Cheukting/mathqa huggingface dataset:
Question (gain): carl is facing very difficult financial times and can only pay the interest on a $ 30,000 loan he has taken . the bank charges him a quarterly compound rate of 5 % . what is the approximate interest he pays annually ?
Options: a ) $ 1200 , b ) $ 2000 , c ) $ 2150 , d ) $ 6000 , e ) $ 12000
Rationale: "usually , you are given the annual rate of interest and it is mentioned that it is annual rate . the bank charges him a quarterly compounded annual rate of 20 % . here you find per quarter rate as ( 20 / 4 ) % = 5 % i have actually never seen a question with quarter rate given but since this question did not mentionannual rate of interestand since the options did not make sense with 5 % annual rate of interest , it is apparent that the intent was a 5 % quarterly rate . so the bank charges 5 % every quarter and compounds it in the next quarter . had it been a simple quarterly rate , we would have just found 4 * 5 % of 30,000 = $ 6000 as our answer . but since , the interest is compounded , it will be a bit more than $ 6000 . option ( d ) looks correct ."
Answer: d
Question (gain): on a certain transatlantic crossing , 20 percent of a ship ’ s passengers held round - trip tickets and also took their cars abroad the ship . if 50 percent of the passengers with round - trip tickets did not take their cars abroad the ship , what percent of the ship ’ s passengers held round - trip tickets ?
Options: a ) 30 % , b ) 40 % , c ) 50 % , d ) 60 % , e ) 65 %
Rationale: "let t be the total number of passengers . let x be the number of people with round trip tickets . 0.2 t had round trip tickets and took their cars . 0.5 x had round trip tickets and took their cars . 0.5 x = 0.2 t x = 0.4 t the answer is b ."
Answer: b
Question (physics): the pinedale bus line travels at an average speed of 60 km / h , and has stops every 5 minutes along its route . yahya wants to go from his house to the pinedale mall , which is 7 stops away . how far away , in kilometers , is pinedale mall away from yahya ' s house ?
Options: a ) 20 km , b ) 35 km , c ) 40 km , d ) 50 km , e ) 60 km
Rationale: "number of stops in an hour : 60 / 5 = 12 distance between stops : 60 / 12 = 5 km distance between yahya ' s house and pinedale mall : 5 x 7 = 35 km imo , correct answer is ` ` b . ' '"
Answer: b
Question (general): if p represents the product of the first 18 positive integers , then p is not a multiple of
Options: a ) 99 , b ) 84 , c ) 72 , d ) 65 , e ) 95
Rationale: "answer is e . since prime factor of 95 is 5 x 19 . 19 is a prime number and not part of the first 18 positive integers . a ) 9 x 11 b ) 12 x 7 c ) 12 x 6 d ) 13 x 5 e ) 3 x 19"
Answer: e
Question (general): the fuel indicator in a car shows 1 / 5 th of the fuel tank as full . when 22 more liters of fuel are poured in to the tank , the indicator rests at the 3 / 4 of the full mark . find the capacity of the tank .
Options: a ) 25 litres , b ) 35 litres , c ) 30 litres , d ) 40 litres , e ) none of these
Rationale: x / 5 + 22 = 3 x / 4 = > x = 40 litres answer : d
Answer: d
Question (other): a salesman sold twice as much pears in the afternoon than in the morning . if he sold $ 450 kilograms of pears that day , how many kilograms did he sell in the afternoon ?
Options: a ) 120 , b ) 180 , c ) 300 , d ) 280 , e ) 320
Rationale: 3 x = 450 x = 150 therefore , the salesman sold 150 kg in the morning and 2 ⋅ 150 = 300 kg in the afternoon . so answer is c .
Answer: c
Question (physics): a and b together can complete work in 10 days . a alone starts working and leaves it after working for 6 days completing only half of the work . in how many days it can be completed if the remaining job is undertaken by b ?
