Thursday, April 30, 2009

Singapore Math: NYPS Prelim 2007

These are questions from the Killer Math paper (from Nanyang Primary Prelim 2007) I talked about in my previous post. Well, at least it killed me, the following last 6 questions in the paper constituted 28 marks, and I got only 1 mark :(

Monica, post your working, I'm sure they're quite different from Brian's.
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Q48. A bus and a car travelled from Town X to Town Y. The bus left Town X at 1048pm and it took 5 hours to reach Town Y. The car started 30 min later than the bus and it took 4 hours to reach Town Y. At what time did the car catch up with the bus?

Let x be the distance from Town X to town Y.

Bus's Speed --> x/5h = 1/10 x/ 30min
Car's Speed --> x/4h = 1/8 x/ 30min

At 1118pm, Car travelled 0 x, Bus travelled 1/10 x
At 1148pm, Car travelled 1/8 x, Bus travelled 1/5 x
At 1218am, Car travelled 1/4 x, Bus travelled 3/10 x
At 1248am, Car travelled 3/8 x, Bus travelled 2/5 x
At 118am, Car travelled 1/2 x, Bus travelled 1/2 x

Car caught up with bus at 118am.

Ans: 1.18 am (answer given in testpaper is 1.18pm)

* I think there should be a better way to present the working than this, if anyone cares to give it a try, I'd very much appreciate it.

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Q47. At a school carnival, there were 520 more girls than boys. 1/8 of the girls and 20% of the boys left the carnival. In the end, there were 488 more girls than boys.
a) Did more girls or boys leave the carnival? How many more?
b) How many children were there at the carnival in the end?


a) 520-488 = 32

Ans: Girls, 32 more.

b) Let G and B be the number of Girls and Boys at first.

G = B + 520 -- (1)
7/8 G = 4/5 B + 488 --(2)

(1) - (2)

1/8 G = 1/5 B + 32 -- (3)

(1) divided by 8

1/8 G = 1/8 B + 65 -- (4)

(4) - (3)

3/40 B = 33
B = 440
G = 440 + 520 = 960

7/8 G + 4/5 B
= 7/8 (960) + 4/5 (440)
= 840 + 352
= 1192

Ans: 1192
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Q46 Azman had 25% more marbles than Chongfu. Chongfu had 60% more marbles than Bala. During a game, Azman and Bala lost some marbles to Chongfu in the ratio of 3:1. In the end, Azman and Bala had 780 and 480 marbles left respectively. How many marbles did Azman have at first?

Let A, B and C, be the number of marbles Azman, Bala and Chongfu had at first.

A = 5/4 C
4/5 A = C -- (1)

C = 8/5 B -- (2)

Substitute (1) into (2)

4/5 A = 8/5 B
A = 2B -- (3)

Let x be the number of marbles lost

A - 3x = 780 -- (4)
B - x = 480 -- (5)

Substitute (3) into (4)

2B - 3x = 780 -- (6)

(5)x2

2B - 2x = 960 -- (7)

(7)-(6)

x = 180

From (4)

A - 3(180) = 780
A = 780 + 540 = 1320

Ans: 1320

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Q45 A rectangular tank measuring 60cm by 35cm by 40cm is half-filled with water. If Tap A is turned on, it will take 6 min to fill the remaining half of the tank to its brim. Tap B drains water from the tank at a rate of 12 litres per min. How long will it take for the tank to be filled to 1/8 of its capacity if both taps are turned on at the same time?

Tap A fills up 1/2 of the tank in
--> ((60x35x40)/2)cm3/6 min
= (84000/2)cm3/6 min
= 42000cm3/6 min
= 7000cm3/min

The tank loses 12000 - 7000 --> 5000cm3 of water every minute.

Difference between 1/2 and 1/8 of the tank
--> 3/8 (84000)
= 10500 x 3
= 31500cm3

Time taken to fill the tank to 1/8 of capacity
--> 31500/5000
= 315/50
= 63/10
= 6.3 min

Ans: 6.3 min
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Q44 The tickets for a show are priced at $10 and $5. The number of ten-dollar tickets available is 1 and 1/2 times the number of five-dollar tickets. 5 out of 6 ten-dollar tickets and all the five-dollar tickets were sold. The ticket sales amounted to $5600. How much more would have been collected if all the tickets were sold?

Let T and F be the number of ten and five-dollar tickets available respectively.

T = 3/2 F -- (1)
50/6 T + 5F = 5600 -- (2)

(1)x2/3

2/3T = F -- (3)

(3)x5

10/3 T = 5F -- (4)

Substitute (4) into (2)

50/6 T + 10/3 T = 5600
70/6 T = 5600
1/6 T = 80
T = 480
F = 320

Amount of money collected if all tickets are sold
--> 10T + 5F
= 4800 + 1600
= 6400

Extra amount collected
--> 6400 - 5600 = 800

Ans: $800
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Q43 The shaded figure is made up of 4 quarter arcs of radius 10cm. Find its area (take pi=3.14).

