36. Step 1: Convert each fraction to a decimal to compare them.

Question

36. Step 1: Convert each fraction to a decimal to compare them.

Asked on April 25, 2026Mathematics

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.

Accepted Answer · 2026-04-25T18:52:30.433Z

36. Step 1: Convert each fraction to a decimal to compare them. $$\frac{5}{11} \approx 0.4545$$ $$\frac{3}{7} \approx 0.4286$$ $$\frac{6}{13} \approx 0.4615$$ $$\frac{2}{5} = 0.4$$ $$\frac{4}{9} \approx 0.4444$$ Step 2: Arrange the decimals in ascending order. $$0.4, 0.4286, 0.4444, 0.4545, 0.4615$$ Step 3: Write the original fractions in ascending order. $$\boxed{\frac{2}{5}, \frac{3}{7}, \frac{4}{9}, \frac{5}{11}, \frac{6}{13}}$$ 37. Step 1: Identify the type of triangle. The markings indicate it is an isosceles triangle, meaning two sides are equal, and the base angles opposite those sides are also equal. Step 2: Calculate the base angles of the isosceles triangle. The sum of angles in a triangle is $180^\circ$. Let the base angles be $x$. $$48^\circ + x + x = 180^\circ$$ $$48^\circ + 2x = 180^\circ$$ $$2x = 180^\circ - 48^\circ$$ $$2x = 132^\circ$$ $$x = \frac{132^\circ}{2}$$ $$x = 66^\circ$$ Step 3: Determine the value of angle $t^\circ$. Angle $t^\circ$ and the base angle $x$ form a linear pair on a straight line, so their sum is $180^\circ$. $$t^\circ + x = 180^\circ$$ $$t^\circ + 66^\circ = 180^\circ$$ $$t^\circ = 180^\circ - 66^\circ$$ $$t^\circ = 114^\circ$$ The value of the angle marked $t^\circ$ is $\boxed{114^\circ}$. 38. Step 1: Use the area of a right-angled triangle formula to find the height. Area $A = \frac{1}{2} \times \text{base} \times \text{height}$ Given $A = 96 \text{ m}^2$ and base $b = 12 \text{ m}$. $$96 \text{ m}^2 = \frac{1}{2} \times 12 \text{ m} \times h$$ $$96 \text{ m}^2 = 6 \text{ m} \times h$$ $$h = \frac{96 \text{ m}^2}{6 \text{ m}}$$ $$h = 16 \text{ m}$$ Step 2: Use the Pythagorean theorem to find the hypotenuse. Let the hypotenuse be $c$. $$c^2 = b^2 + h^2$$ $$c^2 = (12 \text{ m})^2 + (16 \text{ m})^2$$ $$c^2 = 144 \text{ m}^2 + 256 \text{ m}^2$$ $$c^2 = 400 \text{ m}^2$$ $$c = \sqrt{400 \text{ m}^2}$$ $$c = 20 \text{ m}$$ Step 3: Calculate the perimeter of the triangle. Perimeter $P = \text{base} + \text{height} + \text{hypotenuse}$ $$P = 12 \text{ m} + 16 \text{ m} + 20 \text{ m}$$ $$P = 48 \text{ m}$$ The perimeter of the triangle is $\boxed{48 \text{ m}}$. 