P5 ISMC 2018 - W Answers PDF

 

INTERNATIONAL SINGAPORE MATHS COMPETITION

INTERNATIONAL SINGAPORE MATHS COMPETITION 2018 (Primary 5) 1 hour 30 minutes

Instructions to participants 1. Do not open the book booklet let until you are told to do so. 2. Attempt ALL 25 questions. 3. Write y your our an answers swers neatly in the Answer Sheet prov provided. ided. 4. Marks are awarded for correct answers only. 5.  All figures are not drawn to scale.  6. Scientific calculators may be used. Questions in Section A carry 2 marks each, questions in Section B carry 4 marks each and questions in Section C carry between 6 to 10 marks each.

Jointly organised by

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INTERNATIONAL SINGAPORE MATHS COMPETITION Section A:

Each of the questions 1 to 10 carries 2 marks. 1.

How many prime numbers less than 100 can be formed using only the digits 9, 3, 1 and 7 without repetition? There are 25 prime numbers between 1 and 100. Only 3, 7, 13, 17, 19, 31, 37, 71, 73, 79 and 97 can be formed. Hence, only 11 prime numbers can be formed.

2.

The product of of two numbers is 72 and the product of another two numbers is 128. Find the sum of the greatest possible sum of this pair of factors for 128 and the smallest possible sum of this pair of factors for 72. The possible pairs for 72 are 1 × 72, 2 × 36, 3 × 24, 4 × 18, 6 × 12 and 8 × 9. The possible pairs for 128 are 1 × 128, 2 × 74, 4 × 32 and 8 × 16. (1+ 128) + (8 + 9) = 146 The sum is 146.

3.

A bag of carrots weighs 10 kg 25 g, while a bag of onions weighs 2 kg 75 g lighter than the bag of carrots. What much does the bag of onions weigh in kg? 10.025  – 2.075 = 7.950 The bag of onions weighs 7.95 kg

4.

Fill in the blank: 50% of 250 is equal to ______% of 25. 50% of 250 = 125 

  × 100% = 500%



5.









If Benson gives Alison  of his pocket money, Benson will have  as much as Alison. What is the ratio of Alison’s pocket money to Benson’s pocket money?  

If Benson gives Alison  of his pocket money, he would have 2 units and Alison would have 

16 units. Therefore, Benson actually has 3 units andmoney Alisonisactually Ratio of Alison ’s pocket money to Benson’s pocket 5 : 1 has 15 units. 2

 

INTERNATIONAL SINGAPORE MATHS COMPETITION

6.

The length of a piece of pink ribbon was 5.7 m. The length of a piece of blue ribbon was 4.1 m. After the same length was cut off from both ribbons, the length of the pink ribbon was three times that of the blue ribbon. What was the length of the pink ribbon that was cut off? 5.7 – 4.1 = 1.6 2 units → 1.6 m →

1 units unit (pink)  0.8 m→ 2.4 m 3 5.7 – 2.4 = 3.3 m was cut off.

7.

Several pupils were told to stand in position such that they form the shape of a hexagon, with 4 pupils along each of the sides. How many pupils were there altogether?  A single pupil is required to stand at each of the vertices of the hexagon. h exagon. Since every side of the hexagon has 4 pupils, there will be 2 pupils on every side between the two pupils standing at the vertices. 6 + 2 × 6 = 18 There are 18 pupils in total.

8.

The graph below shows the age and the hours of sleep of 6 children A, B, C, D, E and F. Study it carefully and name the child who is younger than C and gets less sleep than C.  A C Hours Hour s of sleep slee p

F

B E

D  Age

 Answer: Child B B   9.

