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What is the name of the polygon with sum of interior angles 1080 degrees? Find the number of sides of a regular polygon if the sum of interior angles is 1080 degrees.

Mathematics

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Answers

4 parts
i

Octagon\text{Octagon}

a

524\frac{5}{24}

b

−1920 sh-1920 \text{ sh}

Step-by-step solution

11 steps

Part 35

Part i

Octagon\text{Octagon}

Give the name of the polygon.

The sum of interior angles of a polygon is given as 1080∘1080^\circ.
The formula for the sum of interior angles of a polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ.

  1. Set up the equation.

    (n−2)×180∘=1080∘(n-2) \times 180^\circ = 1080^\circ
  2. Solve for n−2n-2.

    n−2=1080180n-2 = \frac{1080}{180} n−2=6n-2 = 6
  3. Solve for nn.

    n=6+2n = 6 + 2 n=8n = 8

    A polygon with 8 sides is called an octagon.
    The name of the polygon is Octagon\boxed{\text{Octagon}}.

Part ii8

Find the number of sides of a regular polygon if the sum of interior angles is 1080∘1080^\circ.

From the calculation in part (i), we found that the number of sides, nn, is 8.
The number of sides is 8\boxed{8}.

Part 39

Ethan spent 13\frac{1}{3} of his salary on food, 14\frac{1}{4} on rent, 12\frac{1}{2} of the remainder on school fees and saved the rest.

Part a

524\frac{5}{24}

What fraction of the salary did he save?

Let the total salary be 1.

  1. Calculate the total fraction spent on food and rent.

    Fraction (food + rent)=13+14\text{Fraction (food + rent)} = \frac{1}{3} + \frac{1}{4} Fraction (food + rent)=412+312=712\text{Fraction (food + rent)} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}
  2. Calculate the remainder after spending on food and rent.

    Remainder=1−712=1212−712=512\text{Remainder} = 1 - \frac{7}{12} = \frac{12}{12} - \frac{7}{12} = \frac{5}{12}
  3. Calculate the fraction spent on school fees.

    He spent 12\frac{1}{2} of the remainder on school fees.

    Fraction (school fees)=12×512=524\text{Fraction (school fees)} = \frac{1}{2} \times \frac{5}{12} = \frac{5}{24}
  4. Calculate the fraction saved.

    The fraction saved is the remainder minus the fraction spent on school fees.

    Fraction saved=512−524\text{Fraction saved} = \frac{5}{12} - \frac{5}{24} Fraction saved=1024−524=524\text{Fraction saved} = \frac{10}{24} - \frac{5}{24} = \frac{5}{24}

    The fraction of the salary he saved is 524\boxed{\frac{5}{24}}.

Part b

−1920 sh-1920 \text{ sh}

If he saved sh 4800. How much more was spent on rent than on food?

  1. Use the fraction saved to find the total salary.

    Let SS be the total salary.

    524×S=4800\frac{5}{24} \times S = 4800 S=4800×245S = 4800 \times \frac{24}{5} S=960×24S = 960 \times 24 S=23040S = 23040

    Ethan's total salary is sh 23,040.

  2. Calculate the amount spent on rent.

    Amount (rent)=14×S\text{Amount (rent)} = \frac{1}{4} \times S Amount (rent)=14×23040\text{Amount (rent)} = \frac{1}{4} \times 23040 Amount (rent)=5760 sh\text{Amount (rent)} = 5760 \text{ sh}
  3. Calculate the amount spent on food.

    Amount (food)=13×S\text{Amount (food)} = \frac{1}{3} \times S Amount (food)=13×23040\text{Amount (food)} = \frac{1}{3} \times 23040 Amount (food)=7680 sh\text{Amount (food)} = 7680 \text{ sh}
  4. Calculate how much more was spent on rent than on food.

    Difference=Amount (rent)−Amount (food)\text{Difference} = \text{Amount (rent)} - \text{Amount (food)} Difference=5760−7680\text{Difference} = 5760 - 7680 Difference=−1920 sh\text{Difference} = -1920 \text{ sh}

    The negative sign indicates that sh 1920 less was spent on rent than on food.
    The amount is −1920 sh\boxed{-1920 \text{ sh}}.

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