.0725 = 7.25%
3/40 === 0.075 == 7.5%
31/400 = 0.0775 = 7.75%
2/25 === 0.08 === 8%
33/400 = 0.0825 = 8.25%
17/200 = 0.085 == 8.5%
7/80 === 0.0875 = 8.75%
9/100 == 0.09 === 9%
37/400 = 0.0925 = 9.25%
19/200 = 0.095 == 9.5%
39/400 = 0.0975 = 9.75%
1/10 === 0.1 ==== 10%
41/400 = 0.1025 = 10.25%
21/200 = 0.105 == 10.5%
43/400 = 0.1075 = 10.75%
11/100 = 0.11 === 11%
9/80 === 0.1125 = 11.25%
23/200 = 0.115 == 11.5%
47/400 = 0.1175 = 11.75%
3/25 === 0.12 === 12%
49/400 = 0.1225 = 12.25%
1/8 ==== 0.125 == 12.5% --> 100/8 = 12.5
51/400 = 0.1275 = 12.75%
other interesting fractions with definite patterns:
1/14 = 0.0714
1/15 = 0.0666 (6.66%)
1/20 = 0.05 (5%)
1/22 = 0.04545
1/25 = 0.04 (4%)
10/25 = 0.4 (40%)
24/25 = 0.96 (96%)
1/30 = 0.03333 (3.33%)
1/40 = 0.025 (2.5%)
1/45 = 0.02222
1/49 = 0.020408
1/50 = 0.02 (2%)
49/50 = 0.98 (98%)
1/60 = 0.016666
1/64 = 0.015625
1/66 = 0.0151515 (*)
1/75 = 0.013333
1/90 = 0.011111
1/99 = 0.010101
75/99 = 0.7575
1/100 = 0.01 (1%)
1/120 = 0.0083333
1/128 = 0.0078125
1/256 = 0.004 (0.4%)
1/512 = 0.002 (0.2%)
123/999 = 0.123123
1/1000 = 0.001 (0.1%)
Monday, October 15, 2012
Shortcuts in Multiplication, Division, Addition & Subtraction
These are some of the practical secrets in mental math calculation speed. Knowing these can help you figure out answers to most of life's daily activities involving numbers. There are many useful Math techniques, tricks, and secrets that can be valuable in your day to day work or study. Knowing these shortcuts are key to computation speed and accuracy in your Mental Mathematical Aptitude. This can be useful in applications such as IQ tests, aptitude tests, job application tests, military tests, college entrance tests and so many more uses in the office, workplace, or in your own home.
SHORTCUTS IN MULTIPLICATION:
Multiplication using multiples
12 x 15
= 12 x 5 x 3
= 60 x 3
= 180
Multiplication by distribution
12 x 17
= (12 x 10) + (12 x 7) ---> 12 is multiplied to both 10 & 7
= 120 + 84
= 204
Multiplication by "giving and taking"
12 x 47
= 12 x (50 - 3)
= (12 x 50) - (12 x 3)
= 600 - 36
= 564
Multiplication by 5 --> take the half(0.5) then multiply by 10
428 x 5
= (428 x 1/2) x 10 = 428 x 0.5 x 10
= 214 x 10
= 2140
Multiplication by 10 ---> just move the decimal point one place to the right
14 x 10
= 140 ---> added one zero
Multiplication by 50 ---> take the half(0.5) then multiply by 100
18 x 50
= (18/2) x 100 = 18 x 0.5 x 100
= 9 x 100
= 900
Multiplication by 100 ---> move the decimal point two places to the right
42 x 100
= 4200 ---> added two zeroes
Multiplication by 500 ---> take the half(0.5) then multiply by 1000
21 x 500
= 21/2 x 1000
= 10.5 x 1000
= 10500
Multiplication by 25 ---> use the analogy $1 = 4 x 25 cents
25 x 14
= (25 x 10) + (25 x 4) ---> 250 + 100 ---> $2.50 + $1
= 350
Multiplication by 25 ---> divide by 4 then multiply by 100
36 x 25
= (36/4) x 100
= 9 x 100
= 900
Multiplication by 11 if sum of digits is less than 10
72 x 11
= 7_2 ---> the middle term = 7 + 2 = 9
= place the middle term 9 between 7 & 2
= 792
Multiplication by 11 if sum of digits is greater than 10
87 x 11