Sam and Max are mischievous boys who live next door. Sam always lies lies only on Monday, Tuesday and Wednesday. Max always lies only on Thursday, Friday and Saturday. Just this morning, I met both of them. t hem. Sam said, “Yesterday I was lying.” And Max said, “So was I.” W hich day of the week is today? Since none of them lie on the same day, one of them is lying today. Because both the Sam and Max speak the truth on Sunday, it cannot be Sunday. If today is Thursday, then Sam can truthfully say that he was lying on Wednesday. Because Max lies on Thursdays but tells the truth on Wednesdays, we know Max is lying about lying on Wednesday. Therefore, today is Thursday. 3

 

INTERNATIONAL SINGAPORE MATHS COMPETITION

10. How many 5-digit numbers can be formed with with the digits 0, 1, 2 and 3 that are multiples of 36, if you are allowed to repeat one of the digits?  A number which is a multiple of 36 is also a multiple of 4 and 9. The digit sum of the 5-digit number that is a multiple of 9 has to be equal to 9, that is, the digits are 0, 1, 2, 3 and 3. The last 2-digits of the 5-digit number has to be 12, 20 or 32 for it to be a multiple of 4. Hence, the9possibilities are 30312, 33012, 13320, 31320, 33120, 10332, 13032, 30132, 31032. There are of them altogether.

Section B

Each of the questions 11 to 20 carries 4 marks. 11. The ratio of the sides of Square A to Square B is 1 : 2. The ratio of the sides of Square A to Square C is 1 : 4. What is the ratio of the area of Square B to the area of Square C?  Area of A : Area of B : Area of C 1 : 4 : 16  Area of B : Area of C 1 : 4

12. The table below shows the number of days it took di different fferent number of workmen to paint some houses. Given that all the workmen paint at the same rate, how many days would it take 8 workmen to paint 12 houses? (Leave your answer as a mixed number.)

Workmen 9

Houses 9

Days 9

2

4

18

If it takes 9 days for 9 men to paint 9 houses, then it takes 9 days for 1 man to paint 1 house. Similarly, it will take 9 days for 12 men to paint 12 houses. If the number of men is reduced to 8 men, it would take longer than 12 days to finish the work. Hence,

 



  × 9 = 13  days. 

13. How many different triangles exist where the longest side is 12 cm and the lengths of the remaining sides are different whole numbers? Since side of the triangle cm butthe addlongest up to be more than 12 cm.is 12 cm, the two other sides must each be less than 12 4

 

INTERNATIONAL SINGAPORE MATHS COMPETITION

The other 2 sides could be: 11 and 2; 2; 11 and 3; 3; 11 and 4; 11 and 5; 11 and 6; 11 and 7, 11 and 8; 11 and 9; 11 and 10, 10 and 3; 3; 10 and 4; 10 and 5; 10 and 6; 6; 10 and 7; 10 and 8; 10 and 9; 9 and 4; 4; 9 and 5; 9 and 6; 9 and 7; 9 and 8; 8; 8 and 5; 8 and 6; 8 and 7; 7 and 6. There are 25 such triangles.

14. A, B and C are non-zero whole numbers. If A × A = B and C × C × C = B, and B is less than 100, what is the value of B? The only number less than 100 that is a square number as well as a cube number is 64.

4+

15. Given that − 



=

  − 

  and

− 

= 8 , what is the value of

 

 in its lowest terms?

 = 8 

 = 15   4+ 

=

 − 

 

 = 12    

 =

 4 

 

16. Robert had 105 kg of apples and oranges. 20% of the fruits were apples. He bought some more apples and the percentage of apples increased to 30%. How many kilograms of apples did he buy? 105 ÷ 5 = 21 kg of apples at first Let x be the weight of the new apples Robert bought, (21 + x) ÷ (105 + x) = 0.3 x = 15 Robert bought 15 kg of apples.

17. A boy in a class noted that the ratio of the number o off boys (excluding himself) to the number of girls is 3 : 2. A girl from the same class noted that the ratio of the number of girls (excluding herself) to the number of boys is 3 : 5. How many pupils are there in the class altogether? For ratio 3 : 2, total number of units = 5 5

 

INTERNATIONAL SINGAPORE MATHS COMPETITION

For ratio 3 : 5, total number of units = 8 Since the ratios represent the same number of pupils in class (less 1 person), the common multiple is 40. Therefore, the class has 40 + 1 = 41 pupils

18. The figure below shows my watch at the time I met my friend. Reading the time on my w watch, atch, I 

had thought I was  hour late but my watch turned out to be 5 minutes fast. If my friend had 6

arrived 12 minutes before the time we were supposed to meet, how long would my friend have waited for me? 11 12  1   Actual time I arrived: 9:25 2  10  Meeting time: 9:20 9  3  12 min before 9:20 9:08 8  4  7  6  5  From 9:08 to 9:25 is 17 min

