Cheukting/mathqa huggingface dataset:
Question (general): a rainstorm increased the amount of water stored in state j reservoirs from 180 billion gallons to 200 billion gallons . if the storm increased the amount of water in the reservoirs to 60 percent of total capacity , approximately how many billion gallons of water were the reservoirs short of total capacity prior to the storm ?
Options: a ) 90 , b ) 114 , c ) 125 , d ) 130 , e ) 144
Rationale: after the reservoir is filled to 200 gallons the amount of water is at 60 % - which means that 40 % of the reservoir is empty . to figure out what that 40 % is approximate : 200 gallons / 60 percent = x gallons / 40 percent , therefore , x = 133.33 gallons , answer choices a , b , c , d are below 133.33 . we know that the reservoir must be short more than 133.33 gallons , therefore , the only possible choice is e .
Answer: e
Question (gain): find the simple interest for rs . 2000 at 8 % p / a for 2 years .
Options: a ) rs . 300 , b ) 3 rs . 20 , c ) rs . 420 , d ) rs . 520 , e ) rs . 460
Rationale: "s . i = ( 2000 * 8 * 2 ) / 100 p = 320 answer : b"
Answer: b
Question (geometry): 16 business executives and 7 chairmen meet at a conference . if each business executive shakes the hand of every other business executive and every chairman once , and each chairman shakes the hand of each of the business executives but not the other chairmen , how many handshakes would take place ?
Options: a ) 144 , b ) 131 , c ) 232 , d ) 90 , e ) 45
Rationale: "there are 16 business exec and in each handshake 2 business execs are involved . hence 16 c 2 = 120 also , each of 16 exec will shake hand with every 7 other chairmen for total of 112 handshake . total = 120 + 112 = 232 ans : c"
Answer: c
Question (physics): if two trains are 120 miles apart and are traveling toward each other at constant rate of 30 miles per hour and 40 miles per hour , respectively , how far apart will they be 1 hour before they meet ?
Options: a ) 10 , b ) 30 , c ) 40 , d ) 50 , e ) 70
Rationale: they would meet in 120 / 70 = 12 / 7 hours 1 hour earlier 5 / 7 hours distance between trains = total distance - ( distance covered by train a + distance covered by train b ) = 120 - ( 5 / 7 ( 40 + 30 ) ) [ distance of a = 5 / 7 * 40 ; like wise distance by b ] = 70 km answer : e
Answer: e
Question (general): a computer is programmed to multiply consecutive even integers 2 * 4 * 6 * 8 * … * n until the product is divisible by 2211 , what is the value of n ?
Options: a ) 22 , b ) 134 , c ) 62 , d ) 122 , e ) 672
Rationale: "factorise 2211 . . 3 * 11 * 67 . . so n has to be a multiple of largest prime number , 61 . . so n = 2 * 61 = 134 . . ans : b"
Answer: b
Question (physics): if 30 honey bees make 30 grams of honey in 30 days , then 1 honey bee will make 1 gram of honey in how many days ?
Options: a ) 1 , b ) 3.5 , c ) 30 , d ) 49 , e ) 30
Rationale: "explanation : let the required number days be x . less honey bees , more days ( indirect proportion ) less honey , less days ( direct proportion ) honey bees 1 : 30 : : 30 : x honey 30 : 1 = > 1 x 30 x x = 30 x 1 x 30 = > x = 30 . answer : c"
Answer: c
Question (general): a basket of 1430 apples is divided equally among a group of apple lovers . if 45 people join the group , each apple lover would receive 9 apples less . how many q apples did each person get before 45 people joined the feast ?
Options: a ) 20 . , b ) 21 . , c ) 22 . , d ) 23 . , e ) 24 .
Rationale: "before solving it algebraically , let us prime factorize 1430 = 2 * 5 * 11 * 13 . since number of apples per person * total persons q = 1430 , the answer should be a factor of 1430 . only c is . and that ' s your answer . c"
Answer: c
Question (geometry): if the sides of a triangle are 9 cm , 16 cm and 19 cm , what is its area ?