Options: a ) 4 , b ) 8 , c ) 5 , d ) 6 , e ) 30
Rationale: "explanation : ( a + b ) one day work = 1 / 10 now a does half of the work in 6 days so a can complete the whole work in 12 days a ’ s one day work = 1 / 12 b ’ s one day work = 1 / 10 - 1 / 12 = 2 / 120 = 1 / 60 b alone can complete the work in 60 days so half of the work in 30 days answer : option e"
Answer: e
Question (other): set a consists of the integers from 4 to 16 , inclusive , while set b consists of the integers from 6 to 20 , inclusive . how many distinct integers do belong to the both sets at the same time ?
Options: a ) 5 , b ) 7 , c ) 8 , d ) 9 , e ) 11
Rationale: "a = { 4 , 5,6 , 7,8 , 9,10 , 11,12 , . . . 16 } b = { 6 , 7,8 , 9,10 , 11,12 . . . } thus we see that there are 11 distinct integers that are common to both . e is the correct answer ."
Answer: e
Question (general): find large no . from below question the difference of two numbers is 1365 . on dividing the larger number by the smaller , we get 6 as quotient and the 30 as remainder
Options: a ) 1235 , b ) 1456 , c ) 1567 , d ) 1632 , e ) 1635
Rationale: "let the smaller number be x . then larger number = ( x + 1365 ) . x + 1365 = 6 x + 30 5 x = 1335 x = 267 large number = 267 + 1365 = 1632 d"
Answer: d
Question (general): the sum of the ages of 5 children born at the intervals of 3 years each is 50 years . what is the age of the youngest child ?
Options: a ) 4 years , b ) 8 years , c ) 10 years , d ) 12 years , e ) none of these
Rationale: "let the ages of the children be x , ( x + 3 ) , ( x + 6 ) , ( x + 9 ) and ( x + 12 ) years . then , x + ( x + 3 ) + ( x + 6 ) + ( x + 9 ) + ( x + 12 ) = 50 5 x = 20 = > x = 4 . age of youngest child = x = 4 years . answer : a"
Answer: a
Question (general): paul sells encyclopedias door - to - door . he earns $ 150 on every paycheck , regardless of how many sets he sells . in addition , he earns commission as follows : commission sales 10 % $ 0.00 - $ 10 , 000.00 5 % $ 10 , 000.01 - - - > he does not earn double commission . that is , if his sales are $ 12,000 , he earns 10 % on the first $ 10,000 and 5 % on the remaining $ 2,000 . his largest paycheck of the year was $ 1,320 . what were his sales for that pay period ?
Options: a ) 13,400 , b ) 14,800 , c ) 16,400 , d ) 15,800 , e ) 19,600
Rationale: his pay check was $ 1320 . out of this , $ 150 was his fixed salary so the total commission he earned was $ 1320 - $ 150 = $ 1170 he earns 10 % on the sales of first $ 10,000 which gives a commission of $ 1000 . he earns 5 % on every subsequent dollar . since he earns another $ 170 , he must have had sales of another 170 * ( 100 / 5 ) = 3400 so his total sales must have been $ 10,000 + $ 3400 = $ 13,400 a
Answer: a
Question (geometry): a rectangle with width 8 and diagonal 30 . find the area ?
Options: a ) 240 square units
b ) 180 square units
c ) 100 square units
d ) 150 square units
e ) 160 square units
Rationale: then the area is : 8 ' ' x 30 ' ' = 240 square inches , or 240 square units hence a
Answer: a
Question (physics): train a leaves the station traveling at 30 miles per hour . two hours later train в leaves the same station traveling in the same direction at 35 miles per hour . how many miles from the station was train a overtaken by train b ?
Options: a ) 420 , b ) 450 , c ) 480 , d ) 510 , e ) 540
Rationale: "after two hours , train a is ahead by 60 miles . train b can catch up at a rate of 5 miles per hour . the time to catch up is 60 / 5 = 12 hours . in 12 hours , train a travels another 30 * 12 = 360 miles for a total of 420 miles . the answer is a ."
