answer:First, calculate the total payment without the discount: [ frac{15}{4} times frac{12}{5} = frac{15 times 12}{4 times 5} = frac{180}{20} = frac{9}{1} = 9 text{ dollars} ] Since frac{12}{5} rooms is less than 2 rooms (since frac{12}{5} = 2.4 and 2.4 > 2), apply a 10% discount: [ 9 - (0.1 times 9) = 9 - 0.9 = 8.1 text{ dollars} ] Since the answer should be in fraction form, we convert this decimal back to fraction: [ 8.1 = frac{81}{10} text{ dollars} ] Clara owes Mandy boxed{frac{81}{10}} dollars.

answer:Let's denote the number of days Ram takes to complete the job independently as ( R ). Ram's work rate is ( frac{1}{R} ) jobs per day, and Gohul's work rate is ( frac{1}{15} ) jobs per day. When they work together, their combined work rate is ( frac{1}{R} + frac{1}{15} ) jobs per day. We know that together they take approximately 6 days to complete the job, so their combined work rate is ( frac{1}{6} ) jobs per day. Setting the combined work rate equal to the sum of their individual work rates, we get: [ frac{1}{R} + frac{1}{15} = frac{1}{6} ] To solve for ( R ), we first find a common denominator for the fractions on the left side of the equation, which is ( 15R ): [ frac{15}{15R} + frac{R}{15R} = frac{1}{6} ] [ frac{15 + R}{15R} = frac{1}{6} ] Now we cross-multiply to solve for ( R ): [ 6(15 + R) = 15R ] [ 90 + 6R = 15R ] Subtract ( 6R ) from both sides: [ 90 = 15R - 6R ] [ 90 = 9R ] Divide both sides by 9: [ R = frac{90}{9} ] [ R = 10 ] So, Ram would take boxed{10} days to complete the job independently.

answer:1. **Convert each equation to slope-intercept form (y = mx + b), to determine m (slope):** - **Equation (1):** 4y - 3x = 16 [ 4y = 3x + 16 implies y = frac{3}{4}x + 4 ] Slope m_1 = frac{3}{4}. - **Equation (2):** -3x - 4y = 15 [ -4y = 3x + 15 implies y = -frac{3}{4}x - frac{15}{4} ] Slope m_2 = -frac{3}{4}. - **Equation (3):** 4y + 3x = 16 [ 4y = -3x + 16 implies y = -frac{3}{4}x + 4 ] Slope m_3 = -frac{3}{4}. - **Equation (4):** 3y + 4x = 15 [ 3y = -4x + 15 implies y = -frac{4}{3}x + 5 ] Slope m_4 = -frac{4}{3}. 2. **Check the product of the slopes for each pair of equations:** - **Product of slopes (1) and (4):** [ m_1 cdot m_4 = frac{3}{4} cdot -frac{4}{3} = -1 ] Thus, lines (1) and (4) are perpendicular. - **Other pairs do not meet the perpendicularity condition:** - m_1 cdot m_2 = frac{3}{4} cdot -frac{3}{4} = -frac{9}{16} - m_1 cdot m_3 = frac{3}{4} cdot -frac{3}{4} = -frac{9}{16} - m_2 cdot m_3 = -frac{3}{4} cdot -frac{3}{4} = frac{9}{16} - m_2 cdot m_4 = -frac{3}{4} cdot -frac{4}{3} = 1 - m_3 cdot m_4 = -frac{3}{4} cdot -frac{4}{3} = 1 3. **Conclusion:** The only pair of lines that are perpendicular, as their slopes' product equals -1, is given by equations (1) and (4). Therefore, the correct answer is textbf{(A)}. The final answer is boxed{textbf{(A)} text{(1) and (4)}}

answer:Solution: (Ⅰ) Since the parametric equation of line l is begin{cases}x= dfrac{1}{2}t y=1- dfrac{ sqrt{3}}{2}tend{cases} (where t is the parameter), Therefore, by eliminating the parameter t, we obtain the standard equation of line l as sqrt{3}x+y-1=0, Since the polar equation of circle C is rho=2sin theta, which implies rho^{2}=2rhosin theta, Therefore, from rho^{2}=x^{2}+y^{2} and rhosin theta=y, we get the Cartesian equation of circle C as x^{2}+y^{2}-2y=0. Since the center of the circle (0,1) lies on line l, Therefore, the number of intersection points between line l and circle C is boxed{2}. (Ⅱ) From (Ⅰ), we know that the center of the circle (0,1) lies on line l, Therefore, AB is the diameter of circle C, Since the Cartesian equation of circle C is x^{2}+y^{2}-2y=0, Therefore, the radius r of circle C is dfrac{1}{2} sqrt{4} =1, hence the diameter of circle C is 2, therefore |AB|=2. Thus, the length of segment AB is boxed{2}.