answer:To convert the hexadecimal number 351_{16} to decimal, we need to expand this in terms of powers of 16: [ 351_{16} = 1 cdot 16^0 + 5 cdot 16^1 + 3 cdot 16^2 ] Calculating each term: - (1 cdot 16^0 = 1) - (5 cdot 16^1 = 80) - (3 cdot 16^2 = 768) Adding these up gives us: [ 1 + 80 + 768 = 849 ] So, 351_{16} = boxed{849} in decimal (base ten). Conclusion: The Martian crystal is 849 years old in base ten.

answer:First, we need to find the value of y by solving the equation 2y = 540. Divide both sides by 2 to isolate y: y = 540 / 2 y = 270 Now that we know y is 270 dollars, we can calculate the new price after a 30 percent increase. A 30 percent increase means the new price is 100% of the original price plus an additional 30%, which is a total of 130% of the original price. To find 130% of y, we multiply y by 1.30 (since 130% as a decimal is 1.30): New price = y * 1.30 New price = 270 * 1.30 New price = 351 So the new price of the computer after a 30 percent increase is boxed{351} dollars.

answer:1. **Let A B = 1 and B C = a, and B D = x**: We place the problem in a coordinate system such that A B= 1 and B C = a. Additionally, we let the length of B D be x. 2. **Determine AC using the Pythagorean theorem**: Since triangle A B C is a right triangle at A, we have: [ A C = sqrt{A B^2 + B C^2} = sqrt{1^2 + a^2 - 1} = sqrt{a^2 - 1}. ] 3. **Apply the angle bisector theorem to find A D**: By the angle bisector theorem in triangle A B C, we have: [ frac{A D}{D C} = frac{A B}{B C} = frac{1}{a}. ] Therefore: [ D C = a times A D. ] 4. **Express A C in terms of A D**: The length A D and D C must sum to A C: [ A D + D C = A C = sqrt{a^2 - 1}. ] Substituting D C = a times A D into this equation: [ A D + a cdot A D = sqrt{a^2 - 1}, ] which simplifies to: [ A D (1 + a) = sqrt{a^2 - 1}. ] Solving for A D: [ A D = frac{sqrt{a^2 - 1}}{1 + a}. ] 5. **Calculate A D^2**: [ A D^2 = left( frac{sqrt{a^2 - 1}}{a + 1} right)^2 = frac{a^2 - 1}{(a + 1)^2} = frac{a - 1}{a + 1}. ] 6. **Apply the Pythagorean theorem to triangle BAD**: In triangle B A D, Pythagoras' theorem gives us: [ x^2 = AB^2 + AD^2 = 1 + frac{a - 1}{a + 1}. ] Combining the terms, [ x^2 = 1 + frac{a - 1}{a + 1} = frac{(a + 1) + (a - 1)}{a + 1} = frac{2a}{a + 1}. ] 7. **"If and only if" proof**: - **The first direction**: Assume frac{1}{x} - frac{1}{a} = frac{1}{2}, we find x: [ x = frac{2a}{a + 2}. ] Plugging x back into our expression for x^2: [ frac{2 a}{a + 1} = x^2 = left( frac{2 a}{a + 2} right)^2 = frac{4 a^2}{a^2 + 4 a + 4}. ] Equating both sides and solving: [ a^2 + 4a + 4 = 2a(a + 1), ] [ 0 = a^2 - 2a - 4. ] Using the quadratic formula, we find: [ a = frac{2 pm sqrt{4 + 16}}{2} = 1 pm sqrt{5}. ] Since a > 0, we select the positive root: [ a = 1 + sqrt{5}. ] 8. **It's also true that**: Since x = frac{2a}{a + 2} again: [ x = frac{2 (1 + sqrt{5})}{3 + sqrt{5}} = sqrt{5} - 1. ] Thus: [ a - x = (1 + sqrt{5}) - (sqrt{5} - 1) = 2. ] 9. **The second direction**: Let a - x = 2: [ x = a - 2. ] From previous calculation and comparison: [ frac{2a}{a + 1} = (a - 2)^2, ] [ 2a = (a^2 - 4a + 4)(a + 1), ] [ 0 = a^3 - 3a^2 - 2a + 4. ] Factorizing, [ 0 = (a - 1)(a^2 - 2a - 4). ] We discard a = 1 as it's not suitable. Solving a^2 - 2a - 4 = 0, we again get: [ a = 1 + sqrt{5}. ] 10. **Final statement**: Since frac{1}{sqrt{5} - 1} - frac{1}{1 + sqrt{5}} = frac{1}{2} confirms the two way equivalency. # Conclusion: [ boxed{frac{1}{B D} - frac{1}{B C} = frac{1}{2 A B}} ] is equivalent to ( boxed{B C - B D = 2 A B} ).

answer:When Danny folds a blanket once, the area of the blanket is reduced by half. Since he folds each blanket 4 times, we need to calculate the area after each fold. Starting with an area of 8 x 8 for each blanket, which is 64 square units, we fold it once to get an area of 64 / 2 = 32 square units. Fold it a second time, and the area is 32 / 2 = 16 square units. Fold it a third time, and the area is 16 / 2 = 8 square units. Fold it a fourth time, and the area is 8 / 2 = 4 square units. So after folding one blanket 4 times, its area is 4 square units. Since Danny has three blankets, we multiply this area by 3 to get the total area of the folded blankets: 4 square units/blanket * 3 blankets = 12 square units Therefore, the total area of the folded picnic blankets is boxed{12} square units.