D. In a box, the number of squares is 5 times as much as the number of triangles. The number of circles is 4/5 of the number of squares.
1) Draw a model to compare the numbers of squares, triangles and circles.
2) What fraction of the number of circles is the number of triangles?
3) Find the ratio of the number of squares to the number of triangles to the number of circles.
4) In its simplest form, what is the ratio of the number of triangles and circles to the total number of shapes?
5) Express the number of circles as a fraction of the total number of shapes in its simplest form.
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Answer:
1) Model:
Let's represent squares with the letter "S," triangles with the letter "T," and circles with the letter "C."
Given information:
- The number of squares is 5 times the number of triangles.
- The number of circles is 4/5 of the number of squares.
Based on this information, we can create the following model:
S S S S S
S S S S S
S S S S S
S S S S S
2) Fraction of circles to triangles:
The number of circles is 4/5 of the number of squares. Since the number of squares is 5 times the number of triangles, the number of circles is (4/5) * 5 = 4 times the number of triangles.
To find the fraction of the number of circles to the number of triangles, we divide the number of circles by the number of triangles:
4/1 = 4
So, the fraction of the number of circles to the number of triangles is 4/1 or simply 4.
3) Ratio of squares, triangles, and circles:
The given information shows that:
- The number of squares is 5 times the number of triangles.
- The number of circles is 4/5 of the number of squares.
Let's assign variables:
Number of squares = S
Number of triangles = T
Number of circles = C
From the given information, we can write the following equations:
S = 5T (Equation 1)
C = (4/5)S (Equation 2)
To find the ratio between these quantities, we can express them in terms of a common variable. Let's rewrite Equation 2 using Equation 1:
C = (4/5)*(5T)
C = 4T
So, the ratio of the number of squares to the number of triangles to the number of circles is 5:1:4.
4) Ratio of triangles and circles to the total number of shapes:
To find the ratio of triangles and circles to the total number of shapes, we add the number of triangles and circles and divide it by the total number of shapes.
Total number of shapes = squares + triangles + circles
Total number of shapes = S + T + C
Using the previous equations (Equation 1 and Equation 2), we can substitute the values:
Total number of shapes = (5T) + T + (4T)
Total number of shapes = 10T
So, the ratio of triangles and circles to the total number of shapes is 1:4.
5) Fraction of circles to the total number of shapes:
To express the number of circles as a fraction of the total number of shapes, we divide the number of circles by the total number of shapes:
Fraction of circles = C / (S + T + C)
Substituting the values from the previous equations:
Fraction of circles = (4T) / (5T + T + 4T)
Fraction of circles = 4T / 10T
Fraction of circles = 4/10
Simplifying the fraction, we get 2/5.
So, the number of circles is expressed as a fraction of the total number of shapes in its simplest form as 2/5.
Step-by-step explanation:
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Verified answer
Let's break down the problem step by step and provide a detailed explanation for each question.
1) Drawing a model to compare the numbers of squares, triangles, and circles:
We can use simple shapes to represent each category. Let's use squares (S), triangles (T), and circles (C).
Given the information:
- Number of squares = 5 times the number of triangles.
- Number of circles = 4/5 of the number of squares.
We can start by assigning an arbitrary value to the number of triangles (T). Let's say we have 1 triangle:
Number of triangles (T) = 1
Number of squares (S) = 5 * Number of triangles = 5 * 1 = 5
Number of circles (C) = (4/5) * Number of squares = (4/5) * 5 = 4
In our model, we would represent 1 triangle (T), 5 squares (S), and 4 circles (C).
2) Finding the fraction of the number of circles compared to the number of triangles:
In our model, we have 4 circles and 1 triangle. Therefore, the fraction of the number of circles to the number of triangles is 4/1, which simplifies to 4.
3) Calculating the ratio of the number of squares, triangles, and circles:
In our model, we have 5 squares, 1 triangle, and 4 circles. Therefore, the ratio of squares to triangles to circles can be expressed as 5:1:4.
4) Simplifying the ratio of the number of triangles and circles to the total number of shapes:
To find this ratio, we add the number of triangles (1) and circles (4) and divide it by the total number of shapes (5+1+4 = 10). The ratio of triangles and circles to the total number of shapes is 5/10, which simplifies to 1/2.
5) Expressing the number of circles as a fraction of the total number of shapes:
In our model, the number of circles is 4, and the total number of shapes is 10. Therefore, the number of circles as a fraction of the total number of shapes is 4/10, which simplifies to 2/5 in its simplest form.
By following these explanations and calculations, you should now have a clear understanding of the relationships between the number of squares, triangles, and circles, as well as the fractions and ratios involved.
May this help.