Problems and Solutions

Jai Ganesh · 2005-09-01 23:36:15

Problem # k + 12

In a 4-digit number, the sum of the first two digits is equal to that
of the last two digits. The sum of the first and last digits is equal to the third digit. Finally, the sum of the second and fourth digits is twice the sum of the other two digits. What is the third digit of the number?

Jai Ganesh · 2005-09-02 15:50:24

Solution to problem # k + 9

The side of the square is 2.
Let a length 'c' be cut from the two ends of a side.
We have right angled triangles of sides 'c'.
The hypotenuse would be h² = 2c²
or h = (√2)c
Since it is a regular octagon.
2 - 2c = (√2)c
c = 2/(2+√2)
h is the side of the octagon,
h = 2 (√2)/(2+√2)
or h = 2/(1+√2)

Jai Ganesh · 2005-09-02 16:22:16

Problem # k + 13

Two squares are chosen at random on a chessboard. What is the probability that they have a side in common?

John E. Franklin · 2005-09-02 17:24:18

I guess 1/18 chance if they can't both be the same chess square.

Jai Ganesh · 2005-09-02 17:42:49

You got it right! Well done! Try the earlier unsolved problmes.

The number of ways of choosing the first square is 64. The number of ways of choosing the second square is 63. There are a total of 64 * 63 = 4032 ways of choosing two squares.

If the first square happens to be any of the four corner ones, the second square can be chosen in 2 ways. If the first square happens to be any of the 24 squares on the side of the chess board, the second square can be chosen in 3 ways. If the first square happens to be any of the 36 remaining squares, the second square can be chosen in 4 ways. Hence the desired number of combinations = (4 * 2) + (24 * 3) + (36 * 4) = 224. Therefore, the required probability = 224/4032 = 1/18

Jai Ganesh · 2005-09-03 19:02:29

Problem # k + 14

What is the area of the largest triangle that can be fitted into a rectangle of length 'l' units and width 'w' units?

John E. Franklin · 2005-09-04 14:05:36

Last edited by John E. Franklin (2005-09-04 14:05:54)

John E. Franklin · 2005-09-04 14:40:58

Last edited by John E. Franklin (2005-09-04 14:42:33)

John E. Franklin · 2005-09-04 14:47:29

John E. Franklin · 2005-09-04 15:08:40

John E. Franklin · 2005-09-04 15:34:44

Jai Ganesh · 2005-09-04 16:00:55

Good work, John E. Franklin

Solution to Problem # k + 14

Any triangle you try to draw with the maximum area would have same base and height. The solution is, without doubt, lw/2. You can try all possibilities. You'd get the same answer, both when the base is 'l' and 'w'.

Solution to Problem # k + 12
Yes, you would be one equation short. But, when you get one number is eight times the other, the numbers would have to be 1 and 8, as each number a,b,c, and d is a single digit number!

Solution to Problem # k + 10
Yes, 5121 is the correct answer. As you said, there may not be another as 0242 is not acceptable!

Solution to Problem # k + 11
You have given a different property! I had this is my mind.
It is a number of the form abcd equal to (a^b)*(c^d)

Solution to Problem # k + 6
You are correct. The ages of the grandmother and the grandson would be (61,1), (62,2), (63,3), (64,4), (65,5), and (66,6).

Jai Ganesh · 2005-09-04 16:14:07

Problem # k + 15

A sample of x litres from a container having a 60 litre mixture of milk and water containing milk and water in the ratio of 2 : 3 is replaced with pure milk so that the container will have milk and water in equal proportions. What is the value of x?

John E. Franklin · 2005-09-04 16:46:15

John E. Franklin · 2005-09-04 17:10:54

Thanks ganesh. Yeah, I see now on #k + 12 that you can solve that
"b" is twice "d", and "d" is 4 times bigger than "a".
That's really something.

Jai Ganesh · 2005-09-04 18:26:46

Solution to Problem # k + 6

You are right, John. I don't know the reason. Just as MathsIsFun thought, I too believed 97531 x 86420 would be the highest product!

Problem # k + 16

The pages in a book are serially numbered from 1. If the number of digits required to total all the pages in the book is 972, how many pages are there in the book?

John E. Franklin · 2005-09-05 03:21:41

Jai Ganesh · 2005-09-05 16:05:39

John, you went wrong somewhere; please try again

Jai Ganesh · 2005-09-05 19:27:38

Problem # k + 17

A rectangular pool 20 meters wide and 60 meters long is surrounded
by a walkway of uniform width. If the total area of the walkway is 516 square meters, how wide, in meters, is the walkway?

Jai Ganesh · 2005-09-05 21:35:18

Problem # k + 18

Persons x and y have the following conversation:
x: I forgot how old your three kids are.
y: The product of their ages is 36.
x: I still don't know their ages.
y: The sum of their ages is the same as your house number.
x: I still don't know their ages.
y: The oldest one has red hair.
x: Now I know their ages!

How old are they?

Jai Ganesh · 2005-09-06 17:49:17

Jai Ganesh · 2005-09-08 16:58:33

Problem # k + 19

At his usual rowing rate, Rahul can travel 12 miles downstream
in a certain river in six hours less than it takes him to travel the same
distance upstream. But if he could double his usual rowing rate for his 24
mile round trip, the downstream 12 miles would then take only one hour
less than the upstream 12 miles. What is the speed of the current in miles
per hour?

Jai Ganesh · 2005-09-08 21:22:32

Problem # k + 20

What is the least number that should be multiplied to 100! to make it perfectly divisible by 3^50?

Jai Ganesh · 2005-09-09 18:03:26

Problem # k + 21

If the diagonal and the area of a rectangle are 25 m and 168 m², what is the length of the rectangle?

Jai Ganesh · 2005-09-11 00:16:57

Problem # k + 22

Grandpa: "My grandson is about as many days as my son is weeks, and my grandson is as many months as I am in years. My grandson, my son and I together are 160 years. Can you tell me my age in years?"

Math Is Fun Forum

#126 2005-09-01 23:36:15

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#127 2005-09-02 15:50:24

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#128 2005-09-02 16:22:16

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#129 2005-09-02 17:24:18

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#130 2005-09-02 17:42:49

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#131 2005-09-03 19:02:29

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#132 2005-09-04 14:05:36

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#133 2005-09-04 14:40:58

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#134 2005-09-04 14:47:29

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#135 2005-09-04 15:08:40

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#136 2005-09-04 15:34:44

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#137 2005-09-04 16:00:55

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#138 2005-09-04 16:14:07

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#139 2005-09-04 16:46:15

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#140 2005-09-04 17:10:54

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#141 2005-09-04 18:26:46

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#142 2005-09-05 03:21:41

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#143 2005-09-05 16:05:39

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#144 2005-09-05 19:27:38

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#145 2005-09-05 21:35:18

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#146 2005-09-06 17:49:17

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#147 2005-09-08 16:58:33

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#148 2005-09-08 21:22:32

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#149 2005-09-09 18:03:26

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#150 2005-09-11 00:16:57

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