39. Step 1: Identify the shape. The figure shows a semicircle with a right-angled triangle inscribed within it. The shaded region is the area of the semicircle minus the area of the triangle. The sides $42 \text{ cm}$ and $56 \text{ cm}$ are the legs of the right-angled triangle, and its hypotenuse is the diameter of the semicircle. Step 2: Calculate the diameter of the semicircle using the Pythagorean theorem. $$d^2 = (42 \text{ cm})^2 + (56 \text{ cm})^2$$ $$d^2 = 1764 \text{ cm}^2 + 3136 \text{ cm}^2$$ $$d^2 = 4900 \text{ cm}^2$$ $$d = \sqrt{4900 \text{ cm}^2}$$ $$d = 70 \text{ cm}$$ Step 3: Calculate the radius of the semicircle. $$r = \frac{d}{2} = \frac{70 \text{ cm}}{2} = 35 \text{ cm}$$ Step 4: Calculate the area of the semicircle. Use $\pi = \frac{22}{7}$. $$A_{semicircle} = \frac{1}{2} \pi r^2$$ $$A_{semicircle} = \frac{1}{2} \times \frac{22}{7} \times (35 \text{ cm})^2$$ $$A_{semicircle} = \frac{1}{2} \times \frac{22}{7} \times 1225 \text{ cm}^2$$ $$A_{semicircle} = 11 \times \frac{1225}{7} \text{ cm}^2$$ $$A_{semicircle} = 11 \times 175 \text{ cm}^2$$ $$A_{semicircle} = 1925 \text{ cm}^2$$ Step 5: Calculate the area of the triangle. $$A_{triangle} = \frac{1}{2} \times \text{base} \times \text{height}$$ $$A_{triangle} = \frac{1}{2} \times 42 \text{ cm} \times 56 \text{ cm}$$ $$A_{triangle} = 21 \text{ cm} \times 56 \text{ cm}$$ $$A_{triangle} = 1176 \text{ cm}^2$$ Step 6: Calculate the area of the shaded region. $$A_{shaded} = A_{semicircle} - A_{triangle}$$ $$A_{shaded} = 1925 \text{ cm}^2 - 1176 \text{ cm}^2$$ $$A_{shaded} = 749 \text{ cm}^2$$ The area of the shaded region is $\boxed{749 \text{ cm}^2}$. 40. Step 1: List the number of slips of each color. Number of blue slips = 3 Number of red slips = 4 Number of green slips = 2 Number of yellow slips = 6 Step 2: Calculate the total number of slips. Total slips = $3 + 4 + 2 + 6 = 15$ Step 3: Determine the number of red slips. Number of red slips = 4 Step 4: Calculate the probability of drawing a red slip. $$P(\text{red}) = \frac{\text{Number of red slips}}{\text{Total number of slips}}$$ $$P(\text{red}) = \frac{4}{15}$$ The probability of it being a red slip is $\boxed{\frac{4}{15}}$. That's 2 down. 3 left today — send the next one.

