Two picture frames are similar. The ratio of the perimeters of the two pieces is 3:5. If the area of the smaller frame is 108 square inches. what is the area of the larger frame?
the areas are in the ratio 3^3:5^2 = 9:25 = 108:300
300
To find the area of the larger frame, we need to determine the ratio of their areas.
First, let's find the ratio of the sides of the two frames. Since the ratio of the perimeters is 3:5, we can assume that the ratio of the lengths of corresponding sides is also 3:5.
Let's denote the length of the smaller frame as x. Therefore, the length of the larger frame is (5/3)x, as per the ratio.
The area of a rectangle is given by length multiplied by width.
So, the area of the smaller frame is x * y = 108 square inches, where y denotes the width of the smaller frame.
Similarly, the area of the larger frame is (5/3)x * y = (5/3)xy.
To find the area of the larger frame, we need the value of xy.
Dividing the area equation for the smaller frame by the area equation for the larger frame, we get:
(x * y) / ((5/3)x * y) = 108 / (5/3)xy
Cancelling out the xy terms, we are left with:
1 / (5/3) = 108 / (5/3)
To find the value of 1 / (5/3), we can multiply the numerator and denominator by the reciprocal of the denominator:
1 / (5/3) = 1 * (3/5) = 3/5
Therefore, 3/5 = 108 / (5/3)
We can now solve for xy:
3/5 = 108 / (5/3)
Cross-multiplying, we get:
(3/5) * 108 = (5/3) * xy
Simplifying:
(3 * 108) / 5 = (5/3) * xy
324 / 5 = (5/3) * xy
Multiplying both sides by 5/3:
(5/3) * (324 / 5) = (5/3) * (5/3) * xy
324 / 3 = xy
108 = xy
So, the area of the larger frame is 108 square inches.
To find the area of the larger frame, we need to determine the ratio of their areas and then use the given area of the smaller frame.
Since the ratio of the perimeters is 3:5, we can assume that the ratio of the sides is also 3:5. Let's say the sides of the smaller frame are 3x and 3y, and the sides of the larger frame are 5x and 5y.
The formula for the area of a rectangle is length × width. Therefore, the area of the smaller frame is (3x) × (3y) = 9xy = 108 square inches.
Now, we can solve for xy by dividing both sides of the equation by 9: xy = 108 / 9 = 12.
Since the sides of the larger frame are 5x and 5y, its area will be (5x) × (5y) = 25xy = 25 × 12 = 300 square inches.
Therefore, the area of the larger frame is 300 square inches.