Answer: a
Question (probability): a box contains 25 electric bulbs , out of which 4 are defective . two bulbs are chosen at random from this box . the probability that at least one of these is defective is
Options: a ) 3 / 10 , b ) 7 / 19 , c ) 12 / 19 , d ) 21 / 95 , e ) none
Rationale: "solution p ( none is defective ) = 21 c 2 / 25 c 2 = 7 / 10 p ( at least one is defective ) = ( 1 - 7 / 10 ) = 3 / 10 . answer a"
Answer: a
Question (general): how many prime numbers are between 28 / 3 and 86 / 6 ?
Options: a ) 1 , b ) 2 , c ) 3 , d ) 4 , e ) 5
Rationale: "28 / 3 = 9 . xxx 86 / 6 = 14 . xxx so we need to find prime numbers between 9 ( exclusive ) - 13 ( inclusive ) there are 2 prime numbers 1113 hence answer will be ( b ) 2 b"
Answer: b
Question (gain): in a school of 800 students , 44 % wear blue shirts , 28 % wear red shirts , 10 % wear green shirts , and the remaining students wear other colors . how many students wear other colors ( not blue , not red , not green ) ?
Options: a ) 120 , b ) 132 , c ) 144 , d ) 156 , e ) 168
Rationale: "44 + 28 + 10 = 82 % 100 – 82 = 18 % 800 * 18 / 100 = 144 the answer is c ."
Answer: c
Question (general): a man whose speed is 6 kmph in still water rows to a certain upstream point and back to the starting point in a river which flows at 1.5 kmph , find his average speed for the total journey ?
Options: a ) 7 , b ) 5.6 , c ) 6 , d ) 9 , e ) 3
Rationale: "m = 6.0 s = 1.5 ds = 7.5 us = 4.5 as = ( 2 * 7.5 * 4.5 ) / 12 = 5.6 answer : b"
Answer: b
Question (other): in what ratio must rice of rs . 25 per kg be mixed with rice of rs . 12 per kg so that cost of mixture is rs . 20 per kg ?
Options: a ) 8 : 5 , b ) 5 : 8 , c ) 20 : 25 , d ) 12 : 20 , e ) 25 : 12
Rationale: "( 20 - 12 ) / ( 25 - 20 = 8 / 5 = 8 : 5 answer : a"
Answer: a
Question (physics): 10 men and 15 women together can complete a work in 4 days . it takes 100 days for one man alone to complete the same work . how many days will be required for one woman alone to complete the same work ?
Options: a ) 100 days , b ) 667 days , c ) 177 days , d ) 187 days , e ) 225 days
Rationale: "1 man ' s 1 day work = 1 / 100 ( 10 men + 15 women ) ' s 1 day work = 1 / 4 15 women ' s 1 day work = ( 1 / 4 - 10 / 100 ) = 3 / 20 1 woman ' s 1 day work = 1 / 100 1 woman alone can complete the work in 100 days . answer : a"
Answer: a
Question (general): if john makes a contribution to a charity fund at school , the average contribution size will increase by 50 % reaching $ 75 per person . if there were 4 other contributions made before john ' s , what is the size of his donation ?
Options: a ) $ 100 , b ) $ 175 , c ) $ 200 , d ) $ 250 , e ) $ 450
Rationale: cavg = average contribution before john cavg * 1.5 = 75 , therefore the average cont is $ 50 before john . if he needs to increase the average contribution by $ 25 , he must put in $ 25 for each of the 4 people . so $ 100 . but , he also has to put in the average for himself ( the fiveth person ) , so add $ 75 . so $ 175 is your answer . answer b
Answer: b
Question (general): if 0.5 : x : : 5 : 8 , then x is equal to :
Options: a ) a ) 1.12 , b ) b ) 1.2 , c ) c ) 1.25 , d ) d ) 0.9 , e ) e ) 0.8
Rationale: "( x * 5 ) = ( 0.5 * 8 ) x = 4 / 5 x = 0.8 answer = e"
Answer: e
Question (general): the average weight of 8 people increases by 2.5 kg when a new person comes in place of one of them weighing 75 kg . what is the weight of the new person ?
Options: a ) 75 kg , b ) 85 kg , c ) 95 kg , d ) 65 kg , e ) 55 kg
Rationale: "the total weight increase = ( 8 x 2.5 ) kg = 20 kg weight of new person = ( 75 + 20 ) kg = 95 kg the answer is c ."
Answer: c
Question (physics): the radius of a wheel is 12.6 cm . what is the distance covered by the wheel in making 200 resolutions ?
Options: a ) 158.210 m , b ) 704 m , c ) 153.256 m , d ) 158.256 m , e ) 204 m
Rationale: "in one resolution , the distance covered by the wheel is its own circumference . distance covered in 200 resolutions . = 200 * 2 * 22 / 7 * 12.6 = 15825.6 cm = 158.256 m answer : d"
Answer: d
Question (general): simplify : 250 x 250 - 100 x 100
Options: a ) 761200 , b ) 761400 , c ) 52500 , d ) 761500 , e ) none of them
Rationale: "( 250 ) ^ 2 - ( 100 ) ^ 2 = ( 250 + 100 ) ( 250 - 100 ) = 350 x 150 = 52500 . answer is c ."
Answer: c
Question (physics): on my sister ' s birthday , she was 143 cm in height , having grown 10 % since the year before . how tall was she the previous year ?
Options: a ) 140 cm , b ) 136 cm , c ) 130 cm , d ) 127 cm , e ) 125 cm
Rationale: "let the previous year ' s height be x . 1.1 x = 143 x = 130 the answer is c ."
Answer: c
Question (gain): if the selling price of 50 articles is equal to the cost price of 35 articles , then the loss or gain percent is :
Options: a ) 45 % , b ) 23 % , c ) 30 % , d ) 60 % , e ) 56 %
Rationale: "c . p . of each article be re . 1 . then , c . p . of 50 articles = rs . 50 ; s . p . of 50 articles = rs . 35 . loss % = 15 / 50 * 100 = 30 % answer c"
Answer: c
Question (geometry): the length of each side of an equilateral triangle having an area of 4 â ˆ š 3 cm 2 is ?
Options: a ) 4 / 3 cm
b ) 3 / 4 cm
c ) 3 cm
d ) 4 cm
e ) 5 cm
Rationale: explanation : â ˆ š 3 / 4 a 2 = 4 â ˆ š 3 - > a = 4 answer is d
Answer: d
Question (geometry): if the sides of a rectangle are increased by 25 % , what is the percentage increase in the area ?
Options: a ) 54.25 % , b ) 56.25 $ , c ) 53.25 % , d ) 55.25 % , e ) 58.25 %
Rationale: "if sides are a and b , after increase sides would be 1.25 a and 1.25 b . percentage increase in area = ( 1.25 a * 1.25 b - ab ) * 100 / ab = 56.25 % answer : b"
Answer: b
Question (gain): the charge for a single room at hotel p is 30 percent less than the charge for a single room at hotel r and 10 percent less than the charge for a single room at hotel g . the charge for a single room at hotel r is what percent greater than the charge for a single room at hotel g ?
Options: a ) 28 % , b ) 20 % , c ) 40 % , d ) 50 % , e ) 150 %
Rationale: "let rate in r = 100 x then p = 70 x g = 100 y p = 90 y thus 70 x = 90 y or x = 1.28 y ans r = 128 y so increase = 28 % answer : a"
Answer: a
Question (gain): a trader marked the selling price of an article at 11 % above the cost price . at the time of selling , he allows certain discount and suffers a loss of 1 % . he allowed a discount of :
Options: a ) 10 % , b ) 10.5 % , c ) 9.9 % , d ) 12.5 % , e ) none
Rationale: "sol . let c . p . = rs . 100 . then , marked price = rs . 110 , s . p . = rs . 99 . ∴ discount % = [ 11 / 111 * 100 ] % = 9.9 % answer c"
Answer: c
Question (general): bottle r contains 250 capsules and costs $ 5.25 . bottle t contains 130 capsules and costs $ 2.99 . what is the difference between the cost per capsule for bottle r and the cost per capsule for bottle t ?
Options: a ) $ 0.25 , b ) $ 0.12 , c ) $ 0.05 , d ) $ 0.03 , e ) $ 0.002
Rationale: cost per capsule in r is 5.25 / 250 = 0.525 / 25 = 0.021 cost per capsule in t is 2.99 / 130 = 0.023 the difference is 0.002 the answer is e
Answer: e
Question (general): sushil got thrice as many marks in english as in science . his total marks in english , science and maths are 162 . if the ratio of his marks in english and maths is 4 : 5 , find his marks in science ?
Options: a ) 10.9 , b ) 20.9 , c ) 30.9 , d ) 35 , e ) 45
Rationale: "s : e = 1 : 3 e : m = 4 : 5 - - - - - - - - - - - - s : e : m = 4 : 12 : 15 4 / 31 * 162 = 20.9 answer : b"
Answer: b
Question (general): solve the equation for x : 21 ( x + y ) + 5 = 21 ( - x + y ) + 47
Options: a ) - 2 , b ) - 1 , c ) 0 , d ) 2 , e ) 1
Rationale: "e 1 21 ( x + y ) + 5 = 21 ( - x + y ) + 47 21 x + 21 y + 5 = - 21 x + 21 y + 47 21 x + 5 = - 21 x + 47 42 x = 42 = > x = 1"
Answer: e
Question (physics): area of four walls of a room is 99 m 2 . the length and breadth of the room are 7.5 m and 3.5 m respectively . the height of the room is :
Options: a ) 2.5 m
b ) 3.5 m
c ) 1.5 m
d ) 4.5 m
e ) 5.5 m
Rationale: 2 ( 7.5 + 3.5 ) × h = 99 2 ( 11 ) h = 99 22 h = 99 h = 99 / 22 = 9 / 2 = 4.5 m answer is d .
Answer: d
Question (general): x is a positive integer less than 300 . when x is divided by 7 , the remainder is 1 ; when x is divided by 3 , the remainder is 2 . how many x are there ?
Options: a ) 21 , b ) 22 , c ) 23 , d ) 24 , e ) 25
Rationale: "the nubmer which when divided by 7 leaves remainder 1 should be of the form 7 k + 1 this number when divided by 3 leaves remainder 2 . so , ( 7 k + 1 ) - 2 should be divisible by 3 or 7 k - 1 should be divisible by 3 . we now put the values of k starting from 0 to find first number divisible by 3 we find 1 st number at k = 1 thus smallest number will be 7 ( 1 ) + 1 = 8 now , next number will be = 8 + lcm of 37 i . e 29 now we will find number of all such values less than 500 by using the formula for last term of an a . p 8 + ( n - 1 ) 21 = 300 n = 22.13 or n = 22 answer : - b"
Answer: b
Question (probability): how many different ways can 2 students be seated in a row of 4 desks , so that there is always at least one empty desk between the students ?
Options: a ) 2 , b ) 3 , c ) 4 , d ) 6 , e ) 12
Rationale: "total cases : 12 ( student one has 4 options and student two has three options , 4 x 3 = 12 ) non - favourable cases : 6 ( when two students sit together . students in desk 1 and desk 2 , in desk 2 and desk 3 , in desk 3 and desk 4 ) for each of these cases there are two possibilities because the positions can be interchanged . hence 2 x 3 = 6 . so favourable cases : 12 - 6 = 6 . answer : d"
Answer: d
Question (gain): if each side of a square is increased by 25 % , find the percentage change in its area ?