Options: a ) 69 , b ) 72 , c ) 75 , d ) 89 , e ) 93
Rationale: "the triangle with sides 9 cm , 16 cm and 19 cm is right angled , where the hypotenuse is 19 cm . area of the triangle = 1 / 2 * 9 * 16 = 72 cm 2 answer : option b"
Answer: b
Question (general): the number of positive integers valued pairs ( x , y ) satisfying 4 x - 17 y = 1 and x < = 1000 ( x is less than or equal to 1000 ) is
Options: a ) 59 , b ) 57 , c ) 55 , d ) 58 , e ) 60
Rationale: "we are asked to find integer solutions for x , y . there is no need to apply euclidian algorithm here , we can get particular solutions simply by plugging in some values . as y reaches 3 we get : y ’ = 3 and x ’ = 13 . we need only to solve for x . general solution for x will be : x = x ’ + bn = 13 + 17 n hence lower limit for n is n ≥ 0 . now we ’ ll find upper limit using given restriction x ≤ 1000 . plugging in general solution we ’ ll get : 13 + 17 n ≤ 1000 17 n ≤ 987 n ≤ 58 and our list of possible values of n has following representation : 0 , 1,2 , 3 , … . . , 58 total number of elements is 59 . answer : a"
Answer: a
Question (physics): a monkey ascends a greased pole 20 meters high . he ascends 2 meters in the first minute and then slips down 1 meter in the alternate minute . if this pattern continues until he climbs the pole , in how many minutes would he reach at the top of the pole ?
Options: a ) 30 th minute , b ) 31 st minute , c ) 37 th minute , d ) 32 nd minute , e ) 33 th minute
Rationale: "the money is climbing 1 meter in 2 min . this pattern will go on till he reaches 10 meters . i mean this will continue for first 18 * 2 = 36 mins . he would have reached 18 meters . after that he will climb 2 meters and he will reach the pole . so total time taken = 36 + 1 = 37 mins . so , asnwer will be c"
Answer: c
Question (general): a retailer sells 10 shirts . the first 3 he sells for $ 20 , $ 22 and $ 25 . if the retailer wishes to sell the 10 shirts for an overall average price of over $ 20 , what must be the minimum average price of the remaining 7 shirts ?
Options: a ) $ 14.00 , b ) $ 16.00 , c ) $ 17.00 , d ) $ 19.00 , e ) $ 23.00
Rationale: "first 3 shirts are sold for $ 20 , $ 22 and $ 25 = $ 67 . to get average price of $ 20 , total sale should be 10 * $ 20 = $ 200 so remaining 7 shirts to be sold for $ 200 - $ 67 = $ 133 answer should be 133 / 7 = $ 19.00 that is d"
Answer: d
Question (general): for any positive integer n , the sum of the first n positive integers equals n ( n + 1 ) / 2 . what is the sum z of all the even integers between 99 and 301 ?
Options: a ) 10,100 , b ) 20,200 , c ) 22,650 , d ) 40,200 , e ) 45,150
Rationale: "i would solve it in a different way . first of all , total number of even integers between 99 and 301 are , ( 301 - 99 ) / 2 = 202 / 2 = 101 average = ( 301 + 99 ) / 2 = 400 / 2 = 200 sum z = average * total count = 200 * 101 = 20,200 answer is b"
Answer: b
Question (gain): there are 300 employees in a room . 99 % are managers . how many managers must leave the room to bring down the percentage of managers to 98 % ?
Options: a ) 200 , b ) 175 , c ) 150 , d ) 125 , e ) 100
Rationale: there are 297 managers and 3 others . the 3 others would compose 2 % of the total number of people if there were 150 people in the room . thus 150 managers must leave . the answer is c .
Answer: c
Question (other): the ratio of the number of red cars in a certain parking lot to the number of black cars is 3 to 8 . if there are 75 black cars in the lot , how many red cars are there in the lot ?
