Question

1. what is the ratio of pentagons to stars?

(pentagon, star, pentagon, pentagon, pentagon, pentagon, pentagon, pentagon)
(pentagon, pentagon, pentagon, star, pentagon)

1. what is the ratio of circles to moons?

(circle, moon, moon, circle, circle, moon, moon, circle)
(circle, circle, circle, moon, moon, circle, moon, circle)
(circle, circle, moon, moon, moon, circle, circle, circle)
(moon, circle)

1. what is the ratio of stars to moons?

(moon, moon, star, moon, star, moon, moon, star)
(star)

1. what is the ratio of triangles to pentagons?

(triangle, triangle, pentagon, triangle, triangle, triangle, triangle, triangle)
(triangle, triangle, triangle, pentagon, pentagon, triangle)

Explanation:

Question 8:

Step1: Count pentagons

Looking at the figures for question 8, count the pentagons. First row: 7 pentagons (excluding the star), second row: 4 pentagons (excluding the star). Total pentagons: \(7 + 4= 11\)? Wait, no, let's re - count. First row: shapes are pentagon, star, pentagon, pentagon, pentagon, pentagon, pentagon, pentagon (that's 7 pentagons and 1 star). Second row: pentagon, pentagon, pentagon, star, pentagon (that's 4 pentagons and 1 star). So total pentagons: \(7 + 4=11\)? Wait, no, first row: 8 shapes, 1 star, so 7 pentagons. Second row: 5 shapes, 1 star, so 4 pentagons. Total pentagons: \(7 + 4 = 11\). Total stars: \(1+1 = 2\). Wait, no, first row has 1 star, second row has 1 star, so total stars \(n_{stars}=2\). Total pentagons \(n_{pentagons}=7 + 4=11\)? Wait, no, let's list them:

First row (left to right): pentagon (1), star (1), pentagon (2), pentagon (3), pentagon (4), pentagon (5), pentagon (6), pentagon (7)

Second row: pentagon (8), pentagon (9), pentagon (10), star (2), pentagon (11)

Yes, so pentagons: 11, stars: 2.

Step2: Find the ratio

The ratio of pentagons to stars is \(\frac{n_{pentagons}}{n_{stars}}=\frac{11}{2}\)

Step1: Count circles and moons

For question 9, count the circles (O) and moons (C). Let's count row by row.

First row: O, C, C, O, O, C, C, O (circles: 4, moons: 4)

Second row: O, O, O, C, C, O, C, O (circles: 5, moons: 3)

Third row: O, O, C, C, C, O, O, O (circles: 5, moons: 3)

Fourth row: C, O (circles: 1, moons: 1)

Now sum circles: \(4 + 5+5 + 1=15\)

Sum moons: \(4+3 + 3+1 = 11\)? Wait, no, let's count again.

First row: O (1), C (1), C (2), O (2), O (3), C (3), C (4), O (4) → circles: 4, moons: 4

Second row: O (5), O (6), O (7), C (5), C (6), O (8), C (7), O (9) → circles: 5, moons: 3

Third row: O (10), O (11), C (8), C (9), C (10), O (12), O (13), O (14) → circles: 5, moons: 3

Fourth row: C (11), O (15) → circles: 1, moons: 1

Total circles: \(4 + 5+5 + 1=15\)

Total moons: \(4+3 + 3+1 = 11\)? Wait, no, 4 (first row) + 3 (second) + 3 (third) + 1 (fourth) = 11. So ratio of circles to moons is \(\frac{15}{11}\) (or 15:11)

Step1: Count stars and moons

For question 10, count stars and moons. The shapes are: moon, moon, star, moon, star, moon, moon, star (first row), and star (second row).

Count stars: Let's list them. First row: star (1), star (2), star (3). Second row: star (4). So total stars \(n_{stars}=4\)

Count moons: First row: moon (1), moon (2), moon (3), moon (4), moon (5), moon (6). So total moons \(n_{moons}=6\)

Step2: Find the ratio

The ratio of stars to moons is \(\frac{n_{stars}}{n_{moons}}=\frac{4}{6}=\frac{2}{3}\) (simplified)

Answer:

\(\frac{11}{2}\) (or 11:2)

Question 9: