2009 Science and Maths Society “Maths Quiz”
1.Fresh Mango contains 70% water by weight whereas dry mango contains 10% water by weight. What is the weight of dry mango that can be obtained from 20 kg of Fresh Mango?
Solution:
The correct answer is 6.67 Kg.
From 20 kgs of fresh mango, we get 6 kg of pure solid (no water) i.e. 30 % of 20 kg. Since we know dry mango contains 10% water, just divide 6 kgs by 0.9 to get the answer
2.What is the remainder when 2^1344452457 is divided by 11?
Solution : The correct answer is 7.
Solution:
2^1 divided by 11 gives a remainder of 2
2^2 divided by 11 gives a remainder of 4;
Similarly:
2^3 gives a remainder of 8,
2^4 gives a remainder of 5
2^5 givesa remainder of 10
2^6 gives a remainder of 9
2^7 gives a remainder of 7
2^8 gives a remainder of 3
2^9 gives a remainder of 6
2^10 gives a remainder of 1
2^11 gives a remainder of 2
So the cycle of remainders repeats in cycles of 10.
In the question, 7 values are left after (removing all the cycles of 10); hence the remainder will be the same as 2^7 divided by 11 i.e. 7.
3. If a is a prime number and a[(a-1)! + 1] is divisible by 2a, then a^a is?
(! : factorial)
The correct answer is 4.
Since a[(a-1)! + 1] is divisible by 2a, it means [(a-1)! + 1] is divisible by 2;
which is possible for only two values of a i.e. 1 & 2 {all other values of a, (a-1)! is an even number, and hence [(a-1)! + 1] will be an odd number (which in turn means that it is not divisible by 2)} Now only 2 is the prime number, therefore a=2 and 2^2=4.
4. The sum of seven consecutive integers is 1,617. How many of them are prime?
The correct answer is 2.
1617 divided by 7 is 231; so the numbers are 234,233,232,231,230,229 and 228. The prime numbers are 229 and 233.
5. In my class, there are 99 students other than me. 50 of us play soccer, 45 basketball and 50 play volleyball. Only 15 of us play all three games. Everyone plays at least one game. How many play only two games?
The correct answer is 15.
Let the people who play only two games be :
x (Soccer and basketball)
y(soccer and volleyball)
z(Volleyball and basketball)
Therefore, 100 = 50 + 45 + 50 -(x+15) -(y+15) -(z+15) + 15.
Solving, we get x+y+z=15.
6. What is the number of players in a chess tournament, if a total of 63 matches have been played?
(The tournament is in a knock out format i.e. any player who loses is out of the tournament and there were no 'byes')
The correct answer is 64.
Using the formula of S_n=(〖a(r〗^n-1))/(r-1) , where r > 1. Sn = 63, a = 1, r = 2. Therefore, n = 6. Tn = ar^n-1, therefore, T6 = 32. There are 32 matches. So, there are 64 players.
7. Ace goes to a furniture shop to buy a sofa set and a centre table. He bargains for a 10% discount on the centre table and a 25% discount on the sofa set. However, the shopkeeper, by mistake, interchanged the discount figures while making the bill and Ace paid accordingly. When compared to what he should have paid for his purchases, what percentage did Ace pay extra given that the centre table costs 40% as much as the sofa set?
The correct answer is 8.1%.
Let the cost of sofa set be 100 and the cost of centre table be 40 (for example). Amount payable by Ace = 75 + 36 = 111. Amount calculated by shopkeeper= 90 + 30 = 120. Percentage excess paid by Ace=9/111×100% = 8.1%.
8. A certain number of two digits is decreased by 54 when the digits are interchanged. The tens digit is three times the unit digit. Find the number.
The correct answer is 93 & ninety-three.
Let the tens digit be x and let the unit digit be y. Then, the value of the number is 10x + y because (for example) 23 is equal to 2 * 10 + 3 where 2 is the tens digit and 3 is the unit digit (Aha! That's why I gave that hint!). When the digits are interchanged (reversed), the value of the number is 10y + x.
Therefore:
(10x+y) - (10y+x) = 54
10x + y - 10y - x = 54
9x - 9y = 54.
If you divide the equation throughout by 9, we get
x - y = 6.
We know that the tens digit is three times the unit digit, therefore
x = 3y, 3y - y = 6
2y = 6
y = 3.
Since x = 3y and y = 3, x = 9.
The number is 93.
The number with the digits reversed is 39 (93 - 39 = 54).
Notice that the ten digit in 93 (9) is three times the unit digit (3).
1.Fresh Mango contains 70% water by weight whereas dry mango contains 10% water by weight. What is the weight of dry mango that can be obtained from 20 kg of Fresh Mango?
Solution:
The correct answer is 6.67 Kg.
From 20 kgs of fresh mango, we get 6 kg of pure solid (no water) i.e. 30 % of 20 kg. Since we know dry mango contains 10% water, just divide 6 kgs by 0.9 to get the answer
2.What is the remainder when 2^1344452457 is divided by 11?
Solution : The correct answer is 7.
Solution:
2^1 divided by 11 gives a remainder of 2
2^2 divided by 11 gives a remainder of 4;
Similarly:
2^3 gives a remainder of 8,
2^4 gives a remainder of 5
2^5 givesa remainder of 10
2^6 gives a remainder of 9
2^7 gives a remainder of 7
2^8 gives a remainder of 3
2^9 gives a remainder of 6
2^10 gives a remainder of 1
2^11 gives a remainder of 2
So the cycle of remainders repeats in cycles of 10.
In the question, 7 values are left after (removing all the cycles of 10); hence the remainder will be the same as 2^7 divided by 11 i.e. 7.
3. If a is a prime number and a[(a-1)! + 1] is divisible by 2a, then a^a is?
(! : factorial)
The correct answer is 4.
Since a[(a-1)! + 1] is divisible by 2a, it means [(a-1)! + 1] is divisible by 2;
which is possible for only two values of a i.e. 1 & 2 {all other values of a, (a-1)! is an even number, and hence [(a-1)! + 1] will be an odd number (which in turn means that it is not divisible by 2)} Now only 2 is the prime number, therefore a=2 and 2^2=4.
4. The sum of seven consecutive integers is 1,617. How many of them are prime?
The correct answer is 2.
1617 divided by 7 is 231; so the numbers are 234,233,232,231,230,229 and 228. The prime numbers are 229 and 233.
5. In my class, there are 99 students other than me. 50 of us play soccer, 45 basketball and 50 play volleyball. Only 15 of us play all three games. Everyone plays at least one game. How many play only two games?
The correct answer is 15.
Let the people who play only two games be :
x (Soccer and basketball)
y(soccer and volleyball)
z(Volleyball and basketball)
Therefore, 100 = 50 + 45 + 50 -(x+15) -(y+15) -(z+15) + 15.
Solving, we get x+y+z=15.
6. What is the number of players in a chess tournament, if a total of 63 matches have been played?
(The tournament is in a knock out format i.e. any player who loses is out of the tournament and there were no 'byes')
The correct answer is 64.
Using the formula of S_n=(〖a(r〗^n-1))/(r-1) , where r > 1. Sn = 63, a = 1, r = 2. Therefore, n = 6. Tn = ar^n-1, therefore, T6 = 32. There are 32 matches. So, there are 64 players.
7. Ace goes to a furniture shop to buy a sofa set and a centre table. He bargains for a 10% discount on the centre table and a 25% discount on the sofa set. However, the shopkeeper, by mistake, interchanged the discount figures while making the bill and Ace paid accordingly. When compared to what he should have paid for his purchases, what percentage did Ace pay extra given that the centre table costs 40% as much as the sofa set?
The correct answer is 8.1%.
Let the cost of sofa set be 100 and the cost of centre table be 40 (for example). Amount payable by Ace = 75 + 36 = 111. Amount calculated by shopkeeper= 90 + 30 = 120. Percentage excess paid by Ace=9/111×100% = 8.1%.
8. A certain number of two digits is decreased by 54 when the digits are interchanged. The tens digit is three times the unit digit. Find the number.
The correct answer is 93 & ninety-three.
Let the tens digit be x and let the unit digit be y. Then, the value of the number is 10x + y because (for example) 23 is equal to 2 * 10 + 3 where 2 is the tens digit and 3 is the unit digit (Aha! That's why I gave that hint!). When the digits are interchanged (reversed), the value of the number is 10y + x.
Therefore:
(10x+y) - (10y+x) = 54
10x + y - 10y - x = 54
9x - 9y = 54.
If you divide the equation throughout by 9, we get
x - y = 6.
We know that the tens digit is three times the unit digit, therefore
x = 3y, 3y - y = 6
2y = 6
y = 3.
Since x = 3y and y = 3, x = 9.
The number is 93.
The number with the digits reversed is 39 (93 - 39 = 54).
Notice that the ten digit in 93 (9) is three times the unit digit (3).
