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[Vanilla Cake]

trytry:

I need help in 2 average questions below:

Your 2 questions are from National Mathematical Olympiad of Singapore - Special Round set on 31 July 2010.
Q1 is from Q20 [5 marks]
Q2 is from Q16 [5 marks]

trytry:

Q1. 20 students took a test. The teacher ranked them from the highest score (1st) to the lowest score (20th). The top 10 students had an average of 92 marks and the next 10 students had an average of 80 marks. Also, the top 5 students had an average score that was 12 marks higher than the average of next 15 students. Find the lowest possible score of the student in the 20th rank

Refer to http://psle2010a.blogspot.com/2010/08/nmos-q20.html.
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2. There are 5 members in John's family : Father, Mother, elder sister Helen, John and younger sister Lucy. Just before Lucy was borned, the average age of the family was 19 yrs old and the age of John's father is same as the sum of ages of the rest of family. 2 years ago, the average age of the family was 35.2 yrs and the age of John's mother was 8 years more than the sum of ages of John and Lucy. The current average age of the family, excluding Helen, is 39.5 years. Find the present age of John.

Refer to http://psle2010a.blogspot.com/2010/08/nmos-2010-q16.html

while waiting for KSP members to respond.

VC's mum

[atutor2001]

trytry:

I need help in 2 average questions below:


Q1. 20 students took a test. The teacher ranked them from the highest score (1st) to the lowest score (20th). The top 10 students had an average of 92 marks and the next 10 students had an average of 80 marks. Also, the top 5 students had an average score that was 12 marks higher than the average of next 15 students. Find the lowest possible score of the student in the 20th rank. \t\t\t\t\t\t\t
.

The solution can be broken down into 2 parts :

Part 1

Through normal calculation (model preferred) we can find the average score of the following bands :

1st to 5th : Average = 95
6th to 10th : Average = 89
11th to 20th : Average = 80

Part 2

Concept :

a) If there are odd number of consecutive numbers, the average will be the middle score.
b) If we want the last score to be smallest we must make the front score as big as possible.

Applying concept (a) to band (1st - 5th position) starting with the middle position i.e. 3rd position being equal to the average i.e. 95, we can build the score of the other positions

3rd(95)
2nd(96) 3rd(95) 4th(94)
1st(97) 2nd(96) 3rd(95) 4th(94) 5th(95)

From concept (b), we would like the 5th score to be as high as possible but that is not possible because the question stated \"no 2 scores are the same\". So the highest possible score for the 5th is 95.

Similarly for band (6th - 10th position), we get

8th(89)
7th(90) 8th(89) 9th(88.)
6th(91) 7th(90) 8th(89) 9th(88.) 10th(87)

Similarly, the biggest possible score for the 10th position is 87

For the last band (11th - 20th position), we want the front position to be as big as possible. But the biggest for position 11th must be less than 87 i.e. 11th position is 86. We then built downwards until position 19th

11th(86) 12th(85) ........ 18th(79) 19th(78.)

The total score of 11th to 20th = 10 x 80 = 800

The total score of 11th to 19th position = 86+85+...+79+78= 738

The smallest 20th score = 800 - 738 =62

[atutor2001]

trytry:

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2. There are 5 members in John's family : Father, Mother, elder sister Helen, John and younger sister Lucy. Just before Lucy was borned, the average age of the family was 19 yrs old and the age of John's father is same as the sum of ages of the rest of family. 2 years ago, the average age of the family was 35.2 yrs and the age of John's mother was 8 years more than the sum of ages of John and Lucy. The current average age of the family, excluding Helen, is 39.5 years. Find the present age of John.

Before L was born

F+M+H+J = 19 x 4 = 76

F = 76/2 = 38


Two years ago

F+M+H+J+L = 35.2 x 5 = 176

M = J+L+8


Now

F+M+J+L = 39.5 x 4 = 158 (now)


Computations :

F+M+J+L = 158-2-2-2-2 = 150 (2 yrs ago)
176 - 150 = 26 (H's age 2 yrs ago)

Finding the number of years between \"before L was born\" and \"Two years ago\"

176 - 76 = 100
100/5 = 20 (this is the number of years between \"before L was born\" and \"Two years ago\")

Note : L is 0 years old just before birth

L = 0 + 20 = 20 (2 yrs ago)
F = 38 + 20 = 58 (2 yrs ago)

M = J+L+8 = J + 28

F+M+H+J+L = 176
58 + (J+28 ) + 26 + J + 20 = 176
2J = 176 - 132
2J = 44
J = 22 (2 yrs ago)

J = 22 + 2 = 24 (now)

[trytry]

VC’s mum, thanks for your link.


atutor2001, thanks for your clear and helpful explanation.

[Dharma]

trytry:

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2. There are 5 members in John's family : Father, Mother, elder sister Helen, John and younger sister Lucy. Just before Lucy was borned, the average age of the family was 19 yrs old and the age of John's father is same as the sum of ages of the rest of family. 2 years ago, the average age of the family was 35.2 yrs and the age of John's mother was 8 years more than the sum of ages of John and Lucy. The current average age of the family, excluding Helen, is 39.5 years. Find the present age of John.


Thanks.

Before Lucy was born
F + M + H + J = 19 x 4 = 76

F = M + H + J
F = 38
M + H + J = 38

Just after Lucy was born
F + M + H + J + L = 76
Average age = 76/5 = 15.2

2 years ago
Lucy’s age = 35.2 – 15.2 = 20
Father’s age = 38 + 20 = 58
M = J + L + 8
M = J + 28

Now
F + M + J + L = 39.5 x 4 = 158
M + J = 158 – 60 – 22 = 76

2 yrs ago
M + J = 76 – 4 = 72 ---------------- ( 1 )
M – J = 28 ----------------------------( 2 )
(1 ) – (2 )
2J = 72 - 28 = 44
J = 22

John’s age now = 22 + 2 = 24 yrs old

[Dharma]

trytry:

Q1. 20 students took a test. The teacher ranked them from the highest score (1st) to the lowest score (20th). The top 10 students had an average of 92 marks and the next 10 students had an average of 80 marks. Also, the top 5 students had an average score that was 12 marks higher than the average of next 15 students. Find the lowest possible score of the student in the 20th rank. \t\t\t\t\t\t\t

Thanks.

Top 10 students : Ave = 92
Top 11th to 20th student : Ave = 80

5(1u + 12) + 15(1u) = 920 + 800 = 1720
20u = 1660
1u = 83
Top 5 students : Ave = 83 + 12 = 95
Top 6th to 10th students : Ave = (920 – 5(95)/5 = 89

RANKING
1. 97
2. 96
3. 95
4. 94
5. 93
6. 91
7. 90
8. 89
9. 88
10. 87

86 + 85 + …. + 78 = [ 4(78 + 85) + 86] = 738

Lowest possible score = (80 x 10 ) – 738 = 62

[Muffins]

thanks Dharma, for your crystal clear explanation!!!!

[Dharma]

Muffins:

thanks Dharma, for your crystal clear explanation!!!!

No worries, mate.

[trytry]

Thanks Dharma.

[2DMommy]

Mr Muthu bought a refrigerator worth $1450 with his monthly salary. After spending 1/6 of his remaining salary on a digital camera, he had 5/8 of his salary left. What was Mr Muthu’s salary?