MATH OLYMPIAD QUESTIONS AND SOLUTIONS FOR CLASS 6

Question 1 :

Mrs. Angelina bought 44 cakes. the ratio of the number of butter cakes to the number of chocolate cakes was 6 : 5. After giving some chocolate cakes to her neighbours,  Mrs Angelina had twice as many butter cakes as chocolates cakes. How many chocolate cakes did she give away ?

(A)  7  (B)  8  (C)  9  (D)  11

Solution :

Number of butter cakes  =  6x

Number of chocolate cakes  =  5x

Total number of cakes  =  44

6x + 5x  =  44

11x  =  44

x  =  4

6x  =  24

5x  =  20

Let "a" be the number chocolate cake that she has given.

20 - a = Number of chocolate cakes

Number of butter cakes = 2 (Number of chocolate cakes)

24  =  2(20 - a)

24  =  40 - 2a

2a  =  40 - 24

2a  =  16  ==>  a  =  8

Hence the number of chocolate cakes she gave away is 8.

Question 2 :

In a factory X, the ratio  of male workers to female workers is 5 : 4. In a factory Y, the ratio of male workers to female workers is 2 : 7.

After 10 female workers leave Factory X to join Factory Y, the ratio of male workers to female workers in Factory Y becomes 4 : 15.

How many female workers are in Factory Y now ?

(A)  130  (B)  135  (C)  150  (D)  160

Solution :

Factory X :

Number of male workers  =  5x

Number of female workers  =  4x

Factory Y:

Number of male workers  =  2y

Number of female workers  =  7y

Number of female workers left from Factory X and joined to factory  Y  =  10

Number of female workers in Factory Y  =  7y + 10

2y : (7y+10)  =  4y : 15y

2y / (7y+10)  =  4y / 15y

2y / (7y+10)  =  4 / 15

15(2y)  =  4(7y + 10)

30y  =  28y + 40

30y - 28y  =  40

2y  =  40

y  =  20

Now, number of female workers  =  7y + 10 

=  7(20) + 10

=  150

Question 3 :

Alvin saves 40% as much as Theodor, Simon saves 30% more than Theodor. What is the ratio of Alvin's saving to Simon's to Theodor's savings ?

(A)  4 : 11 : 10  (B)  3 : 13 : 10  (C)  4 : 13 : 10  (D)  2 : 7 : 9

Solution :

Let Thedor saves 100% 

Then, Alvin's saving  =  40%

Simon saves  =  100 + 30  =  130%

=  40 : 130 : 100

=  4 : 13 : 10

Question 4 :

A Math clinic has all its employees working an equal number of hours each day. The total number of working hours for a day is 133. How many employees are there ?

(A)  7  (B)  8  (C)  11  (D)  19

Solution :

Let "x" be the number of workers and "y" be the number of working hours.

xy  =  133

x  =  133/y

If y = 7, then x  =  133/7  ==> 19

Hence there are 19 employees working 7 hours per day.

Question 5 :

(999, 999 x 222, 222) + (333, 333 x 333, 334)  =  ?

Solution :

  =  3 x 333, 333 x 222, 222 + 333, 333 x 333, 334 

  =  333, 333 x 666, 666 + 333, 333 x 333, 334 

  =  333, 333 (666, 666 + 333, 334)

  =  333, 333 (1000000)

  =  333333000000

Question 6 :

The difference in mass between two boys is 8 kg. If their total mass is 58 kg, what is the ratio of the mass of the lighter boy to the mass of the heavier boy ?

(A)  25 : 33  (B)  12 : 33  (C)  11 : 32  (D)  7 : 31

Solution :

Let "x" and "y" be the mass of lighter boy and heavier boy respectively.

y - x  =  8  ----(1)

y + x  =  58 ----(2)

(1) + (2)

2y  =  66

y  =  33

By applying the value of y in (1), we get

33 - x  =  8

x  =  33 - 8

x  =  25

The ratio of the mass of the lighter boy to the mass of the heavier boy :

  =  25 : 33

Question 7 :

Sales for a business were 3 million dollars more the second year than the first, and sales for the third year were double the sales for the second year. If sales for the third year were 38 million dollars, what were sales, in millions of dollars, for the first year?

(A)  16  (B)  17.5  (C)  20.5  (D)  22

Solution :

Note :

Each number represents million dollars.

Let "x" be the sales in first year.

x + 3  =  Sales in second year

Sales in third year  =  2(x + 3)

Actual sales in third year  =  38 million dollar

2(x + 3)  =  38

x + 3  =  19

x  =  19 - 3  ==> 16

Hence the answer is 16 million dollar.

Question 8 :

A typical high school student consumes 67.5 pounds of sugar per year. As part of a new nutrition plan, each member of a track team plans to lower the sugar he or she consumes by at least 20% for the coming year. Assuming each track member had consumed sugar at the level of a typical high school student and will adhere to this plan for the coming year, what is the maximum number of pounds of sugar to be consumed by each track team member in the coming year?

(A)  54 pounds   (B)  14 pounds 

(C)  41.6 pounds  (D)  13.5 pounds

Solution :

According the new nutrition plan, they need to lower the sugar at least by 20% what they already consumed

Let 67.5 pounds be the 100% of sugar.

  =  80% of 67.5 pounds

  =  0.80 (67.5)

  =  54 pounds.

Question 9 :

There are 160 white and blue balls, 180 blue and yellow balls. and 170 white and yellow balls. Find the number of each color balls.

Solution :

Let x, y and z be the number of white, blue and yellow balls respectively.

x + y  =  160  ---(1)

y + z  =  180 ---(2)

x + z  =  170 ---(3)

(1) - (2)

x - z  =  160 - 180

x - z  =  -20  -----(4)

(3) + (4)

2x  =  150

x  =  150/2  ==>  75

By applying the value of x in (1), we get

75 + y  =  160

y  =  160 - 75 ==>  85

By applying the value of x in (3), we get

75 + z  =  170

z  =  170 - 75

z  =  95

Hence the number of white, blue and yellow balls are 75, 85 and 95 respectively.

Question 10 :

Find the unit digit of the following 

185¹⁶³

Solution :

The unit digit of the given number is 5. Now let us consider the cyclicity of 5.

5 x 5  =  25 (unit digit is 5)

5 x 5 x 5  =  125 (unit digit is 5)

5 x 5 x 5 x 5  =  625 (unit digit is 5)

5 x 5 x 5 x 5 x 5  =  3125 (unit digit is 5)

By observing the above, we decide that the unit digit of the given number is 5.

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