There's actually a diagram of the kinda cut-out circle; if anyone's really interested, just go to this link to the testpaper, it's Q43.

Area of circle
--> 3.14 x 10 x 10
= 314

Side of square
--> Sqrt(10x10 + 10x10)
= Sqrt200

Area of square
--> Sqrt200 x Sqrt200
= 200

Area of 4 arcs
--> 314 - 200
= 114

Area of 1 arc
--> 114/4
= 28.5

Area of shaded part
--> 200 - 2(28.5)
= 200 - 57
= 143

Ans: 143cm2

*Not sure if that cut-out part of the circle is called an arc though, is it an arc or sector?

13 comments:

monlim said...

Of the questions you posted, as mentioned, I couldn't do the speed one, will ask L-A if she knows. Got Q46 and 43 wrong. I haven't examined my answer to find out why I got Q46 wrong, I tried to use ratio. For Q43, I just didn't understand the point about the arcs, like what portion of an arc is a circle? I think I get it now on hindsight though...

Lilian said...

I don't get any of them so can't help you with the arc bit.

I remember getting No47 right the first time I did this, but this time got stuck, dunno why. Brain is rusty.

Whoa, No45 looks confusing to me and you did it, I told you when I see anything about speed and rate, mind switches off! Now you know how bad I am at this.

Anonymous said...

The speed question -

When they meet, it's the same distance travelled. D = ST. Note T for car is 1/2 hour less.


x/5 * (y + 1/2) = x/4 * y
giving you y = 2 hours (11:18 + 2 = 1:18)

There!


There!

Lilian said...

Whoaaaa Anonymous, brilliant stuff! Thanks! Do we know you?

monlim said...

that answer is brilliant but you know, I think simplifying that equation to get 2 is not that easy for many kids. Tried it with L-A, she couldn't do it...

Lilian said...

That's cos L-A hasn't had much exposure to algebra. It's all about practice, I guess, just like using models. The hard part isn't really the simplifying, but coming up with the equation...cheeem.

monlim said...

PS spotted error, your last answer shd be 143 not 142 lah... and hooray, L-A got this sum right!!

Lilian said...

Hurray for L-A, you trumped your mummy! With more practice, I'm sure you'll be able to do the other questions in no time at all. Keep up the good work!

ps, will correct the error now.

Anonymous said...

Q46 - you can use models

1st, draw 5 boxes for A
Then draw 4 boxes for C (A got 25% more than C)
Then divide C's boxes into 8
Cause B will require 5 boxes (C got 60% more than B)

So that will give you 10 units for A, 8 for C and 5 for B

780 + 3 blocks = 10 units
480 + 1 block = 5 units

(block refers to marbles lost)

780 + 3 blocks = 10 units
960 + 2 blocks = 10 units

Thus 1 block of marbles lost = 180

Thus Azman will have 780 + 3 x 180 = 1320

There!

Lilian said...

Hi there There! I'll call you There! from now on :)

Thanks so much There! Me and some mummies think you're just da bomb, please keep the solutions coming.

See you around here again There!

Anonymous said...

You may want to ask ml to draw those boxes and post it up in case it's not so clear to some mummies.

i left out one step and that is to also divide A's boxes into the same unit (as those when you divide C's boxes into 8), thus from 5 boxes it becomes 10 smaller ones.

Lilian said...

Okay Monica, get to work!

Anonymous said...

Hi
Qn 48 (non algebra method -- analyzing method)

Understand that:
1) Bus took 5 hours to complete journey and Car took 4 hours to complete journey.
2) Car is faster than Bus.
3) They travelled the same distance.


Working:
Since they travelled the same distance, we shall find the LCM of 4 and 5 = 20.
So the dist betwn the two towns are 20 parts.
Bus took 5 hours to complete (implying that Bus covered 4 parts in each hour).
Car took 4 hours to complete (implying that Car covered 5 parts in each hour).
note: Your understanding of the Car is faster than the Bus is proven in your last step shown above (Car travelled 1 more part than the Bus in an hour).

Bus is given a headstart of 30 mins.
Within that 30 mins, how much head start dist (aka how many parts) does the Bus gain?
1hour = 4 parts (for Bus)
30 mins = 2 parts (for Bus)

So....The Car NEEDS to catch up this 2 parts that he lost.

Recall that the Car travels 1 more part than the Bus within each hour. How long will the Car take to catch up those 2 parts?
1 hour = Car catches up by 1 part

To catch up 2 parts, the car needs 2 HOURS. ... which is 1.18am (10.48pm + 30mins + 2hr)


Sorry for the lengthy post....


Hope this is serves as an alternative way of explaning rather than using algebra...


Jan