19. I have a 1¢, a 2¢, a 5¢ and a 10¢ coin. Different amounts of money can be formed from different numbers of these coins. The least amount is 1 ¢ and the largest amount is 18 ¢. List the amounts  less than 18¢ that cannot be formed by any number of these coins.  Ans: 4¢, 9¢ and 14¢ 

20. Merrill has 30 more stamps than Barry. If Barry gives 30 stamps to Merrill, Merrill will have twice as many stamps as Barry. What percentage per centage of his stamps must Barry give g ive to Merrill such that Merrill has 4 times as many stamps as Barry? Let x be the number of stamps that Barry has, 2(x  30) = x + 30 + 30 x = – 120 Let y be the number of stamps that Barry has to give Merrill such that Merrill has 4 times as many stamps as Barry, 4(x – y) = x +30 + y 5y = 330 y = 66 (66 ÷ 120) × 100% = 55% Barry has to give Merrill 55% of his stamps.

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INTERNATIONAL SINGAPORE MATHS COMPETITION Section C

Questions 21, 22, 23, 24 and 25 carry 6, 7, 8, 9 and 10 marks respectively. 21. What is the largest largest number less than 10 000 that has only 3 factors?  A number that has only 3 factors is the square of a prime. pr ime. Since 10 000 = 100 × 100, the largest prime number less than 100 is 97, then the largest square of prime less than 10 000 is 97 × 97 = 9409.

22. Ted and Esther had money in in the ratio of 5 : 3. After Ted earned $20 and Esther spent $6, the ratio became 8 : 3. How much money did each of them have at first? 3 units – $6 = 3 new units 1 unit – $2 = 1 new unit 8 new units = 8 units  – $16 But 8 new units = 5 units + $20 Therefore, 8 units – $16 = 5 units + $20

     9    Q

3 $36 (Esther) 1 units unit ==$12 5 units = $60 (Ted)

     8    Q

Ted had $60 and Esther had $36.

     7    Q  

23.

A group of 10 pupils was given a quiz   6 that has 10 questions. The graph below shows the    Q number of students who answered each question correctly. If the passing standard is to   answer 6 out of 10 questions correctly, what is the greatest possible number of pupils who    5    Q passed?  

10 9    h   o   s   w  r   s   e    t   n   w   e   s    d   n   a   u   t    t   s   c    f   e   r   o   r   r   o   e   c    b   t   m  o   u   g    N

8 7 6 5 4 3 2 1  

   4    Q      3    Q      2    Q      1    Q

 

 

 

 

 

 

 

 

   8   7   6   5   4   3   2   1 Q1

Q2

Q3

Q4

Q5

Q6

Question Number

7

Q7

Q8

Q9

Q1 0

 

INTERNATIONAL SINGAPORE MATHS COMPETITION

There is a total of 42 wrong answers. If 10 different pupils got 4 questions wrong, that will only account for 40 wrong answers. Since there are 42 wrong answers, then 2 of the pupils must have given 5 wrong answers. Therefore, the greatest possible number of pupils who passed is 10  – 2 = 8 pupils.

24. A florist is selling some pink pink and red roses on Valentine’s Day. After selling an equal number of pink and red roses, the florist has 0.75 of the pink roses and 0.6 of the red roses left. What fraction of the roses left were pink roses? Pink roses sold: Red roses sold:





4

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 =  



 

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Number of units of pink roses left = 6 units Number of units of red roses left = 3 units 6

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



Fraction of the roses left that are pink =  =  

25. How many different ways are there to rearrange the letters of the following following words? a)

POLE 4! = 4 × 3 × 2 × 1 = 24

b)

PEEL 4! ÷ 2! = (4 × 3 × 2 × 1) ÷ (2 × 1) = 12

c)

PEEP 4! ÷ (2! × 2!) = (4 × 3 × 2 × 1) ÷ 4 = 6

d)

PEOPLE 5! ÷ (2! × 2!) = (5 × 4 × 3 × 2 × 1) ÷ 4 = 30

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INTERNATIONAL SINGAPORE MATHS COMPETITION

End of Paper

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