= 8_7 ---> the middle term = 8 + 7 = 15
because the middle term is greater than 10, use 5 then
add 1 to the first term 8, which leads to the answer of
= 957
Multiplication of 37 by the 3, 6, 9 until 27 series of numbers --> the "triple effect"
solve 37 x 3
multiply 7 by 3 = 21, knowing the last digit (1), just combine two more 1's giving the triple digit answer 111
solve 37 x 9
multiply 7 by 9 = 63, knowing the last digit (3), just combine two more 3's giving the triple digit answer 333
solve 37 x 21
multiply 7 by 21 = 147, knowing the last digit (7), just combine two more 7's giving the triple digit answer 777
Multiplication of the "dozen teens" group of numbers --
(i.e. 12, 13, 14, 15, 16, 17, 18, 19) by ANY of the numbers within the group:
solve 14 x 17
4 x 7 = 28; remember 8, carry 2
14 + 7 = 21
add 21 to whats is carried (2)
giving the result 23
form the answer by combinig 23 to what is remembered (8)
giving the answer 238
Multiplication of numbers ending in 5 with difference of 10
45 x 35
first term = [(4 + 1) x 3] = 15; (4 is the first digit of 45 and 3 is the first digit of 35 --> add 1 to the higher first digit which is 4 in this case, then multiply the result by 3, giving 15)
last term = 75
combining the first term and last term,
= 1575
75 x 85
first term = (8 + 1) x 7 = 63
last term = 75
combining first and last terms,
= 6375
15 x 25
= 375
Multiplication of numbers ending in 5 with the same first terms (square of a number)
25 x 25
first term = (2 + 1) x 2 = 6
last term = 25
answer = 625 ---> square of 25
75 x 75
first term = (7 + 1) x 7 = 56
last term = 25
answer = 5625 ---> 75 squared
SHORTCUTS IN DIVISION:
Division by parts ---> imagine dividing $874 between two persons
874/2
= 800/2 + 74/2
= 400 + 37
= 437
Division using the factors of the divisor: "double division"
70/14
= (70/7)/2 ---> 7 and 2 are the factors of 14
= 10/2
= 5
Division by using fractions:
132/2
= (100/2 + 32/2) ---> break down into two fractions
= (50 + 16)
= 66
Division by 5 ---> divide by 100 then multiply by 20
1400/5
= (1400/100) x 20
= 14 x 20
= 280
Division by 10 ---> move the decimal point one place to the left
0.5/10
= 0.05 ---> 5% is 50% divided by ten
Division by 50 ---> divide by 100 then multiply by 2
2100/50
= (2100/100) x 2
= 21 x 2
= 42
700/50
= (700/100) x 2
= 7 x 2
= 14
Division by 100 ---> move the decimal point two places to the left
25/100
= 0.25
Division by 500 ---> divide by 100 then multiply by 0.2
17/500
= (17/100) x 0.2
= 0.17 x 0.2
= 0.034
Division by 25 ---> divide by 100 then multiply by 4
500/25
= (500/100) x 4
= 5 x 4
= 20
750/25
= (750/100) x 4
= 7.5 x 2 x 2
= 30
SHORTCUTS IN ADDITION:
Addition of numbers close to multiples of ten (e.g. 19, 29, 89, 99 etc.)
116 + 39
= 116 + (40 - 1)
= 116 + 40 - 1 ---> add 40 then subtract 1
= 156 - 1
= 155
116 + 97
= 116 + (100 - 3)
= 116 + 100 - 3 ---> add 100 then subtract 3
= 216 - 3
= 213
Addition of decimals
12.5 + 6.25
= (12 + 0.5) + (6 + 0.25)
= 12 + 6 + 0.5 + 0.25 ---> add the integers then the decimals
= 18 + 0.5 + 0.25
= 18.75
SHORTCUTS IN SUBTRACTION:
Subtraction by numbers close to 100, 200, 300, 400, etc.
