Math Problems & Logic Puzzles

Let the number be 10x + y where x is tens digit, y is units digit. Square: (10x+y)² = 100x² + 20xy + y². Dividing by half the number (5x + y/2) and adding 36 then dividing by 2 gives the reversed number 10y + x. Also x = 2(y-x). Testing option B: 46² = 2116, half of 46 = 23, 2116/23 = 92, 92+36 = 128, 128/2 = 64 ✓ And 4 = 2(6-4) = 4 ✓ The number is 46.

To solve this question, the user needs to know how to manipulate equations with variables and fractions.

We are given the equation:

$$(a-b)/3.5 = 4/7$$

We can simplify this equation by cross-multiplying and solving for a-b:

$$(a-b) = (4/7)*3.5$$

$$(a-b) = 2$$

Now, we have the equation a-b = 2, which means that a is two more than b. Therefore, b < a.

Option B is correct because b is always less than a in this equation.

Option A is incorrect because b is not equal to a.

Option C is incorrect because b is always less than a in this equation.

Option D is incorrect because b is always less than a in this equation.

The Answer is: B. b < a.

To solve this question, the user needs to know algebraic manipulation and simplification. The user must use the given equation to find the value of w and then substitute it into the expression (2w)2.

First, let's solve for w using the given equation:

13 = 13w/(1-w)

Multiplying both sides by (1-w), we get:

13 - 13w = 13w

Adding 13w to both sides, we get:

13 = 26w

Dividing both sides by 26, we get:

w = 1/2

Now that we have the value of w, we can substitute it into the expression (2w)2:

(2w)2 = (2(1/2))2 = 12

Thus, the correct answer is:

The Answer is: D. 1

To solve this question, the user needs to know the concept of encoding and decoding using a specific pattern or rule.

Each letter in the word 'CODING' has been shifted one letter forward in the English alphabet. For example, 'C' becomes 'D', 'O' becomes 'P', and so on.

Using the same pattern, we can decode the word 'CURFEW' as follows:

• 'C' becomes 'D'
• 'U' becomes 'V'
• 'R' becomes 'S'
• 'F' becomes 'G'
• 'E' becomes 'F'
• 'W' becomes 'X'

Therefore, the encoded form of the word 'CURFEW' is 'DVSGFX'.

Thus, the correct answer is:

The Answer is: A. DVSGFX.

To solve this question, the user needs to know the coding pattern that is being used to encode the words. In this case, it seems that each letter in the word is being replaced by a unique digit. To find the code for the word 'MANGO', we need to first identify the digits that correspond to each letter in the word.

Looking at the given codes, we can see that the letters 'R', 'A', 'N', 'G', 'O', and 'M' are represented by the digits 1, 2, 3, 4, 5, and 6, respectively. Using this pattern, we can determine that the code for the word 'MANGO' is:

M A N G O
6 2 3 4 5

Therefore, the answer is:

The Answer is: D. 82347

The pattern alternates: 32+6=38, 38-27=11, 11+8=19, 19+5=24, 24+10=34, 34-8=26, 26+12=38, 38+10=48, 48+14=62.

Boiling eggs is a parallel process - multiple eggs can be cooked simultaneously in the same pot. The cooking time depends on water temperature and egg size, not on the number of eggs. Therefore, 4 eggs take the same 5 minutes as 1 egg.

Walking speed and time are inversely proportional. At 5/6 of usual speed, time becomes 6/5 of usual time. Let T be usual time. New time - T = 40 minutes. So (6/5)T - T = 40, which gives T/5 = 40, so T = 200 minutes = 3 hours 20 minutes.

Given x/y = 4, we have x = 4y. Then 2x - y = 2(4y) - y = 7y. Since x = 4y, y = x/4. So 2x - y = 7(x/4) = 1.75x, which equals 175% of x.

Set up equation: 68000 - 80t = 42000 + 120t, where t is years. Solving: 26000 = 200t, so t = 130 years. In 130 years, both cities will have population 68000 - 80(130) = 68000 - 10400 = 57600.

When 0 < x < 1, we have x > x^2 > x^3 (since multiplying by x less than 1 decreases value), but 1/x > 1 (since denominator < numerator). Therefore 1/x is the largest among x, x^2, x^3, and 1/x.

For a square of side 6cm, the inscribed circle has diameter 6cm (radius 3cm). Circle area = πr² = 9π. Square area = 6² = 36. The ratio is 9π/36 = π/4 ≈ 0.785. Among options, 11/14 ≈ 0.786 is the closest approximation to π/4.

Original contribution per person = D/M. After 3 men drop out, M-3 men remain, each contributing D/(M-3). The extra amount = D/(M-3) - D/M. Taking LCM: = DM/M(M-3) - D(M-3)/M(M-3) = 3D/M(M-3).

Let t = time per non-math question. Math questions get 2t each. Total time: 50(2t) + 150(t) = 180 minutes. 250t = 180, so t = 0.72 min. Time on math problems = 50 × 2t = 100 × 0.72 = 72 minutes.

To form a three-digit number with digits in ascending order from 1-9, we simply need to choose any 3 distinct digits - there's exactly one way to arrange them in ascending order. This is a combination problem: 9C3 = 9!/(3!×6!) = (9×8×7)/(3×2×1) = 84. The key insight is that choosing digits determines the arrangement uniquely.

Let A start with x coins. After the sequence of transfers, we can track that B ends with x/4-2 and D ends with x/8+7. Setting them equal: x/4-2 = x/8+7 gives x = 72. Verifying: A gives 40 to B (keeps 32), B gives 24 to C (keeps 16), C gives 16 to D (keeps 8). Both B and D have 16 coins, so the answer is 72.

The sum of first n natural numbers is n(n+1)/2. We need this close to 177. Testing values: 18×19/2 = 171 (too low), 19×20/2 = 190 (too high). The student got 177, so they were adding 1-19 but missed a number. 190 - 177 = 13, which is the missed number. The correct sum of 1-19 is 190.

To cut rods into equal parts of maximum length, we need the HCF of all lengths. Prime factorization: 78=2×3×13, 104=2³×13, 117=3²×13, 169=13². HCF=13 cm. Number of pieces = 78/13 + 104/13 + 117/13 + 169/13 = 6+8+9+13 = 36. Option D (480) is incorrect because it's just the total length, not the number of pieces.