Options: a ) 11 , b ) 28 , c ) 24 , d ) 29 , e ) 32
Rationale: "b is correct r / b = 3 / 8 and b = 75 r = 75 * 3 / 8 = 28"
Answer: b
Question (general): a total of 50 percent of the geese included in a certain migration study were male . if some of the geese migrated during the study and 20 percent of the migrating geese were male , what was the ratio of the migration rate for the male geese to the migration rate for the female geese ? [ migration rate for geese of a certain sex = ( number of geese of that sex migrating ) / ( total number of geese of that sex ) ]
Options: a ) 1 / 4 , b ) 7 / 12 , c ) 2 / 3 , d ) 7 / 8 , e ) 8 / 7
Rationale: let ' take the number of geese to be 100 . male = 50 . female = 50 . now the second part of the q , let ' s take the number migrated to be 20 . so we have 20 geese that migrated and out of that 20 % are male i . e 20 / 100 * 20 = 4 geese ( males ) and now we know out of the total 20 geese , 4 are male , then 16 have to be female . now the ratio part , male geese ratios = 4 / 50 = 2 / 25 . - a female geese ratios = 16 / 50 = 8 / 25 - b cross multiply equations a and b and you get = 1 / 4 . ans a
Answer: a
Question (other): mukesh weight 56.7 kg . if he reduces his weight in the ratio 7 : 6 . new weight of mukesh is ?
Options: a ) 47.6 , b ) 48.6 , c ) 49.6 , d ) 50.6 , e ) 51.6
Rationale: given ratio is 7 : 6 ; let weight before and after reduction be 7 x and 6 x respectively . a / q 7 x = 56.7 = > x = 56.7 / 7 = > x = 8.1 reduced weight = 6 x = 6 * 8.1 = 48.6 ; hence reduced weight = 48.6 answer : b
Answer: b
Question (gain): a shopkeeper fixes the marked price of an item 30 % above its cost price . the percentage of discount allowed to gain 8 % is
Options: a ) 15.92 % , b ) 16.92 % , c ) 17.92 % , d ) 18.92 % , e ) none of these
Rationale: "explanation : let the cost price = rs 100 then , marked price = rs 130 required gain = 8 % , so selling price = rs 108 discount = 130 - 108 = 22 discount % = ( 22 / 130 ) * 100 = 16.92 % option b"
Answer: b
Question (physics): maxwell leaves his home and walks toward brad ' s house at the same time that brad leaves his home and runs toward maxwell ' s house . if the distance between their homes is 36 kilometers , maxwell ' s walking speed is 3 km / h , and brad ' s running speed is 6 km / h . what is the distance traveled by maxwell when they meet in the middle ?
Options: a ) 12 , b ) 14 , c ) 15 , d ) 13 , e ) 11
Rationale: consider max starts from point a and brad starts from point b and move towards each other . assume they shall meet at point o after time ' t ' . the question asks us to find oa . from the question stem we can make out : - distance oa = 50 km - distance ob = > 3 xt = 36 - 6 xt ( i . e distance = speed x time ) = > 9 t = 36 hence t = 4 oa = 3 x 4 = 12 km answer : a
Answer: a
Question (general): a meal cost $ 32.50 and there was no tax . if the tip was more than 10 pc but less than 15 pc of the price , then the total amount paid should be :
Options: a ) 40 - 42 , b ) 39 - 41 , c ) 38 - 40 , d ) 37 - 39 , e ) 36 - 37
Rationale: "10 % ( 32.5 ) = 3.25 15 % ( 32.5 ) = 4.875 total amount could have been 32.5 + 3.25 and 32.5 + 4.875 = > could have been between 35.75 and 37.375 = > approximately between 36 and 37 answer is e ."
Answer: e
Question (physics): stacy and heather are 15 miles apart and walk towards each other along the same route . stacy walks at constant rate that is 1 mile per hour fast than heather ' s constant rate of 5 miles / hour . if heather starts her journey 24 minutes after stacy , how far from the original destination has heather walked when the two meet ? .
