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Quadratic Equations

Long Answer Type

171. The denominator of a fraction exceeds its numerator by 2. If one is added to both numerator and denominator, the difference between the new and the original fractions is 1/21. Find the original fractions.

Let numerator of the fraction be x
So, denominator = x + 2
Hence, fraction (original) =

space space fraction numerator straight x over denominator straight x plus 2 end fraction

If 1 is added to both numerator and denominator then,
Numerator = x + 1
and denominator = x + 2 + 1 = x + 3
Hence, fraction =

fraction numerator straight x plus 1 over denominator straight x plus 3 end fraction

According to the given condition,

space fraction numerator straight x plus 1 over denominator straight x plus 3 end fraction minus fraction numerator straight x over denominator straight x plus 2 end fraction equals 1 over 21

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fraction numerator left parenthesis straight x plus 1 right parenthesis left parenthesis straight x plus 2 right parenthesis minus straight x left parenthesis straight x plus 3 right parenthesis over denominator left parenthesis straight x plus 3 right parenthesis left parenthesis straight x plus 2 right parenthesis end fraction equals 1 over 21

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fraction numerator left parenthesis straight x squared plus 2 straight x plus straight x plus 2 right parenthesis minus straight x squared minus 3 straight x over denominator straight x squared plus 2 straight x plus 3 straight x plus 6 end fraction equals 1 over 21

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fraction numerator straight x squared plus 3 straight x plus 2 minus straight x squared minus 3 straight x over denominator straight x squared plus 5 straight x plus 6 end fraction equals 1 over 21

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fraction numerator 2 over denominator straight x squared plus 5 straight x plus 6 end fraction equals 1 over 21

rightwards double arrow space space space space space space space space x squared plus 5 x plus 6 space equals space 42 rightwards double arrow space space space space space space space space x squared plus 5 x minus 36 space equals space 0 rightwards double arrow space space space space space space space space x squared plus 9 x minus 4 x minus 36 space space equals space 0 rightwards double arrow space space space space space space space space x left parenthesis x plus 9 right parenthesis minus 4 left parenthesis x plus 9 right parenthesis space space equals space 0 rightwards double arrow space space space space space space space space space space space space space space space space space space left parenthesis x minus 4 right parenthesis space left parenthesis x plus 9 right parenthesis space equals space 0 rightwards double arrow space space space space space space space space space space space space space space space space space space space space space space space space space x space minus space 4 space equals space 0

straight x plus 9 space equals space 0

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x = 4
or x = -9
x = -9 is not possible
Therefore, x = 4
Now, numerator = x = 4
and denominator = x + 2 = 6
So, required fractions are

4 over 6 space or space space 2 over 3.

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172. A two-digit number is four times the sum and three times the product of its digits. Find the number.

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173. O Girl! Out of a group of swans,

space 7 over 2

times the square root of the number are playing on the shore of a tank. The two remaining ones are playing, with amorous fight, in the water. What is the total number of swans?

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174. Out of a number of Saras birds, one-fourth the number are moving about in lotus plants;

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7 times the square root of the number move on a hill; 56 birds remain in vakula trees. What is the total number of birds? (Mahavira)

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175. The angry Arjun carried some arrows for fighting with Bheeshm. With half the arrows, he cut down the arrows thrown by Bheeshm on him and with six other arrows, he killed the rath driver of Bheeshm. With one arrow each, he knocked down respectively the rath, flag and the bow of Bheeshm. Finally, with one more than four times the square root of arrows he laid Bheeshm unconscious of on arrow bed. Find the total number of arrows Arjun had.

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176. A motorboat whose speed in still water is 5 km/h, takes 1 hour more to go 12 km upstream than to return downstream to the same spot. Find the speed of the stream.

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177. A plane left 30 minutes later than the schedule time and in order to reach its destination 1500 km away in time it has to increase its speed by 250 km/hr from its usual speed. Find its usual speed.

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178. Two trains leave a railway station at the same time. The first train travels due west and the second train due north. The first train travels 5 km/hr faster than the second train. If after two hours, they are 50 km apart, find the average speed of each train.

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179. A fast train takes 3 hours less than a slow train for a journey of 600 km. If the speed of the slow train is 10 km/hr less than that of the fast train, find the speeds of the two trains.

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180. In a flight of 600 km an aircraft was slowed down due to bad weather. Its average speed for the trip was reduced by 200 km/hr and the time of flight increased by 30 minutes. Find the duration of flight.

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