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Two persons are awarded pensions in proportion to the square root of the number of years they have served. The second person's service period is 9 years longer than that of the first. If the length of service of the first person had exceeded that of the second by 4.25 years, their pensions would have been in the respective ratio of 9 : 8. How long have the two served?
Let n be the number of years served by the second.
^{} = ^{}
^{}
64n + 272 = 81n
n = 16 years
First person's service = 16  9 = 7 years
The monthly incomes of Vasu and Ravi are in the ratio 7 : 8 and their monthly expenditures are in the ratio 3 : 4, respectively. If the respective savings of Vasu and Ravi are Rs. 2,400 and Rs. 1,600, then find the monthly income of Ravi.
Ratio of monthly incomes of Vasu and Ravi = 7 : 8
Let their incomes be 7x and 8x, respectively.
Ratio of their expenditures = 3 : 4
Expenditure = Income  Savings
Savings of Vasu = Rs. 2,400
His expenditure = (7x  2,400)
Savings of Ravi = Rs. 1,600
His expenditure = (8x  1,600)
According to question:
^{}
⇒ 28x  9,600 = 24x  4,800
⇒ 4x = 4,800
⇒ x = 1,200
Monthly income of Ravi = 8x = (8 × 1,200) = Rs. 9,600
Suman began a business with a capital of Rs. 50,000 and after 3 months, Manu entered into the partnership with a capital of Rs. 75,000. Three months later, Amandeep joined the firm with a capital of Rs. 1,25,000. At the end of the year, the firm made a profit of Rs. 99,495. How much of this sum did Amandeep receive?
Money invested by Suman for 12 months = Rs. 50,000
Money invested by Manu for 9 months = Rs. 75,000
Money invested by Amandeep for 6 months = Rs. 1,25,000
Share of Suman : Manu : Amandeep
= 50,000 × 12 : 75,000 × 9 : 1,25,000 × 6 = 6,00,000 : 6,75,000 : 7,50,000
= 600 : 675 : 750 = 8 : 9 : 10
Total profit = Rs. 99,495
Profit of Amandeep = ^{} × 99,495 = ^{} × 99,495 = Rs. 36,850
The only people in each of the auditoriums A and B in a school are students, and each student in each of the auditoriums A and B is either from Grade 8 or from Grade 7. The ratio of number of students from Grade 8 to those from Grade 7 in auditorium A is 4 : 5. The ratio of number of students from Grade 8 to those from Grade 7 in auditorium B is 3 : 17, and the ratio of the total number of students from Grade 8 in both auditoriums to the total number of students from Grade 7 in both auditoriums A and B is 5 : 7. What is the ratio of the total number of students in auditorium A to the total number of students in auditorium B?
Let number of students of Grade 7 in auditorium A = 5a
Thus, number of students of Grade 8 in auditorium A = 4a
Let number of students of Grade 7 in auditorium B = 17b
Thus, number of students of Grade 8 in auditorium B = 3b
Total number of students of Grade 8 = 4a + 3b
Total number of students of Grade 7 = 17b + 5a
Now, (4a + 3b)/(17b + 5a) = 5/7
Or, a = (64/3)b
Let b = 3c
We get a = 64c
Total number of students in auditorium A = 9a = 576c
Total number of students in auditorium B = 20b = 60c
Required ratio = 576 : 60 = 48 : 5
Hence, answer option 4 is correct.
Three friends count the number of coins they have. They found that the ratio of the total number of coins of 25 p, 50 p and Re 1 is 11 : 6 : 13. They have 25 p coins in the ratio 2 : 4 : 5, 50 p coins in the ratio 6 : 3 : 9 and Re 1 coins in the ratio 7 : 2 : 4. Find the amount with the second person, if they have money only in the denomination of 25 p, 50 p and Re 1 and the total number of 25 p coins with them is 132.
Let the total number of coins = x
Total number of 25 coins = 132
The ratio of 25 p, 50 p and Re 1 coins is 11 : 6 : 13.
So, 25 p coins = x ^{} ^{}
132 = x ^{} ^{}
x = ^{} = 360
So, the total number of coins = 360
Total 50 p coins = 360 ^{}^{} = 72
Total 1 Re coins = 360 ^{}^{} = 156
Number of 25 p coins with second person = 132 ^{}^{} = 48
Number of 50 p coins with second person = 72 ^{} ^{} = 12
Number of 1 Re coins with second person = 156 ^{} ^{} = 24
That is, the second person have 48 coins of 25 p, 12 coins of 50 p and 24 coins of 1 Re.