Options: a ) 56.29 % , b ) 56.25 % , c ) 56.95 % , d ) 53.25 % , e ) 36.25 %
Rationale: "let each side of the square be a , then area = a x a new side = 125 a / 100 = 5 a / 4 new area = ( 5 a x 5 a ) / ( 4 x 4 ) = ( 25 a ² / 16 ) increased area = = ( 25 a ² / 16 ) - a ² increase % = [ ( 9 a ² / 16 ) x ( 1 / a ² ) x 100 ] % = 56.25 % answer : b"
Answer: b
Question (gain): a dishonest dealer professes to sell goods at the cost price but uses a weight of 720 grams per kg , what is his percent ?
Options: a ) 15 % , b ) 25 % , c ) 38.9 % , d ) 45 % , e ) 35 %
Rationale: "explanation : 720 - - - 280 100 - - - ? = > 38.9 % answer : c"
Answer: c
Question (general): if the mean of numbers 28 , x , 42 , 78 , 82 and 104 is 62 , then what is the mean of 128 , 255 , 511 , 1023 and x ?
Options: a ) 395 , b ) 275 , c ) 391 , d ) 415 , e ) 365
Rationale: "the mean of numbers 28 , x , 42 , 78 and 104 is 62 : 28 + x + 42 + 78 + 82 + 104 = 62 * 6 - - > x = 38 ; so , the mean of 128 , 255 , 511 , 1023 and x is ( 128 + 255 + 511 + 1023 + 38 ) / 5 = 391 . answer : c ."
Answer: c
Question (gain): machine a produces 100 parts twice as fast as machine b does . machine b produces 100 parts in 80 minutes . if each machine produces parts at a constant rate , how many parts does machine a produce in 20 minutes ?
Options: a ) 60 , b ) 80 , c ) 70 , d ) 40 , e ) 50
Rationale: machine b produces 100 part in 80 minutes . machine a produces 100 parts twice as fast as b , so machine a produces 100 parts in 80 / 2 = 40 minutes . now , machine a produces 100 parts in 40 minutes which is 100 / 40 = 10 / 4 parts / minute . 10 / 4 parts x a total of 20 minutes = 10 / 4 * 20 = 50 e
Answer: e
Question (general): if x and y are integers such that | y + 3 | ≤ 3 and 2 y – 3 x + 6 = 0 , what is the least possible value t of the product xy ?
Options: a ) - 12 , b ) - 3 , c ) 0 , d ) 2 , e ) none of the above
Rationale: "how to deal with inequalities involving absolute values ? first example shows us the so callednumber case in this case we have | y + 3 | ≤ 3 which is generalized | something | ≤ some number . first we solve as if there were no absolute value brackets : y + 3 ≤ 3 y ≤ 0 so y is 0 or negative second scenario - remove the absolute value brackets . put a negative sign around the other side of the inequality , andflip the sign : y + 3 > = - 3 y > = - 6 therefore we have a possible range for y : - 6 = < y < = 0 ok , so far so good , we ' re half way through . what about x ? here ' s the formula : 2 y – 3 x + 6 = 0 , rewrite it as 2 y + 6 = 3 x . you can say that 2 y + 6 is a multiple of 3 ( = 3 x ) . so all values which must be integer must also satisfy this constraint . i ' m just saying that , so it ' s easier to evaluate all the possible numbers ( - 6 , - 3 , 0 ) . if you plug in y = 0 , x will be 2 and xy = 0 as the lowest possible value t . hence , answer choice c is the one to go ."
Answer: c
Question (general): a certain characteristic in a large population has a distribution that is symmetric about the mean m . if 66 percent of the distribution lies within one standard deviation d of the mean , what percent of the distribution is less than m + d ?
