💡First, I need to identify the two consecutive odd numbers whose sum is 48. Let's denote the smaller odd number as ( n ). Since consecutive odd numbers differ by 2, the larger odd number can be represented as ( n + 2 ).Next, I'll set up an equation based on the given information:[n + (n + 2) = 48]Combining like terms:[2n + 2 = 48]Subtracting 2 from both sides to isolate the term with ( n ):[2n = 46]Finally, dividing both sides by 2 to solve for ( n ):[n = 23]Therefore, the smaller odd number is 23.

💡First, I recognize that the total amount of work remains constant regardless of the number of men working on it. I calculate the total work by multiplying the number of men by the number of days they take to complete the work. With 36 men taking 18 days, the total work is 36 multiplied by 18, which equals 648 man-days.Next, I set up an equation where the number of men in the larger group multiplied by the number of days they take (6 days) equals the total work of 648 man-days.Solving for the number of men in the larger group, I divide 648 by 6, resulting in 108 men.

💡First, Georgie has 8 choices for the window to enter the house.After entering, Georgie must choose a different window to exit, leaving 7 options.Therefore, the total number of ways Georgie can enter through one window and exit through a different window is 8 multiplied by 7, which equals 56.

💡First, I need to determine how many gallons the Ultraflow pump can pump in 75 minutes. Since the pump's rate is given in gallons per hour, I'll convert 75 minutes to hours by dividing by 60. This gives me 1.25 hours. Then, I'll multiply the pumping rate by the time in hours to find the total gallons pumped.Next, I'll calculate the total water pumped by both the Ultraflow and MiniFlow pumps in 50 minutes. I'll start by converting 50 minutes to hours, which is approximately 0.8333 hours. I'll multiply each pump's rate by this time to find the gallons pumped by each individually. Finally, I'll add the two amounts together to get the total water pumped by both pumps in 50 minutes.

💡First, I need to solve the equation ( a + frac{a}{3} - frac{a}{9} = frac{10}{3} ) for ( a ).To simplify the left side of the equation, I'll find a common denominator for the terms involving ( a ). The common denominator for 1, 3, and 9 is 9.Converting each term to have the denominator of 9:- ( a = frac{9a}{9} )- ( frac{a}{3} = frac{3a}{9} )- ( frac{a}{9} = frac{a}{9} )Now, substitute these back into the equation:[frac{9a}{9} + frac{3a}{9} - frac{a}{9} = frac{10}{3}]Combine the terms on the left side:[frac{9a + 3a - a}{9} = frac{10}{3}][frac{11a}{9} = frac{10}{3}]To solve for ( a ), cross-multiply:[11a = 9 times frac{10}{3}][11a = 30]Finally, divide both sides by 11:[a = frac{30}{11}]

💡First, I recognize that the expression (cos^2 15^circ - sin^2 15^circ) resembles the double angle formula for cosine, which states that (cos 2theta = cos^2 theta - sin^2 theta).Applying this formula, I substitute (theta) with (15^circ), resulting in:[cos^2 15^circ - sin^2 15^circ = cos(2 times 15^circ) = cos 30^circ]Next, I recall the exact value of (cos 30^circ), which is (frac{sqrt{3}}{2}).Therefore, the value of (cos^2 15^circ - sin^2 15^circ) is (frac{sqrt{3}}{2}).