ARYABHATTA INTER-SCHOOL MATHS COMPETITION 2010

CLASS V

PART I: ARITHMETIC

 

Q2. There are __ two-digit numbers where the tens place digits is greater than the unit’s digit.

Solution. Let’s consider a number with 9 as tens digit. Then the number on its unit’s digits can be 0, 1, 2, 3 …, 8. A total of 9 numbers

Similarly, consider a number with 8 as tens digit. Then the number on its unit’s digit can be 0, 1, 2, …, 7. A total of 8 numbers

Again, consider a number with 7 as tens digit. Then the number on its unit’s digit can be 0, 1, 2, …, 6. A total of 7 numbers

Now, proceeding in the same manner till a number with 1 as its tens digit.

Total such numbers = 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45

Q3. If 10% of a number is 100, then 100% of the same number is _____.

Solution. 10% of a number is 100

To convert 10% to 100%, we have to multiply it by 10. Since, we are multiplying 10% of a number by 10, the value of the number will also get increased by 10 times (multiply by 10).

10% of a number is 10

100% of a number is 100 × 10 = 1000 

Q5. The product of three whole numbers is 60 and their sum is 13 and the numbers are all different and greater than 1. The numbers are

Solution. Prime factorization of 60 = 1 × 2 × 2 × 3 × 5

Now, we have to find three whole numbers from these prime factors. The following pair of three whole numbers can be formed.

(2, 5, 6), (3, 4, 5), (2, 3, 10)

Now, in each pair the product of the numbers is 60. But in only one pair the sum of the numbers is 13 i.e. (2, 5, 6).

Q8. The greatest of three consecutive numbers whose sum is 24 is

Solution. Let’s consider the pairs given below

Sum of 1, 2, 3 = 6

Sum of 2, 3, 4 = 9

Sum of 3, 4, 5 = 12

Sum of 4, 5, 6 = 15

Sum of 5, 6, 7 = 18

Sum of 6, 7, 8 = 21

Sum of 7, 8, 9 = 24

Thus, the greatest of the three numbers is 9.

Q9. 0.127 is greater than 1/8 by (answer in fraction in lowest term)

Solution. 1/8 = 0.125

Difference = 0.127 – 0.125 = 0.002 or 2/1000 or 1/500

Q10. Priya got either 90 or 100 in each of her five math tests. The average of all her math tests is 98. The number of times she got 90 is

Solution. Since, her average marks scored are given, we can easily find out the total marks scored by her in five tests.

Total marks scored = 5 × 98 = 490

But she got either 90 or 100 in each of her test. The distribution of her marks can be 100, 100, 100, 100, 90. hence, she scored 90 only once in the given tests.

Q11. A ribbon is twice as long as a cane 2 m 7 cm long. The cane is 3 times longer than the stick. The total length of the ribbon, cane, and the stick is

Solution. Length of cane = 2 m 7 cm = 200 cm + 7 cm = 207 cm or 2.07 m

Length of ribbon = 2 × 207 = 414 cm

Length of stick = Length of cane ÷ 3 = 207 ÷ 3 = 169 cm

Total length of the ribbon, stick and cane = 207 + 414 + 169 = 790 cm

Q12. If a ∆ b means a + a – b, then the value of (7 ∆ 3) ∆ 2 is

Solution. First solve the term given in bracket

(7 D 3) = 7 + 7 – 3 = 11

Now, the given expression becomes

(7 D 3) D 2 = 11 D 2

11 D 2 = 11 + 11 – 2 = 20

Q14. Dhruv has less than 30 marbles. When he puts them in piles of 3 he has no marbles left over. When puts them in piles of 2 he has 1 left. When he puts them in piles of 5 he has 1 left. He has ____ marbles.

Solution. After putting the marbles in piles of 3 he has no marble left over. This statement implies that the number of marbles is in the multiple of 3.

Multiples of 3 less than 30 = 3, 6, 9, 12, 15, 18, 21, 24, 27

When he puts them in piles of 2 he has 1 left. This statement excludes 6, 12, 18, 24 from the above set of numbers.

Remaining numbers = 3, 6, 9, 15, 21, 27

When he puts them in plies of 5 he has 1 left. This statement left us with only one number from the remaining set of numbers i.e. 21.

Q15. Rohan gave half of his stamps to Nikita. Nikita gave half of her stamps to Anu.  Anu gave 1/4^th of the stamps given to her to Vikram and kept the remaining 12. Rohan had ___ stamps.

Solution. Anu gave 1/4^th of the stamps to Vikram and kept remaining 12. This statement implies that she kept 3/4^th of the stamps with her after giving 1/4^th to the Vikram.

Þ 3/4^th of the stamps is 12

Þ 1 complete part of the stamps = 12 × 4/3 = 16

Þ Number of stamps with Anu = 16

Nikita gave half of her stamps to Anu i.e. if Anu has 16 stamps, then Nikita must have 32 stamps.

Similarly, Rohan must have 64 stamps. 