Accepted Answer · 2026-04-25T18:52:30.433Z

36. Step 1: Convert each fraction to a decimal to compare them. $$\frac{5}{11} \approx 0.4545$$ $$\frac{3}{7} \approx 0.4286$$ $$\frac{6}{13} \approx 0.4615$$ $$\frac{2}{5} = 0.4$$ $$\frac{4}{9} \approx 0.4444$$ Step 2: Arrange the decimals in ascending order. $$0.4, 0.4286, 0.4444, 0.4545, 0.4615$$ Step 3: Write the original fractions in ascending order. $$\boxed{\frac{2}{5}, \frac{3}{7}, \frac{4}{9}, \frac{5}{11}, \frac{6}{13}}$$ 37. Step 1: Identify the type of triangle. The markings indicate it is an isosceles triangle, meaning two sides are equal, and the base angles opposite those sides are also equal. Step 2: Calculate the base angles of the isosceles triangle. The sum of angles in a triangle is $180^\circ$. Let the base angles be $x$. $$48^\circ + x + x = 180^\circ$$ $$48^\circ + 2x = 180^\circ$$ $$2x = 180^\circ - 48^\circ$$ $$2x = 132^\circ$$ $$x = \frac{132^\circ}{2}$$ $$x = 66^\circ$$ Step 3: Determine the value of angle $t^\circ$. Angle $t^\circ$ and the base angle $x$ form a linear pair on a straight line, so their sum is $180^\circ$. $$t^\circ + x = 180^\circ$$ $$t^\circ + 66^\circ = 180^\circ$$ $$t^\circ = 180^\circ - 66^\circ$$ $$t^\circ = 114^\circ$$ The value of the angle marked $t^\circ$ is $\boxed{114^\circ}$. 38. Step 1: Use the area of a right-angled triangle formula to find the height. Area $A = \frac{1}{2} \times \text{base} \times \text{height}$ Given $A = 96 \text{ m}^2$ and base $b = 12 \text{ m}$. $$96 \text{ m}^2 = \frac{1}{2} \times 12 \text{ m} \times h$$ $$96 \text{ m}^2 = 6 \text{ m} \times h$$ $$h = \frac{96 \text{ m}^2}{6 \text{ m}}$$ $$h = 16 \text{ m}$$ Step 2: Use the Pythagorean theorem to find the hypotenuse. Let the hypotenuse be $c$. $$c^2 = b^2 + h^2$$ $$c^2 = (12 \text{ m})^2 + (16 \text{ m})^2$$ $$c^2 = 144 \text{ m}^2 + 256 \text{ m}^2$$ $$c^2 = 400 \text{ m}^2$$ $$c = \sqrt{400 \text{ m}^2}$$ $$c = 20 \text{ m}$$ Step 3: Calculate the perimeter of the triangle. Perimeter $P = \text{base} + \text{height} + \text{hypotenuse}$ $$P = 12 \text{ m} + 16 \text{ m} + 20 \text{ m}$$ $$P = 48 \text{ m}$$ The perimeter of the triangle is $\boxed{48 \text{ m}}$. 39. Step 1: Identify the shape. The figure shows a semicircle with a right-angled triangle inscribed within it. The shaded region is the area of the semicircle minus the area of the triangle. The sides $42 \text{ cm}$ and $56 \text{ cm}$ are the legs of the right-angled triangle, and its hypotenuse is the diameter of the semicircle. Step 2: Calculate the diameter of the semicircle using the Pythagorean theorem. $$d^2 = (42 \text{ cm})^2 + (56 \text{ cm})^2$$ $$d^2 = 1764 \text{ cm}^2 + 3136 \text{ cm}^2$$ $$d^2 = 4900 \text{ cm}^2$$ $$d = \sqrt{4900 \text{ cm}^2}$$ $$d = 70 \text{ cm}$$ Step 3: Calculate the radius of the semicircle. $$r = \frac{d}{2} = \frac{70 \text{ cm}}{2} = 35 \text{ cm}$$ Step 4: Calculate the area of the semicircle. Use $\pi = \frac{22}{7}$. $$A_{semicircle} = \frac{1}{2} \pi r^2$$ $$A_{semicircle} = \frac{1}{2} \times \frac{22}{7} \times (35 \text{ cm})^2$$ $$A_{semicircle} = \frac{1}{2} \times \frac{22}{7} \times 1225 \text{ cm}^2$$ $$A_{semicircle} = 11 \times \frac{1225}{7} \text{ cm}^2$$ $$A_{semicircle} = 11 \times 175 \text{ cm}^2$$ $$A_{semicircle} = 1925 \text{ cm}^2$$ Step 5: Calculate the area of the triangle. $$A_{triangle} = \frac{1}{2} \times \text{base} \times \text{height}$$ $$A_{triangle} = \frac{1}{2} \times 42 \text{ cm} \times 56 \text{ cm}$$ $$A_{triangle} = 21 \text{ cm} \times 56 \text{ cm}$$ $$A_{triangle} = 1176 \text{ cm}^2$$ Step 6: Calculate the area of the shaded region. $$A_{shaded} = A_{semicircle} - A_{triangle}$$ $$A_{shaded} = 1925 \text{ cm}^2 - 1176 \text{ cm}^2$$ $$A_{shaded} = 749 \text{ cm}^2$$ The area of the shaded region is $\boxed{749 \text{ cm}^2}$. 40. Step 1: List the number of slips of each color. Number of blue slips = 3 Number of red slips = 4 Number of green slips = 2 Number of yellow slips = 6 Step 2: Calculate the total number of slips. Total slips = $3 + 4 + 2 + 6 = 15$ Step 3: Determine the number of red slips. Number of red slips = 4 Step 4: Calculate the probability of drawing a red slip. $$P(\text{red}) = \frac{\text{Number of red slips}}{\text{Total number of slips}}$$ $$P(\text{red}) = \frac{4}{15}$$ The probability of it being a red slip is $\boxed{\frac{4}{15}}$. That's 2 down. 3 left today — send the next one.

36. Step 1: Convert each fraction to a decimal to compare them.