250 - 96
= 250 - (100 - 4)
= 250 - 100 + 4 ---> subtract 100 then add 4
= 150 + 4
= 154
250 - 196
= 250 - (200 - 4)
= 250 - 200 + 4 ---> subtract 200 then add 4
= 50 + 4
= 54
216 - 61
= 216 - (100 - 39)
= 216 - 100 + 39
= 116 + (40 - 1) ---> now the operation is addition, which is much easier
= 156 - 1
= 155
Subtraction of decimals
47 - 9.9
= 47 - (9 + 0.9) ---> "double subtraction"
= 47 - 9 - 0.9 ---> subtract the integer first then the decimal
= 38 - 0.9
= 37.1
18.3 - 0.8
= 18 + 0.3 - 0.8
= (18 - 0.8) + 0.3 ---> subtract 0.8 from 18 first
= 17.2 + 0.3
= 17.5
WORKING ON PERCENTAGES:
30% of 120
= 10% x 3 x 120 ---> it is much easier working with tens (10%)
= 10% x 120 x 3
= 12 x 3
= 36
five percent of a number: 5%
360 x 5%
= 360 x 10%/2 ---> take the 10% and divide by 2
= 36/2
= 18
360 x 5%
= 360 x 50%/10 ---> take the half(0.5) and divide by 10
= (360/2)/10
= 180/10
= 18
ninety percent of a number: 90%
90% of 700
= (100% - 10%) x 700
= (100% x 700) - (10% x 700) ---> 100% minus 10% of the number
= 700 - 70
= 630
What is 18 as a percentage of 50?
= 18/50
= (18/100) x 2 ---> method: division by 50 (explained above)
= 0.18 x 2
= 0.36
= 36%
What is 132 as a percentage of 200?
= 132/200
= (132/2)/100
= [100/2 + 32/2]/100 ---> solution by "double division"
= (50 + 16)/100
= 66/100
= 0.66
= 66%
What is 270 as a percentage of 300?
= 270/300
= [(270/3)/100] ---> "double division" (using the factors of 300)
= 90/100
= 90%
What is 17 as percentage of 500?
= 17/500
= (17/50)/10
= (17/100) x 2/10 ---> solution using the procedure: division by 50
= (0.17 x 2)/10
= 0.34/10
= 0.034
= 3.4 %
percentages close to 100:
95% of 700
= (100% - 5%) x 700
= (100% x 700) - (5% x 700)
= 700 - (10% x 700/2) -------> 5% is 10%/2
= 700 - 70/2
= 700 - 35
= 665
percentages less than 10 percent:
3% of 70
= (3/100) x 70
= (70/100) x 3 ---> divide by 100 then multiply the percent value
= 0.7 x 3
= 2.1
DECIMALS:
To convert or express percentages as decimals, divide by 100 or simply just move the decimal point by two places to the left of the given number, thus:
1% = 1/100 = 0.01
2% = 2/100 = 0.02 = 1/50
3% = 3/100 = 0.03
4% = 4/100 = 0.04 = 1/25
5% = 5/100 = 0.05 = 1/20
6.25% = 6.25/100 = 0.0625 = 1/16
7% = 7/100 = 0.07
7.5% = 7.5/100 = 0.075
10% = 10/100 = 0.1 = 1/10
12.5% = 12.5/100 = 0.125 = 1/8
20% = 0.2 = 1/5
21% = 0.21
25% = 0.25 = 1/4
30% = 0.3 = 3/10
33.33% = 33.33/100 = 0.3333 = 1/3
37.5% = 0.375 = 3/8
40% = 0.4 = 2/5
50% = 0.5 = 1/2
60% = 0.6 = 3/5
62.5% = 0.625 = 5/8
66.66% = 66.66/100 = 2/3
75% = 0.75 = 3/4
80% = 0.8 = 4/5
87.5% = 0.875 = 7/8
100% = 1
125% = 1.25 = 1 1/4
150% = 1.5 = 1 1/2
200% = 2
FRACTIONS:
What is three quarters of 80?
= 3/4 x 80
= (80/4) x 3 ---> divide by 4 then multiply by 3
= 20 x 3
= 60
How many quarters in two and a half?
2.5/.25
= 10 ---> there are 10 quarters in $2.50
Improper fractions:
3/2 = 1 1/2 = 1.5 = 150%
4/3 = 1 1/3 = 1.3333 = 133.33% ---> useful number for volume of sphere, etc.
9/5 = 1 4/5 = 1.8 = 180% ---> conversion factor for Celsius/Fahrenheit temperatures
V = 4/3 pi * r^3
where:
V = volume of sphere
r = radius of sphere
F = (1.8 C) + 32
where:
F = temperature in Fahrenheit
C = temperature in Celsius
See Also:
IQ TEST: Math, Mixture, Solutions, Concentration, Coin Problems
MATHEMATICS: Ratio, Proportion, Variation - Power, Resistance of Conductor, Volume, Turbine Flowrate
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