A boy on being asked 13/14 of a certain fraction had made the mistake of dividing the fraction by 14/13 and so got an answer that exceeded the correct answer by 3/65. The correct fraction is?

0.3+ 0.6 + 0.7 + 0.8 = x,
? x = 3/9 + 6/9 + 7/9 + 8/9 = 24/9 = 8/3 = 2²/₃

Length of each piece = 3¹/₄ = 13/4 m
Maximum pieces that can be cut = 30/(13/4) = 9
? Rope left = 30 - 13/4 x 9 = 120 - 117/4 = 3/4 m
? Required fraction = 3/4/30 = 1/4

Time required for service before overhauling = 2/3h = 40 min in 45 days
and time required for service after overhualing =2/3h= 40 min in 60 days
Now,LCM of 45 and 60 = 180 days
time required for service before overhauling is 180 days = 40 x 4 = 160 min
and time required for service before overhauling is 180 days =40 x 3 = 120 min
? fraction of time saved = 160 - 120/160 =40/160 = 1/4

Total number of students participated = (B/4 + 3G/4)
? Required percentage= (B/4 + 3G/4) / (B+G) x 100%
Clearly, given data is inadequate.

Let the population of village = N
Then, adult's population = 5N/8
and adult females = 1/2 x 5N/8 = 5N/16
? illiterate females = 4/5 of adult female population
? 400 = 4/5 x 5N/16
? N = (400 x 16)/4
? N = 1600

Let the total property = N
Then, property of his daughter = P/7
? Remaining property = P - P/7 = 6P/7
Now, this property is divided among his sons, so that each son gets twice the property of his daughter.
Let number of sons be K.
Then, 6N/7 x K = 2N/7
? K = 3

Let the weight of vesel = x kg
Then, 1/4 of volume of water = (5.25 - x)kg
Full volume of water = (16.5 - x)
? 4(5.25 - x) = (16.5 - x)
? 21 - 4x = 16.5 - x
? 21 - 16.5 = 3x
? x = 4.5/3 = 1.5 kg

Let A has ? 200 at the starting of game, then B has ? 100 at the starting of game.
After the game, money left with A = 200 - 200/3 = ? 400/3
? Total money with B = 100 + 200/3 = ? 500/3
Let the fraction of money lost by B = N
Then, 500/3 - N x 500/3 = 400/3 + N x 500/3
? 500/3 - 400/3 = 500N/3 + 500N/3
? 100/3 = 1000N/3
? N = (100/3) x (3/1000) = 1/10
so, the fraction of money lost is 1/10.

Let the fraction be x, y and z respectively in decreasing order.
Then, according to the question,
x/z = 7/6
? x = 7z/6

And y = 7/6 - 1/3 = 7-2/6 = 5/6

Now, x + y + z = 2¹¹/₂₄
? 7z/6 + 5/6 + z = 59/24
? (13z + 5)/ 6 = 59/24
? 13z = 59/4 - 5 =39/4
? z = 3/4
? x =7z/6 = 7/8
Hence, the fraction are 7/8, 5/6, 3/4.

Let the greatest, middle and smallest fraction be P, Q and R respectively in decreasing order.

According to the question.
(Greatest fraction) / (Smallest fraction) = 7/6
? P/R = 7/6
? P = 7R/6 ....(i)
and Q = 7/6 - 1/3 = 7 - 2/6 = 5/6 (ii)

Now, P + Q + R = 2¹¹/₂₄ (iii )

Substitute the value of P and Q from Eqs. (i), (ii) and put in Eqs (iii), we get
? 7R/6 + 5/6 + R = 59/24
? (7R + 5 + 6R)/6 = 59/24
? (13R + 5) / 6 = 59/24
? 13R = (59/4) - 5 = 39/4
? R = 39 / (4 x 13)
? R = 3/4

On putting the value of R in Eq. (i), we get
P = 7R/6 = (7/6) x (3/4) = 7/8