Options: a ) 6.5 mile , b ) 5.7 mile , c ) 7.0 mile , d ) 8.0 mile , e ) 12 mile
Rationale: "ss - stacy ' s speed = 6 m / hr sh - heather ' s speed = 5 m / hr in 24 minutes stacy will cover = ( 24 / 60 ) * 6 = 2.4 miles now since both are walking in opposite directions , add their speeds - 6 + 5 = 11 m / hr and distance to cover is 15 - 2.4 = 12.6 time taken = distance / speed = 12.6 / 11 = 1.145 hrs heather will cover = 5 * 1.145 = 5.72 miles . answer b"
Answer: b
Question (general): a group of students decided to collect as many paise from each member of group as is the number of members . if the total collection amounts to rs . 13.69 , the number of the member is the group is :
Options: a ) 57 , b ) 67 , c ) 37 , d ) 87 , e ) 97
Rationale: "money collected = ( 13.69 x 100 ) paise = 1369 paise numbers of members = 1369 squareroot = 37 answer c"
Answer: c
Question (physics): find the area of trapezium whose parallel sides are 10 cm and 18 cm long , and the distance between them is 15 cm .
Options: a ) 287 cm 2 , b ) 785 cm 2 , c ) 285 cm 2 , d ) 785 cm 2 , e ) 210 cm 2
Rationale: "explanation : area of a trapezium = 1 / 2 ( sum of parallel sides ) * ( perpendicular distance between them ) = 1 / 2 ( 10 + 18 ) * ( 15 ) = 210 cm 2 answer : option e"
Answer: e
Question (general): how many paying stones , each measuring 2 1 / 2 m * 2 m are required to pave a rectangular court yard 40 m long and 16 1 / 2 m board ?
Options: a ) 99 , b ) 132 , c ) 16 , d ) 10 , e ) 15
Rationale: "40 * 33 / 2 = 5 / 2 * 2 * x = > x = 132 answer : b"
Answer: b
Question (general): when a number is divided by 13 , the remainder is 11 . when the same number is divided by 17 , the remainder is 9 . what is the number ?
Options: a ) 389 , b ) 356 , c ) 380 , d ) 350 , e ) 349
Rationale: "x = 13 p + 11 and x = 17 q + 9 13 q + 11 = 17 q + 9 17 q - 13 q = 2 q = 2 + 13 q / 17 the least value of p for which q = 2 + 13 q / 17 is a whole number is p = 26 x = ( 13 * 26 * 11 ) = ( 338 + 11 ) = 349 . answer is e"
Answer: e
Question (general): if 1 / ( x + 5 ) + 1 / ( x - 5 ) = 1 / ( x - 5 ) , what is the value of x ?
Options: a ) - 1 , b ) 3 , c ) 1 , d ) 2 , e ) 5
Rationale: "if we solve the question , we get x = 5 . option : e"
Answer: e
Question (gain): how many liters of water must be added to 20 liters of milk and water containing 10 % water to make it 60 % water ?
Options: a ) 12 , b ) 15 , c ) 18 , d ) 22 , e ) 25
Rationale: "by rule of alligation : 60 % - 10 % = 50 % 100 % - 60 % = 40 % quantity of pure water : quantity of the mixture = 5 : 4 there are 20 liters of mixture , so we need to add 25 liters of pure water . the answer is e ."
Answer: e
Question (gain): the value of a machine depreciates at 20 % per annum . if its present value is $ 1 , 50,000 , at what price should it be sold after two years such that a profit of $ 24,000 is made ?
Options: a ) $ 250000 , b ) $ 430000 , c ) $ 120000 , d ) $ 170000 , e ) $ 150000
Rationale: "c $ 1 , 20,000 the value of the machine after two years = 0.8 * 0.8 * 1 , 50,000 = $ 96,000 sp such that a profit of $ 24,000 is made = 96,000 + 24,000 = $ 1 , 20,000"
Answer: c
Question (other): a certain university will select 1 of 7 candidates eligible to fill a position in the mathematics department and 2 of 12 candidates eligible to fill 2 identical positions in the computer science department . if none of the candidates is eligible for a position in both departments , how many different sets of 3 candidates are there to fill the 3 positions ?