Q17. The average of seven numbers is 49. If 1 is added to the first number, 2 is added to the second number, 3 is added to the third number and so on upto seventh number, the new average will be

Solution. As the average of the numbers is given, we can easily calculate the sum of the numbers.

Total sum of the numbers = 49 × 7 = 343

As per the given conditions,

If 1 is added to the first number, 2 is added to the second number, 3 is added to the third number and so on upto seventh number. This statement implies that 1, 2, 3, …, 7 are also added to the sum of the numbers.

New total sum of the numbers = 343 + (1 + 2 + 3 + … + 7) = 343 + 28 = 371

Now average of the numbers = 371 ÷ 7 = 53

Thus, the new average will be 53. 

Q19. Arrange in descending order.

7/8       2.66     262%  25/8

Solution. 7/8 = 0.875

262% = 262/100 = 2.62

25/8 = 3.125

This question can also be solved without calculating the decimal values.

7/8 is less than 1            262% means 2.62          25/8 is greater than 3

7/8 < 262% < 2.66 < 25/8

Q20. Akhilesh takes 20 min to jog around the race course one time and 25 min to jog around a second time. If the course is 3 km long, his average speed in km/hr for the whole jog is

Solution. His speed for first time = Distance ÷ Time = 3 km ÷ 20 min = 9 km/hr (change ‘min’ in ‘hour’)

His speed for second time = 3 km ÷ 25 min = 7.2 km/hr

Now, it is easy to calculate the average for the whole jog.

Q21. There are 300 fishes in the tank. Their number is increased by 20%. Then 20% of the fishes die. The number of fishes in the tank now is

Solution. 20% is 20/100 or 1/5

Number of fish increased = 300 × 1/5 = 60

New total number of fish = 300 + 60 = 360

Similarly, number of fish decreased = 360 × 1/5 = 72

New total number of fish = 360 – 72 = 288

Q22. In Class 5 of a school, 40% of the students like music, 3/15 like dance, 1/10 like swimming, 5% like Cricket and not painting. The remaining 40 students like painting. The number of students who like to dance is

Solution. At first, we should convert all of the given values to fraction.

Fraction of students who like music = 40% or 40/100 or 2/5

Fraction of students who like dance = 3/15 or 1/5

Fraction of students who like swimming = 1/10

Fraction of students who like cricket = 5% or 5/100 or 1/20

Remaining fraction of students like painting

Remaining fraction of students = 1 – (2/5 + 1/5 + 1/10 + 1/20) = 1 – ¾ = ¼

Now, remaining 40 students i.e. ¼ of total students like painting

Þ Total students = 4 × 40 = 160

Number of students who like to dance = 1/5 × 160 = 32

Q23. Write an 8-digit number divisible by 8, 9 and 11 without repeating any digit

Solution. I am not providing solution to this question as from my perspective it is too much to ask from a Class V student. 

Q24. Look at the time table and answer the following questions.

			Train 1		Train 2		Train 3
Gremlin Square		a	19:45		11:30		6:05
		d	20:27		13:25		7:32
Fairy Town		a	21:32		14:15		9:30
		d	21:47		14:25		9:35
Angles Port		a	22:10		15:17		10:10
		d	22:25		15:49		10:25
Demons Coast		a	1:15		16:10		11:15

a.       Which is the fastest train between Gremlin Square and Demons Coast?

Solution. To answer this question we should consider the least time taken by a train after departing from Gremlin square and arriving Demons coast.

Time taken by Train 1 from 20:27 to 1:15 = 4:48

Time taken by Train 2 from 13:25 to 16:10 = 2:45

Time taken by Train 3 from 7:32 to 11:15 = 3:43

Thus, Train 2 takes least time or is fastest train to between Gremlin square and Demons coast.

b.      Which train takes the least time to reach Angles Port from Gremlin Square?

Solution. Time taken by Train 1 from 20:27 to 22:10 = 1:43

Time taken by Train 2 from 13:25 to 15:17 = 1:32

Time taken by Train 3 from 7:32 to 10:10 = 2:38

Thus, Train 2 takes the least time between Angles port from Gremlin square.

c.       Which is the fastest train from Fairy Town to Demons Coast?

Solution. Time taken by Train 1 from 21:47 to 1:15 = 3:28

Time taken by Train 2 from 14:25 to 16:10 = 3:45

Time taken by Train 3 from 9:35 to 11:15 = 1:40

Thus, Train 3 is the fastest train between Fairy town to Demons coast.

d.      Which is the slowest train between Fairy Town and Angles Port?

Solution. Time taken by Train1 from 21:47 to 22:10 = 00:23

Time taken by Train 2 from 14:25 to 15:17 = 00:52

Time taken by Train 3 from 9:35 to 10:10 = 00:35

Thus, Train 2 takes the maximum time or is slowest train between Fairy town and Angles port.

 

PART 2: GEOMETRY

 

Q7. The number of degrees that the hour hand of a clock moves through between noon and 02:30 pm of the same day is

Solution. The number of degrees hour hand moves between noon and 2 = 30° + 30° = 60°

The number of degrees hour hand moves between 2 and 2:30 = 15°

Total number of degrees moved = 60° + 15° = 75°