Total money = 48 x 0.25 + 12 ^{} 0.50 + 24 ^{} 1 = 12 + 6 + 24 = Rs. 42
A balloon can go up in the air at the rate of 24 kmph and its speed diminishes by a quantity, which varies as the square root of the number of stones attached to it. With four stones, its speed is 20 kmph. The highest number of stones with which the balloon can go up in the air is
Diminishing speed ^{}^{}
Let speed be S_{1} when no stones are attached.
S_{1} = 24
Let S_{2} be the speed when stones are attached (where N be the number of stones)
S_{2} = 24  ^{ }(where k is proportionality constant)
According to given condition;
20 = 24  ^{}
2k = 4
k = 2
S_{2 }= 24 ^{ }
For S_{2} = 0 , we have
^{}
N = 144
It means that if we tie 144 stones, the balloon cannot go up in the air.
So, the highest number of stones that can be attached with the balloon is 143.
Tanya and Sanya invested some amount of money in the ratio 7 : 6 for the same period. At the end of the year, 20% of the profit was donated to an old age home. Out of the remaining profit, 50% is reinvested and the rest is divided in the ratio of their initial contribution. If the difference between their shares of profit is Rs. 4400, what was the total profit?
Let the total profit be Rs. x.
After donating 20% profit to old age home, remaining profit = 0.8x.
Out of the remaining profit, 50% is reinvested.
So, the remaining profit = 0.4x
Now this 0.4x has to be divided in the ratio 7 : 6.
Tanya's share = 0.4x × ^{}
Sanya's share = 0.4x × ^{}
Difference = 0.4x × ^{}– 0.4x × ^{ }= 4400
= 0.4x ×^{}= 4400
x = ^{}= Rs. 1,43,000
Three persons Akbar, Birbal and Tansen invested in a business Rs. 10,000, Rs. 25,000 and Rs. P, respectively. Akbar withdrew Rs. 1,000 after 3 months and Rs. 1000 again after 5 months. Birbal invested Rs. 7,000 more than his initial investment after 9 months. Tansen maintained his capital for the whole year. If at the end of the year, they made a profit of Rs. 12,400, and Tansen got Rs. 2,400 as his share in the profit, find the initial investment made by Tansen.
Total investment made by Akbar = 10,000 × 3 + 9,000 × 2 + 8,000 × 7
= 30,000 + 18,000 + 56,000 = 1,04,000
Total investment made by Birbal = 25,000 × 9 + 32,000 × 3
= 2,25,000 + 96,000 = 3,21,000
Total investment made by Tansen = P × 12
Total profit = 12,400
Share of Tansen in total profit
= ^{}
2,400 = ^{}
2,400 × 4,25,000 + 2,400 × 12P = 12,400 × 12P
2,400 × 4,25,000 = 10,000 × 12P
P = ^{} = 8,500
So, initial investment made by Tansen = Rs. 8,500
Mr. Alaram designed a new clock, which has 12 divisions as usual, represented by the numbers 1 to 12. But the speed of the minute hand is directly proportional to the digit, which it is going to approach, i.e. if the minute hand is between 1 and 2, its speed is proportional to 2 and if it is between 11 and 12, its speed is proportional to 12. The speed of the minute hand, when it is between 12 and 1 is equal to that of an ordinary clock and both the clocks were set at 9 : 00 am.
What will be the time in the ordinary clock, if the new clock shows 9 : 30 am?
Since, in between 12 and 1, the speed of minutes hand is similar to that of an ordinary clock, it take 5 minutes to cover first division.
Then its speed will become twice, because, the hand is between 1 and 2.
So, it takes ^{} minutes. Between 2 and 3, the speed becomes thrice and takes ^{} minutes.
So, in total it takes ^{} minutes to reach 6 (9 : 30 am).
= 5^{} = ^{} minutes = 12¼ minutes.
So, ordinary clock shows 9 hours 12 minutes 15 seconds.
Amounts deposited by two friends Harpreet and Gagan are in the ratio 7 : 9. Total amount deposited is Rs. 72,000. After some time, Harpreet withdraws Rs. 7500. Find the money withdrawn by Gagan, so that the new ratio of their deposits may become 4 : 5.
Amount deposited by Harpreet and Gagan = Rs. 72,000
Amounts are in the ratio 7 : 9.
Let common ratio = x
Money deposited by Harpreet = 7x
Money deposited by Gagan = 9x
Now, 7x + 9x = Rs. 72,000
16x = Rs. 72,000
x = Rs. 4500
Money deposited by Harpreet = 4500 × 7 = Rs. 31,500
Money deposited by Gagan = 4500 × 9 = Rs. 40,500
Money left with Harpreet after withdrawal of Rs. 7500 = 31,500  7500 = Rs. 24,000
Ratio of amounts of Harpreet and Gagan is 4 : 5.
So, amount of Gagan = 5/4 × 24,000 = Rs. 30,000
Amount withdrawn by Gagan = 40,500  30,000 = Rs. 10,500
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