answer:To solve for A, we need to treat the equation as a simple algebraic equation. The equation given is: 1A + 4B3 = 469 We can see that B3 is a three-digit number where B is the tens digit and 3 is the units digit. Since we are only solving for A, we can ignore the value of B for now and focus on the structure of the equation. The equation can be rewritten as: 10A + 4B3 = 469 Now, we need to find a value for A such that when added to 4B3, the result is 469. Since 4B3 is a three-digit number and the last digit is 3, we know that the last digit of 469 must be a result of adding 3 to a multiple of 10 (since A is a single digit and we are adding 10A to 4B3). Looking at the last digit of 469, which is 9, we can determine that A must be 6 because: 3 (from 4B3) + 6 (from 10A) = 9 (the last digit of 469) Therefore, A should be 6 to satisfy the equation 1A + 4B3 = boxed{469} .

answer:1. **Identify Symmetry Properties**: - Note that point A_1 is the reflection of A across line BD. - Point A_2 is the reflection of A_1 across line AC and lies on diagonal BD. - Therefore, segment lengths A_2A = A_2A_1 and AA_2 = AA_1 because they are reflections. 2. **Prove Equilateral Triangle**: - From the symmetry properties: AA_2 = AA_1 and A_2A = A_2A_1 imply triangle AA_1A_2 is equilateral. - AA_1 = AA_2 (reflection over BD) - AA_1 = A_1A_2 (reflection over AC) 3. **Relate the Intersection of Diagonals**: - The intersection of the diagonals of a parallelogram divides the diagonals into two equal parts. - Let O be the intersection point and midpoint of the diagonals BD and AC. 4. **Length Definitions and Goal**: - Define BD = 5x. - Given: BA_2 = frac{4}{5}BD = 4x. 5. **Apply Cosine Rule** (Cosine Theorem): - Given that AD = 7 and AD is a side of triangle AOD where angle AOD = 120^circ (since ABD is proportionate to angle A being obtuse), use: [ AD^2 = AO^2 + OD^2 - 2 cdot AO cdot OD cdot cos(120^circ) ] Where AO = OD (since diagonals bisect each other): [ 49 = left(frac{5x}{2}right)^2 + left(frac{5x}{2}right)^2 - 2 cdot left(frac{5x}{2}right) cdot left(frac{5x}{2}right) cdot (-0.5) ] 6. **Simplify and Solve for x**: - Simplify the equation: [ 49 = 2 cdot left(frac{25x^2}{4}right) + frac{25x^2}{4} ] Simplifies to: [ 49 = frac{75x^2}{4} ] [ 196 = 75x^2 ] [ x^2 = frac{196}{75} approx 2.6133 ] [ x = 2 ] 7. **Calculate Actual Lengths**: - Substituting x = 2: - BD = 5x = 10 - AC = 2AO = 6 8. **Calculate Area of Parallelogram**: - Find the area using the formula for the area of a parallelogram given two diagonals and the angle between them (which here is angle O = 120^circ): [ text{Area} = frac{1}{2} times BD times AC times sin(120^circ) ] Substituting known values: [ text{Area} = frac{1}{2} times 10 times 6 times frac{sqrt{3}}{2} = 15sqrt{3} ] # Conclusion [ boxed{15 sqrt{3}} ]

answer:Let's start by calculating the total value of the coins Ravi has in terms of nickels, dimes, and quarters. We know that Ravi has 6 nickels. Since each nickel is worth 5 cents, the total value of the nickels is: 6 nickels * 0.05/nickel = 0.30 Let's denote the number of quarters as Q. According to the information given, Ravi has 2 more quarters than nickels, so: Q = 6 nickels + 2 = 8 quarters Since each quarter is worth 25 cents, the total value of the quarters is: 8 quarters * 0.25/quarter = 2.00 Now, let's denote the number of dimes as D. We know that Ravi has a certain number more dimes than quarters, but we don't know this number yet. Since each dime is worth 10 cents, the total value of the dimes is: D dimes * 0.10/dime The total value of all the coins is 3.50, so we can write the equation: 0.30 (nickels) + 2.00 (quarters) + (D * 0.10) (dimes) = 3.50 Now we can solve for D: 0.30 + 2.00 + (0.10 * D) = 3.50 0.10 * D = 3.50 - 0.30 - 2.00 0.10 * D = 1.20 Divide both sides by 0.10 to find the number of dimes: D = 1.20 / 0.10 D = 12 dimes Now we know Ravi has 12 dimes and 8 quarters. To find out how many more dimes he has than quarters, we subtract the number of quarters from the number of dimes: 12 dimes - 8 quarters = 4 more dimes than quarters So, Ravi has boxed{4} more dimes than quarters.

answer:To calculate Craig's total commission for the week, we need to calculate the commission for each type of appliance and then sum them up. For refrigerators: Commission per refrigerator = 75 + (8% of selling price) Total commission for refrigerators = 3 * (75 + (8% of 5280)) For washing machines: Commission per washing machine = 50 + (10% of selling price) Total commission for washing machines = 4 * (50 + (10% of 2140)) For ovens: Commission per oven = 60 + (12% of selling price) Total commission for ovens = 5 * (60 + (12% of 4620)) Now let's calculate each one: Refrigerators: Total commission for refrigerators = 3 * (75 + (0.08 * 5280)) Total commission for refrigerators = 3 * (75 + 422.40) Total commission for refrigerators = 3 * 497.40 Total commission for refrigerators = 1492.20 Washing machines: Total commission for washing machines = 4 * (50 + (0.10 * 2140)) Total commission for washing machines = 4 * (50 + 214) Total commission for washing machines = 4 * 264 Total commission for washing machines = 1056 Ovens: Total commission for ovens = 5 * (60 + (0.12 * 4620)) Total commission for ovens = 5 * (60 + 554.40) Total commission for ovens = 5 * 614.40 Total commission for ovens = 3072 Now, we add up all the commissions: Total commission = Total commission for refrigerators + Total commission for washing machines + Total commission for ovens Total commission = 1492.20 + 1056 + 3072 Total commission = 5620.20 Craig's total commission for that week is boxed{5620.20} .