IMSO 2009
ESSAY PROBLEMS
General Remarks
1.A full mark for each problem is 3 points.
2.When the student gives final answer only, the
mark is 1 point.
1. Bob bought a coat and a shirt. The normal prices of both items are the same, but when Bob bought them, the shirt was discounted by 50% and the coat was discounted by 25%. If he bought them for $130 what was the normal price of the shirt?
2. The area of a rhombus is 36 cm2. One of its diagonals has length twice of the other diagonal. What is the length of the smaller diagonal of the rhombus?
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Answer: $104
Let price = original price of one coat or one shirt.
50% x price + (100% - 25%) x price = 130... 1 point
The original price of the shirt is $104... 1 point
Answer: 6 cm
... 1 point
Area of rhombus 36 cm2
4a2 = 36 ... 1 point
a2 = 9 a = 3
3. I plan to travel by car from City A to City B. I travel the first half of the distance with the speed of 30 km/hour. In order to get my average traveling speed of 40 km/hour, what speed should I take for the second half of the distance?
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Answer:
Marking Scheme:
If the distance = 120 km.
2 hours T hour A o---o---o B 60 km C 60 km
A half of the distance = 120/2 = 60 km.
Time needed from A to C = 60/30 = 2 hour. ………. 1point
In order get 40 km//hour from A to B:
120
40 = --- ……….……….………. ½ point 2 + T
3 1 = --- 2 + T 2 + T = 3
T = 1. ……….…….. 1 point
Speed at the next half of the distance is
4. The mean value of the mathematics marks of the nine students is 70. When Lambert’s mark is added, the mean value will be 69. When John’s mark is added, the mean value will be 72. What will the mean value be, if Lambert’s and John’s marks are both added?
5. Andy bought three packages of goods, each worth $ 35, $ 30, and $ 40. The first package contains 2 books, 1 pencil, and 1 eraser. The second package contains 1 book, 1 pencil, 2 erasers. The third package contains 3 books and 2 erasers. Andy wants to buy the fourth package containing 2 books, 1 pencil, and 3 erasers. What is the price of the fourth package?
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Answer :
Scheme:
Finding Lambert’s mark (1 point) Finding John’s mark (1 point)
Calculating the total mark sum of eleven students (½ point)
Calculating the mean (½ point)
Sum=70*9 Sum+L=10*69 Sum+J=10*72
2*Sum+L+J=10*141 or Sum+L+J=10*141-70*9=780 Thus, (Sum+L+J)/11=780/11
Answer :
Marking scheme
Formulating the system of equations (1 point)
Finding the prices of a book, a pencil, and an eraser (½ point for each)
Calculating the final answer (½ point)
Package # Books # Pencils # Erasers Price
I 2 1 1 $35
II 1 1 2 $30
III 3 0 2 $40
IV 2 1 3 ?
Firstly, comparing (I) & (II): 1 Books - 1 Eraser = $5 or 2 Books = $10 + 2 Erasers
Secondly, combining (II) & (III) and comparing with (I): 2 Books + 3 Erasers = $35
Combining first & second results: 5 erasers = $25 so Eraser = $5.
The price of Package IV is $35 + 2 x price of eraser
6. The following figure shows a regular hexagon. On each side of the hexagon, there is an isosceles right triangle. One side forming a right angle is a side of the hexagon. Determine the angle x.
7. Bob and Ivan have tasks of mowing (cutting) grass in their yard. One day they do their work at the same time. After that, Bob works every 8 days, while Ivan every 6 days. Each time Bob works, he is paid $15 while Ivan gets $17.5. If they work again on the same day, how much money will each of them earn at the end of the day?
Answer:
Day 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Bob v v v v
Ivan v v v v v
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x
Answer:
Internal angle of hexagonal is 1200 ... 1 point
Internal angle of right triangle are 900 dan 450 ... 1 point
Determining of angle x is 360-1200-900- 450=1050 ... 1 point
They meet again at 24th day. At this time: ………1 point
Bob’s work 4 times ……… ½ point
Ivan’s work 5 times ……… ½ point
Hence
Bob’s earn = 4 x $15 = $ 60 ……… ½ point
8. Mum’s kitchen scale is set incorrectly, but otherwise it works fine. When she weighs a bag of sugar, it shows 1.5 kg. When she weighs a bag of flour, it shows 1.3 kg. However, when she weighs both items together, it shows 2.5 kg.
If she weighs a piece of butter of weight 0.3 kg, what number does the scale show?
9. Alan, Billy, Candy, and David are queuing (lining up) in alphabetical order. Alan is in the 7th position from the front while David is in the 9th position from the back. The number of persons between Alan and Billy is the same as those between Candy and David. In total, there are 48 persons in the queue, and six of them are between Billy and Candy. How many persons are there between Alan and Candy?
Answer:
From A to D, there are all together 48-14=34 persons. ……… 1 point
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Answer:
Solution: 0.6 kg
Observe that kg is the real weight of the bag of sugar. In a similar way,
kg is the real weight of the bag of flour. So the scale shifts 0.3 kg to the
right. Hence, the scale shows kg for the butter.
Scheme:
Finding the real weight of sugar or flour ………..1 point Finding the shift of the scale ………..1 point
A B D
Since the number of persons between A and B is the same as those between C and D, the number
of persons between A and B is: 34 6 2
2
- - =12 ………... 1 point
The number of persons between A and C is 12+7=19 ……… 1 point
10.A rectangle has two axes of symmetry, the vertical axis and the horizontal axis. If the rectangle is folded with respect to the vertical axis, we obtain a rectangle with perimeter 40 cm. If the rectangle is folded with respect to the horizontal axis, we obtain a rectangle with perimeter 50 cm. What is the original perimeter of the rectangle? km/hour and left the bicycle at a certain place. Then he walked 5 km/hour to reach B. On the other hand, Brad walked at 4 km/hour, then took Jake's bicycle, and rode at 10 km/hour to reach B. If they started to travel and arrived at B at the same time, how many minutes did Jake leave his bike before it was used by Brad?
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Answer:
Suppose Jake was cycling for x km, then walked for (9-x) km
Jake was cycling at 8 km/h and walking with speed of 5 km/h, so the time needed by Jake to get the
town B is 5
Brad walked at 4 km/h and cycling at 10 km/h, so the time needed Brad to get to the town B is
10
Because of Brad and Jake start traveling and arrive at B at the same time, then it must satisfies
5
This means that Jake has cycled for 4 km before leaving his bike and then walk for 5 km to reach B. Because he was cycling with a speed of 8 km/hour, so it take 4/8 = 1/2 hours. (½ point)
Because Brad walk at 4 km/hour, so he walked on foot for 4/4 = 1 hour before he was cycling to reach B ……… (½ point)
A
13.Tom has a contract to dig out some foundations and it must be done in 30 days. His own machine, which he wishes to use as much as possible, would take 50 days to do all the work. He can hire a bigger machine which would finish the job in 21 days. There is only enough room for one machine at a time. What is the least number of days for which he must hire the bigger machine?
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Performance of small machine is 1/50
Performance of big machine is 1/21 ………..…… 1 point
big machine 14 days 15 days
rest of work 1 – 14/21 = 1/3 1 - 15/21 = 2/7
small machine
needed 1/3 . 50 = 16,66 2/7. 50 = 14,3 ………… 1 point
total day needed 14+16,66 = 30,66 days 15+14,3 = 29,3 days
so Tom should hire a big machine for 15 days. ………..………… 1 point
