When an object is dropped, the number of feet F it falls is given by the formula F = 3⁄2dt2 where t is time in seconds since it was dropped and d is a numerical constant with value 10. If it takes 7 seconds for an object to reach the ground, approximately how many feet does it fall during last 3 seconds?
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Solution
Distance in last 3 seconds would be distance covered in 7 seconds - distance covered in 1st 4 seconds.
Fall in 7 seconds is given by
F7 = 3⁄2 × (10) × (7)2
Fall in 4 seconds = F4 = 3⁄2 × (10) × (4)2
F7 - F4 = Fall in last 3 seconds.
∴ F7 - F4 = 3⁄2 × (10) × [49 - 16]
= 3⁄2 × 10 × 33 = 495
There are 120 televisions on display at an electronics showroom. The number of televisions displaying sports programs is 50% less than 4 times the number of televisions displaying educational programs 1⁄4 of the televisions are displaying neither sports nor educational programs. If all of the 120 televisions are displaying programs, how many televisions are displaying sports programs?
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Solution
Let S, E, 0 denote the number of TVs displaying sports programs, educational programs and other programs respectively.
S + E + 0 = 120
1⁄4 of the TVs are displaying neither sports programs nor educational programs, 0 is 30.
Thus S + E + 30 = 120 ∴ S + E = 90
Also S = 1⁄2 × (4E) ∴ S = 2E
∴ 3E = 90 ∴ E = 30 ∴ S = 60
A Fibonacci sequence is one in which each term after the second is the sum of two preceding terms. For example 2, 5, 7, 12, 19 is a portion of Fibonacci sequence. If a, b, c, d, e is a portion of Fibonacci sequence, express b in terms of d and e.
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Solution
Method 1:
e - d = c
Similarly, d - c = b
∴ d - (e - d) = b ∴ 2d - e = b
12!-11!=
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Solution
12!- 11! = 12 × 11! - 11!
= 11! (12 - 1) = 11! × 11 = 11! (10 + 1)
A group of friends went on a trip to Mauritius. 60% of the women and 40% of the men went dancing at a club. The remaining friends went to a casino. If 60% of the group are women, what % of the group went to a casino?
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Solution
Let the group contain 100 people, 60 women and 40 men. 60% of women went dancing \(□\) 40% of women to casino = 40% of 60 = 24.
Similarly, 40% of men went dancing. Therefore 60% of men to casino = 60% of 40 = 24 men went to casino.
∴ Out of 100 people, 24 men and 24 women i.e. 48 people went to casino which is 48%.
Paul and Mark are racing their sailboats from point A. Paul starts at 9 a.m. and passes a cruise ship that is anchored 20 miles away from point A. Paul is travelling at 36 miles per hour. Mark starts at 10 a.m. and also passes the anchored cruise ship. Mark is travelling at 48 miles per hour. If they continue to travel at the same rates, at what time will Mark overtake Paul?
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Solution
Paul leaves 1 hour before Mark. Therefore he is 36 miles ahead of Mark.
∴ Mark will have to overtake 36 miles to meet Paul.
∴ He will overtake at 12 miles per hour (48 - 36).
∴ He will take \(\frac{36}{12}\) = 3 hours after 10.00 a.m. i.e. 1.00 p.m.
Note: Passing the cruise sheep has no relevance to the problem.
ABC enterprises has 3 CNC machines named x, y, z; producing the same item. Machine x produces <<sup>1⁄4 as many of the items as machine y produces in the same time, and machine y produces 5⁄3 times as many of the item as machine z does it in same time. If all the machines work simultaneously for the same period, then machine y produces what fraction of total number of items produced by machine x and machine z?
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Solution
If machine z produces 12 units (4 × 3), machine y will produce 20 units and therefore machine x will produce 5 units.
∴. Required fraction = \(\frac{y}{x-z}=\frac{20}{17}\)
Area of circle with centre 0 is 64π. What is ratio of perimeter of OPRQ to that of OPSQ (π =3)?
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Solution
Area = 64π ∴ Radius = 8
∴ OP = OQ = 8
Arc PRQ = \(\frac{2\pi\times 8}{360}\times 120=2\times 3\times 8\times \frac{1}{3}=16\)
Arc PSQ = 32
∴ Required ratio = \(\frac{8+8+16}{8+ 8 + 32}=\frac{32}{48}=\frac{2}{3}\)
Dexterous Corporation designed an automatic water feeding mechanism for a water tank. An iron water tank with a capacity of 10,000 gallons was used for this experiment. It had a supply water inlet which fed water at a rate of 300 gallons per hour. The water inlet started automatically when water level went below 5000 gallon mark. But there was a hole formed at the bottom of tank when the tank was filling. Water leaked through this hole at a rate of 500 gallons per hour. How long will it take for whole tank to be emptied?
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Solution
The water supply starts after the water level reaches 5000 gallon mark. This will take \(\frac{5000}{500}\) = 10 hrs.
Once the 5000 gallon mark is reached, there is an inflow at rate of 300 gallons per hour and an outflow at rate of 500 gallons per hour.
So effective emptying rate = 200 gallons per hour.
∴ d 5000 At s rate time require =\(\frac{5000}{200}\)= 25 hours.
∴ Total time = 25 + 10 = 35 hours.
The price of necklace varies as the cube of the number of diamonds it contains. However, the jeweler Paul divided the necklace into 2 parts in the ratio 3 : 7 and thereby made 2 necklaces, thus resulting in a loss of $ 6300 for him. What was the original price of the necklace?
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Solution
Let the original necklace consists of 10x diamonds.
∴ Price = 1000x3
∴ Price of necklace with 3x diamonds = (3x)3 = 27x3
Similarly, price of necklace with 7x diamonds = (7x)3 = 343x3
Total price of 2 necklaces = 27x3 + 343x3 = 370x3
1000x3 - 370x3 = 630x3
If 630x3 = 9450
∴ x3 = \(\frac{6300}{630}\)=$10
∴ Original price = 1000x3= $10,000