Options: a ) 42 , b ) 70 , c ) 140 , d ) 165 , e ) 462
Rationale: "ans : 462 7 c 1 * 12 c 2 answer e )"
Answer: e
Question (gain): an inspector rejects 10 % of the meters as defective . how many will he examine to reject 20 ?
Options: a ) a ) 100 , b ) b ) 120 , c ) c ) 250 , d ) d ) 200 , e ) e ) 160
Rationale: "then , 10 % of x = 20 ( 10 / 100 ) x = 20 x = ( 20 * 100 * ) / 10 = 200 answer is d"
Answer: d
Question (physics): pipe a can fill a tank in 3 hours . due to a leak at the bottom , it takes 6 hours for the pipe a to fill the tank . in what time can the leak alone empty the full tank ?
Options: a ) 67 , b ) 88 , c ) 4.5 , d ) 26 , e ) 12
Rationale: "let the leak can empty the full tank in x hours 1 / 3 - 1 / x = 1 / 6 = > 1 / x = 1 / 2 - 1 / 6 = 1 / 4.5 = > x = 4.5 answer : c"
Answer: c
Question (geometry): the area of a rectangular field is equal to 200 square meters . its perimeter is equal to 60 meters . find the width of this rectangle .
Options: a ) 5 , b ) 10 , c ) 15 , d ) 20 , e ) 25
Rationale: "l * w = 200 : area , l is the length and w is the width . 2 l + 2 w = 60 : perimeter l = 30 - w : solve for l ( 30 - w ) * w = 200 : substitute in the area equation w = 10 and l = 20 : correct answer b"
Answer: b
Question (geometry): carmen made a sculpture from small pieces of wood . the sculpture is 2 feet 10 inches tall . carmen places her sculpture on a base that is 2 inches tall . how tall are the sculpture andbase together ?
Options: a ) 3.1 feet , b ) 3.2 feet , c ) 3.3 feet , d ) 3.0 feet , e ) 3.5 feet
Rationale: "we know 1 feet = 12 inch then 2 feet = 24 inch 24 + 10 = 34 then 34 + 2 = 36 36 / 12 = 3.0 feet answer : d"
Answer: d
Question (gain): if the simple interest on a sum of money for 2 years at 5 % per annum is rs . 56 , what is the compound interest on the same sum at the rate and for the same time ?
Options: a ) s . 57.40 , b ) s . 57.22 , c ) s . 51.219 , d ) s . 56.18 , e ) s . 53.11
Rationale: "explanation : sum = ( 56 * 100 ) / ( 2 * 5 ) = rs . 560 amount = [ 560 * ( 1 + 5 / 100 ) 2 ] = rs . 617.4 c . i . = ( 617.4 - 560 ) = rs . 57.4 . answer : a"
Answer: a
Question (physics): 18 men working 8 hours per day dig 30 m deep . how many extra men should be put to dig to a depth of 50 m working 6 hours per day ?
Options: a ) 22 , b ) 66 , c ) 88 , d ) 100 , e ) 281
Rationale: "( 18 * 8 ) / 30 = ( x * 6 ) / 50 = > x = 40 40 – 18 = 22 answer : a"
Answer: a
Question (geometry): the perimeter of a rectangle is equal to 280 meters . the ratio of its length to its width is 5 : 2 . find the area of the rectangle .
Options: a ) 400 square meters
b ) 4000 square meters
c ) 20 square meters
d ) 2000 square meters
e ) 1000 square meters
Rationale: solution if the ratio of the length to the width is 5 : 2 , then the measure l of the length and and the measure w of the with can be written as l = 5 x and w = 2 x we now use the perimeter to write 280 = 2 ( 2 l + 2 w ) = 2 ( 5 x + 2 x ) = 14 x solve for x 280 = 14 x x = 280 / 14 = 20 the area a of the rectangle is given by a = l × w = 5 x × 2 x = 10 x ^ 2 = 10 × 20 ^ 2 = 4000 square meters answer is b
Answer: b
Question (general): at the end of year x , automobile installment credit accounted for 36 % of all outstanding consumer installment credit . at that time automobile finance companies extended $ 35 billion of credit , or 1 / 3 of the automobile installment credit . how many billion dollars of consumer installment credit was outstanding at that time ?
