Explanation

The phrase "All but nine died" means that 9 sheep survived.

So, the shepherd is left with 9 sheep.

Explanation

Let's analyze the formation requirements:

1. Two ducks in front of a duck

2. Two ducks behind a duck

3. A duck between two ducks

The smallest number that satisfies all these conditions is *3* ducks in a line:

Duck 1 (front), Duck 2 (middle), Duck 3 (back)

For Duck 2: Duck 1 and Duck 3 satisfy the "two ducks" condition in relation to their positions.

Duck 1 has two ducks (2 and 3) behind it.

Duck 3 has two ducks (1 and 2) in front of it.

Duck 2 is between Duck 1 and Duck 3.

Explanation

The dial of a standard telephone has numbers 0 through 9.

The product of all these numbers is:

0 × 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9 = 0

Since any number multiplied by 0 is 0, the product is 0.

Explanation

Let's calculate D:

D = 1 * 10 * 100 = 1000

Now, calculate each term:

D/8 = 1000 / 8 = 125

D/5 = 1000 / 5 = 200

D/4 = 1000 / 4 = 250

D/2 = 1000 / 2 = 500

Sum = 125 + 200 + 250 + 500 = 1075

The sum of the digits of 1075:

1 + 0 + 7 + 5 = 13

Explanation

The formula for the sum of the first n natural numbers is:

S(n) = n(n + 1) / 2

Given S(100) = 5050, let's calculate S(200):

S(200) = 200(200 + 1) / 2

= 200 * 201 / 2

= 100 * 201

= 20100

Explanation

Let's denote the length as L and width as W.

Given W = 0.75L or W = (3/4)L

Perimeter = 2(L + W) = 42

2(L + 3L/4) = 42

2(7L/4) = 42

7L/2 = 42

7L = 84

L = 12

W = (3/4)*12 = 9

Area = L * W = 12 * 9 = 108

Explanation

Let's denote Akram's age as 10a + b and his wife's age as 10b + a, where a > b.

Given the difference between their ages is 1/11 of their sum:

(10a + b) - (10b + a) = 1/11 * ((10a + b) + (10b + a))

9a - 9b = 1/11 * (11a + 11b)

9(a - b) = a + b

9a - 9b = a + b

8a = 10b

4a = 5b

Possible values for (a, b) given a > b:

(5, 4)

Akram's age: 10_5 + 4 = 54

Wife's age: 10_4 + 5 = 45

Explanation

Given -1 < X < Y < 0, both X and Y are negative.

Let's analyze the expressions:

xy²: Since y² is positive (because squaring a number makes it positive), and x is negative, xy² is negative.

x²y: Since x² is positive, and y is negative, x²y is negative.

xy: Since both x and y are negative, xy is positive.

Comparing the signs:

xy is positive.

Both xy² and x²y are negative.

Since xy is positive and the others are negative, xy must be the greatest.

Now, comparing xy² and x²y:

Let's consider the magnitude:

|x| > |y| (since X < Y < 0)

xy² = x * y² (x is more negative, y² is positive)

x²y = x² * y (x² is positive, y is negative)

Given |x| > |y|, x² > y² (since both are positive after squaring)

x²|y| > |x|y² (since |y| < |x| and both x² and y² are positive)

x²y < xy² (both are negative, but x²y is more negative)

So, the order from least to greatest is:

x²y, xy², xy

Explanation

Let's break it down:

(x + y + z) / 3 = 15

x + y + z = 45

(y + z + w) / 3 = 16

y + z + w = 48

Given w = 19:

y + z + 19 = 48

y + z = 29

Now, substitute y + z in the first equation:

x + 29 = 45

x = 16

Explanation

Let's calculate each option:

1/9 of 504 = 504 / 9 = 56

1/4 of 236 = 236 / 4 = 59

1/3 of 741 = 741 / 3 = 247

1/16 of 1028 = 1028 / 16 = 64.25

Comparing the results:

56, 59, 247, 64.25

The greatest number is 247, which corresponds to 1/3 of 741.