1.A:

Assuming 6 legs for each insect, 16 insects will have 96 legs. But, there are 104 - 96 = 8 extra legs.

It is important to note that spiders have 8 legs and flies have 6 legs. So, there are 2 extra legs per spider on the web. Thus, the 8 extra legs belong to 4 spiders.

2.A:

The number is 36, simply because 36 = 4 (3 + 6).

How does one find this number?

Let t be the digit in the tens place and u be the digit in the units place. Then, the number is 10 t + u, and the sum of its digits is t + u. The following equation can be readily written:

10 t + u = 4 t + 4 u or 6 t = 3 u.

Thus, t / u = 3 / 6. Since t and u are digits, t must be 3 and u must be 6.

3.A:

The matches in the championship may be visualized (starting backwards from the finals) as follows:

Match Total Matches Total Players Finals 1 2 Semi Finals 1 + 2 = 3 4 Quarter Finals 1 + 2 + 4 = 7 8 Pre-Quarter Finals 1 + 2 + 4 + 8 = 15 16

Note that half the players lose their matches at each stage and are out of the tournament.

More importantly, the above table shows that the total number of matches played is always one less than the number of players participating in the tournament. This is basically because all players except the champion have to lose one match.

One can now easily answer the two questions posed.

(a) If 127 matches are totally played, then 128 players participate in the tournament.

(b) If 123 players participate in the tournament, then 122 matches are totally played.

4.A:

The merchant has 57 gold coins. It is easy to check this... Let's divide the 57 coins into two unequal numbers, say, 49 and 8.

Then, 57 (49 - 8) = (49 x 49) - (8 x 8).

5.A:

"Let x be my present age. My age 2 years from now will be x + 2, and 2 times that will be 2 (x + 2). Similarly, my age 3 years from now will be x + 3, and 3 times that will be 3 (x + 3).

Adding the two gives 6 times my current age. This gives the equation:

2 (x + 2) + 3 (x + 3) = 6 x.

Solving the above equation gives x = (2 x 2) + (3 x 3) = 13.

Thus, I will be 15 years old 2 years from now."

6.A:

It it important to note that Average speed = Total distance / Total time.

Total distance = 2 x 70 miles.

Time for uphill journey (from Oakland to Pinewood) = 70 / 28 hours.

Time for downhill journey (from Pinewood to Oakland) = 70 / 70 hours.

Total time = (70 / 28) + (70 / 70) = 2 x 70 / 40 hours.

Average speed = Total distance / Total time = 40 miles per hour.

7.A:

It it important to note that Distance = Speed x Time.

(a) The snake has entered the hole when its tail just comes into the hole. It is given that 13 seconds elapse from the time the snake's head enters the hole till its tail enters. Thus, Length of the snake = Distance traveled in 13 seconds = (5 cm/s) x (13 s) = 65 centimetres.

(b) The snake has exited the hole when its tail just leaves the hole. The distance traveled from the time the snake's tail just enters the hole till its tail just leaves the hole equals the length of the hole.

Thus, Time for the snake to exit the hole after entering it = Length of the hole / Speed = = (5.15 m) / (5 cm/s) = (515 cm) / (5 cm/s) = 103 seconds.

8.A:

Grandpa:

"This problem is conveniently solved by writing down the necessary equations.

Let m be my age in years. If s is my son's age in years, then my son is 52s weeks old. If g is my grandson's age in years, then my grandson is 365g days old. Thus, 365g = 52s. Since my grandson is 12g months old, 12g = m. Since my grandson, my son and I together are 100 years, g + s + m = 100. The above system of 3 equations in 3 unknowns (g, s and m) can be solved as follows. m / 12 + 365 m / (52 x 12) + m = 100 or 52 m + 365 m + 624 m = 624 x 100 or m = 624 x 100 / 1041 = 60. So, I am 60 years old."

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