This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Step 1: Evaluate the exponents. 600 + 5^2 + 25 × 3 - 45 + 2^2 = 600 + 25 + 25 × 3 - 45 + 4 Step 2: Perform the multiplication. 600 + 25 + 25 × 3 - 45 + 4 = 600 + 25 + 75 - 45 + 4 Step 3: Perform addition and subtraction from left to right. 600 + 25 + 75 - 45 + 4 = 625 + 75 - 45 + 4 625 + 75 - 45 + 4 = 700 - 45 + 4 700 - 45 + 4 = 655 + 4 655 + 4 = 659 The simplified value is 659. 2. Step 1: Calculate the original price per apple. Price per apple = Total costNumber of apples = Rs. 1320110 = Rs. 12 per apple Step 2: Calculate the new price per apple after a 10% increase. Increase amount = 10\% × Rs. 12 = 0.10 × Rs. 12 = Rs. 1.20 New price per apple = Rs. 12 + Rs. 1.20 = Rs. 13.20 Step 3: Calculate the cost of 130 apples at the new price. Cost of 130 apples = 130 × Rs. 13.20 = Rs. 1716 The cost of 130 apples will be Rs. 1716. 3. Step 1: Determine the lengths of the sides of the rectangle. The vertices are A(12,3), B(12,7), C(15,7), D(15,3). Length AB (vertical side): The x-coordinate is constant (12). The length is the difference in y-coordinates: 7 - 3 = 4 units. Length BC (horizontal side): The y-coordinate is constant (7). The length is the difference in x-coordinates: 15 - 12 = 3 units. So, the length of the rectangle is 4 units and the width is 3 units. Step 2: Calculate the perimeter in units. Perimeter in units = 2 × (Length + Width) = 2 × (4 + 3) = 2 × 7 = 14 units Step 3: Convert the perimeter from units to cm. Given that 1 unit = 2 cm. Perimeter in cm = 14 units × 2 cm/unit = 28 cm The perimeter of the rectangle is 28 cm. 4. Step 1: Set up a proportion for the ratio of orange juice to pineapple juice. Orange juicePineapple juice = (5)/(2) Step 2: Substitute the given amount of orange juice and solve for pineapple juice. 450 mlP = (5)/(2) 5P = 450 ml × 2 5P = 900 ml P = 900 ml5 P = 180 ml You need 180 ml of pineapple juice. 5. Step 1: Calculate the total shift time in minutes. Total shift time = 10 hours × 60 minutes/hour = 600 minutes Step 2: Calculate the time spent on each activity in minutes. • Checking emails: (1)/(30) × 600 minutes = 20 minutes • Data entry: 24\% × 600 minutes = 0.24 × 600 minutes = 144 minutes • Web designing: (1)/(2) × 600 minutes = 300 minutes • Project discussion: 4.06\% × 600 minutes = 0.0406 × 600 minutes = 24.36 minutes Step 3: Calculate the total time spent on these activities. Total activity time = 20 + 144 + 300 + 24.36 = 488.36 minutes Step 4: Calculate the remaining time for rest and lunch break. Remaining time = Total shift time - Total activity time Remaining time = 600 minutes - 488.36 minutes = 111.64 minutes Harish spends 111.64 minutes on rest and lunch break. 6. Step 1: Identify the relationship between the angles. The angle 125^ is an exterior angle of the triangle. The interior angle adjacent to it is 180^ - 125^ = 55^. Step 2: Apply the angle sum property of a triangle. The sum of the interior angles of a triangle is 180^. x + 80^ + 55^ = 180^ x + 135^ = 180^ x = 180^ - 135^ x = 45^ Alternatively, using the exterior angle theorem: The exterior angle of a triangle is equal to the sum of the two opposite interior angles. 125^ = x + 80^ x = 125^ - 80^ x = 45^ The value of x is 45^. 7. Step 1: Set up a proportion for the ratio of flour to sugar. FlourSugar = (4)/(5) Step 2: Substitute the given amount of sugar and solve for flour. (F)/(300 g) = (4)/(5) 5F = 4 × 300 g 5F = 1200 g F = 1200 g5 F = 240 g Jay used 240 g of flour. 8. Step 1: Calculate the area of the circular garden. Given radius r = 7 cm and = 3.14. Area of circle = r^2 = 3.14 × (7 cm)^2 = 3.14 × 49 cm^2 = 153.86 cm^2 Step 2: Calculate the area of the rectangular garden. Given length l = 20 cm and width w = 8 cm. Area of rectangle = l × w = 20 cm × 8 cm = 160 cm^2 Step 3: Compare the areas. Circular garden area = 153.86 cm^2 Rectangular garden area = 160 cm^2 Since 160 cm^2 > 153.86 cm^2, the rectangular garden has more area. Option b) A rectangular garden with dimensions 20 cm × 8 cm gives more area. 9. To find the coldest temperature, we need to find the smallest number in the given list: -3°C, -23°C, 56°C, -47°C, -45°C, 0°C. Comparing the numbers: Positive temperatures: 56°C, 0°C Negative temperatures: -3°C, -23°C, -47°C, -45°C The smallest number among these is the one furthest from zero in the negative direction. -47 is smaller than -45, -23, and -3. The coldest temperature is -47°C. 10. Step 1: Calculate the time taken using the formula Time = (Distance)/(Speed). Time = 97.5 miles26 mph = 3.75 hours Step 2: Convert the decimal part of the hours into minutes. 0.75 hours × 60 minutes/hour = 45 minutes So, the total time is 3 hours and 45 minutes. The journey will be completed in 3 hours and 45 minutes. 11. Step 1: Substitute the given values of x and y into the expression. Given x = 35 and y = 3.5. 24x + 350y = 24(35) + 350(3.5) Step 2: Perform the multiplications. 24 × 35 = 840 350 × 3.5 = 1225 Step 3: Perform the addition. 840 + 1225 = 2065 The value of 24x + 350y is 2065.