Options: a ) 16 % , b ) 32 % , c ) 83 % , d ) 84 % , e ) 92 %
Rationale: "this is easiest to solve with a bell - curve histogram . m here is equal to µ in the gaussian normal distribution and thus m = 50 % of the total population . so , if 66 % is one st . dev , then on either side of m we have 66 / 2 = 33 % . so , 33 % are to the right and left of m ( = 50 % ) . in other words , our value m + d = 50 + 33 = 83 % goingfrom the mean m , to the right of the distributionin the bell shaped histogram . . this means that 83 % of the values are below m + d . like i said , doing it on a bell - curve histogram is much easier to fullygethow this works , or you could apply gmat percentile jargon / theory to it c"
Answer: c
Question (other): in a class of 54 students , 12 enrolled for both english and german . 22 enrolled for german . if the students of the class enrolled for at least one of the two subjects , then how many students enrolled for only english and not german ?
Options: a ) 30 , b ) 10 , c ) 18 , d ) 28 , e ) 32
Rationale: "total = english + german - both + neither - - > 54 = english + 22 - 12 + 0 - - > english = 44 - - > only english = english - both = 44 - 12 = 32 . answer : e ."
Answer: e
Question (physics): usc invited each south carolina high school to send up to 39 students to watch a football game . a section which has 199 seats in each row is reserved for those students . what is the least number of rows needed to guarantee that if 2006 students show up , then all students from the same high school can be seated in the same row ?
Options: a ) 11 , b ) 12 , c ) 13 , d ) 14 , e ) 15
Rationale: the answer is 12 rows . if 59 schools send 34 students each , then we can sit at most 5 groups of students in the same row , so we will need 12 rows . next , 12 rows are sufficient . assume that this is not the case . suppose the groups of students are seated like this : first the largest group , then the second largest group , then the third largest group , etc . suppose we run out of space - there are not enough seats in any row to seat together the next group . suppose the first group that can not be seated together is the kth group and it consists of n students . then k 61 since any row fits at least 5 groups . also , n 2006 / k 2006 / 61 < 33 ( all groups already seated are no smaller than the kth group ) . so , n 32 . since there is not enough space in any of the 12 rows to seat the kth group , then there must be at least 168 students seated in each of the 12 rows . now , 12 × 168 = 2016 > 2006 a contradiction . so , 12 rows are sufficient . correct answer b
Answer: b
Question (geometry): a rectangular photograph is surrounded by a border that is 1 inch wide on each side . the total area of the photograph and the border is m square inches . if the border had been 3 inches wide on each side , the total area would have been ( m + 64 ) square inches . what is the perimeter of the photograph , in inches ?
Options: a ) 16
b ) 24
c ) 32
d ) 40
e ) 48
Rationale: let x and y be the width and length of the photograph . ( x + 2 ) ( y + 2 ) = m and so ( 1 ) xy + 2 x + 2 y + 4 = m ( x + 6 ) ( y + 6 ) = m and so ( 2 ) xy + 6 x + 6 y + 36 = m + 64 let ' s subtract equation ( 1 ) from equation ( 2 ) . 4 x + 4 y + 32 = 64 2 x + 2 y = 16 , which is the perimeter of the photograph . the answer is a .
Answer: a
Question (general): a “ palindromic integer ” is an integer that remains the same when its digits are reversed . so , for example , 43334 and 516615 are both examples of palindromic integers . how many 6 - digit palindromic integers are both even and greater than 300,000 ?
Options: a ) 200 , b ) 216 , c ) 300 , d ) 400 , e ) 2,500
Rationale: "the first digit and last digit are the same so the 3 possibilities are 4 , 6 , or 8 . the second and third digits can be any digit from 0 to 9 . the total number of palindromic integers is 3 * 10 * 10 = 300 the answer is c ."
Answer: c
Question (general): the unit digit in the product 4556 * 3432 * 4581 * 2784 is ?
Options: a ) 8 , b ) 1 , c ) 2 , d ) 3 , e ) 4
Rationale: unit digit in the given product = unit digit in 6 * 2 * 1 * 4 = 8 answer is a
Answer: a