Options: a ) 291.67 , b ) 105 , c ) 301.05 , d ) 288 , e ) 249.55
Rationale: "system of equations a = ( 36 / 100 ) c ( 1 / 3 ) a = 35 - - > a = 105 substitution 105 = ( 36 / 100 ) c c = ( 100 / 36 ) 105 - you can do the ugly calculation 105 / 36 * 100 the correct answer is a . the correct answer is 291.67"
Answer: a
Question (physics): the cross - section of a tunnel is a trapezium in shape . if the tunnel is 15 m wide at the top and 5 m wide at the bottom and the area of cross - section is 400 sq m , the depth of tunnel is ?
Options: a ) 200 m , b ) 600 m , c ) 400 m , d ) 800 m , e ) 900 m
Rationale: 1 / 2 * d ( 15 + 5 ) = 400 d = 400 answer : c
Answer: c
Question (general): if j is the least positive integer that is divisible by every integer from 10 to 15 inclusive , then j / 26 is =
Options: a ) 1155 , b ) 2310 , c ) 4620 , d ) 30030 , e ) 60060
Rationale: the integer should be divisible by : 10 , 11 , 12 , 13 , 14 and 15 , that is 5 * 211 , 3 * 2 ^ 2 , 13 , 2 * 7 , and 3 * 5 . the least common multiple of these integers is the product of 13 , 11 , 7 , 5 , 3 and 2 ^ 2 . then , r / 26 is ( 13 * 11 * 7 * 5 * 3 * 2 ^ 2 ) / 2 * 13 = 11 * 7 * 5 * 3 * 2 = 2310 . answer : b .
Answer: b
Question (gain): john purchased a grinder and a mobile for rs . 15000 & rs . 8000 respectively . he sold the grinder at a loss of 4 % and the mobile phone at a profit of 20 % . overall how much he make aprofit .
Options: a ) s . 190 , b ) s . 1120 , c ) s . 1200 , d ) s . 1250 , e ) s . 1290
Rationale: "let the sp of the refrigerator and the mobile phone be rs . r and rs . m respectively . r = 15000 ( 1 - 4 / 100 ) = 15000 - 600 m = 8000 ( 1 + 20 / 100 ) = 8000 + 1600 total sp - total cp = r + m - ( 15000 + 8000 ) = - 600 + 1600 = rs . 1200 as this is positive , an overall profit of rs . 1200 was made . c"
Answer: c
Question (geometry): the area of sector of a circle whose radius is 18 metro and whose angle at the center is 42 â ° is ?
Options: a ) 52.6 , b ) 52.9 , c ) 52.8 , d ) 118.8 , e ) 52.2
Rationale: "42 / 360 * 22 / 7 * 18 * 18 = 118.8 m 2 answer : d"
Answer: d
Question (physics): the amounts of time that three secretaries worked on a special project are in the ratio of 1 to 2 to 5 . if they worked a combined total of 120 hours , how many hours did the secretary who worked the longest spend on the project ?
Options: a ) 80 , b ) 70 , c ) 75 , d ) 16 , e ) 14
Rationale: "8 x = 120 = > x = 15 therefore the secretary who worked the longest spent 15 x 5 = 75 hours on the project option ( c )"
Answer: c
Question (general): a number of 44 marbles is to be divided and contain with boxes . if each box is to contain 3 , 4 , or 5 marbles , what is the largest possible number of boxes ?
Options: a ) 10 , b ) 14 , c ) 15 , d ) 16 , e ) 17
Rationale: "to maximize # of boxes we should minimize marbles per box : 13 * 3 + 1 * 5 = 44 - - > 13 + 1 = 14 . answer b"
Answer: b
Question (geometry): danny is sitting on a rectangular box . the area of the front face of the box is half the area of the top face , and the area of the top face is 1.5 times the area of the side face . if the volume of the box is 192 , what is the area of the side face of the box ?
Options: a ) 13 , b ) 16 , c ) 18 , d ) 29 , e ) 32
Rationale: "lets suppose length = l , breadth = b , depth = d front face area = l * w = 1 / 2 w * d ( l = 1 / 2 d or d = 2 l ) top face area = w * d side face area = w * d = 1.5 d * l ( w = 1.5 l ) volume = l * w * d = 192 l * 1.5 l * 2 l = 192 l = 4 side face area = l * d = l * 2 l = 4 * 2 * 4 = 32 e is the answer"
Answer: e
Question (general): diana is painting statues . she has 1 / 2 of a gallon of paint remaining . each statue requires 1 / 4 gallon of paint . how many statues can she paint ?
Options: a ) 2 , b ) 20 , c ) 28 , d ) 14 , e ) 19
Rationale: "number of statues = all the paint ÷ amount used per statue = 1 / 2 ÷ 1 / 4 = 1 / 2 * 4 / 1 = 2 answer is a"
Answer: a
Question (general): find the average of first 15 natural numbers .
Options: a ) 10 , b ) 120 , c ) 15 , d ) 8 , e ) 16
Rationale: explanation : sum of first n natural numbers = n ( n + 1 ) / 2 hence , sum of first 15 natural numbers = ( 15 x 16 ) / 2 = 120 therefore , required average of = 120 / 15 = 8 answer : d
Answer: d
Question (general): the “ a - number ” of a number x is defined as the ones digit of 2 ^ x . antony rolls a die with 6 sides labeled with the integers from 1 to 6 , each of which has an equal probability of landing face - up . he then takes 3 ^ a , where a is the a - number of the result of his die roll , and plots 3 ^ a on a number line as the point a . finally , he repeats this entire process , this time plotting the result as the point b . what is the probability that the distance between a and b is greater than the value of b ?
Options: a ) 3 / 8 , b ) 13 / 36 , c ) 17 / 36 , d ) 19 / 36 , e ) 23 / 36
Rationale: if you calculate 3 ^ a for 1 st roll , all 6 results will be 9 , 81 , 6561 , 729 , 9 , 81 . this result is the same for 2 nd roll . 9 , 81 , 6561 , 729 , 9 , 81 . about distance : if the first result is 9 and the second is also 9 , the distance is 9 - 9 = 0 which is smaller than 9 . if the first result is 9 and the second is 81 , the distance is 81 - 9 = 72 which is also smaller than b which has the value of 81 . if the first result is 81 and the second is 9 , the distance will be greater than b . distance 81 - 9 = 72 > 9 . on the first roll , the probability of getting result 9 is 2 / 6 . in this case no other alternative values for second roll which would make the distance greater than b . so probability is 0 . so next estimations are : probability of getting 81 on the first roll ( 2 / 6 ) * probability of getting 9 on the second roll ( 2 / 6 ) = 1 / 9 probability of getting 729 on the first roll ( 1 / 6 ) * probability of getting 9 , 81 on the second roll ( 4 / 6 ) = 1 / 9 probability of getting 6561 on the first roll ( 1 / 6 ) * probability of getting 9 , 81 , 729 on the first roll ( 5 / 6 ) = 5 / 36 all together : 1 / 9 + 1 / 9 + 5 / 36 = 13 / 36 = b
Answer: b
Question (general): you and your friend spent a total of $ 17 for lunch . your friend spent $ 3 more than you . how much did your friend spend on their lunch ?
Options: a ) $ 9 , b ) $ 3 , c ) $ 4 , d ) $ 10 , e ) $ 5
Rationale: "my lunch = l , my friends lunch = l + 3 ( l ) + ( l + 3 ) = 17 l + l + 3 - 3 = 17 - 3 2 l = 14 l = 7 my friends lunch l + 3 = 7 + 3 = $ 10 , the answer is d"
Answer: d