Question

this is a 2-page documentdirections: write the ratio in simplest form.1. 28 elementary schools to 16 middle schools2. 30 treadmills to 36 elliptical machines3. 18 buses to 66 cars4. 180 red marbles to 145 blue marbles5. the hockey team played 82 regular-season games last year. if they won 44 games, what is the ratio of wins to losses?6. in the word flashlight what is the ratio of vowels to total letters?directions: use the given ratios to solve each problem.7. the ratio of the measures of two complementary angles is 7:8. what is the measure of the smaller angle?8. the ratio of the measures of the three angles in a triangle is 2:9:4. find the measures of the angles.9. the ratio of the measures of the three angles in a triangle is 10:3:7. find the measure of the largest angle.10. the ratio of the measure of the vertex angle to the base angle of an isosceles triangle is 8:5. find the measure of the vertex angle.11. the ratio of the measures of the sides of a triangle is 21:8:14. if the perimeter of the triangle is 215 feet, find the length of each side.12. the ratio of the measures of the sides of a triangle is 4:7:5. if the perimeter of the triangle is 128 yards, find the length of the longest side.

Explanation:

Step1: Simplify the ratio 28:16

Find GCD of 28 and 16, divide both by 4: $\frac{28}{4}:\frac{16}{4}=7:4$

Step2: Simplify the ratio 30:36

Find GCD of 30 and 36, divide both by 6: $\frac{30}{6}:\frac{36}{6}=5:6$

Step3: Simplify the ratio 18:66

Find GCD of 18 and 66, divide both by 6: $\frac{18}{6}:\frac{66}{6}=3:11$

Step4: Simplify the ratio 180:145

Find GCD of 180 and 145, divide both by 5: $\frac{180}{5}:\frac{145}{5}=36:29$

Step5: Calculate losses, simplify wins:losses

Losses: $82-44=38$. GCD of 44 and 38 is 2: $\frac{44}{2}:\frac{38}{2}=22:19$

Step6: Count vowels, total letters, simplify

Vowels (A,U) =2; total letters=9. Ratio: $2:9$

Step7: Find smaller complementary angle

Sum of parts: $7+8=15$. Smaller angle: $\frac{7}{15}\times90^\circ=42^\circ$

Step8: Find triangle angles

Sum of parts: $2+9+4=15$. Angles: $\frac{2}{15}\times180^\circ=24^\circ$, $\frac{9}{15}\times180^\circ=108^\circ$, $\frac{4}{15}\times180^\circ=48^\circ$

Step9: Find largest triangle angle

Sum of parts: $10+3+7=20$. Largest angle: $\frac{10}{20}\times180^\circ=90^\circ$

Step10: Find isosceles vertex angle

Sum of parts: $8+5+5=18$. Vertex angle: $\frac{8}{18}\times180^\circ=80^\circ$

Step11: Find triangle side lengths

Sum of parts: $21+8+14=43$. Scale factor: $\frac{215}{43}=5$. Sides: $21\times5=105$ ft, $8\times5=40$ ft, $14\times5=70$ ft

Step12: Find longest triangle side

Sum of parts: $4+7+5=16$. Scale factor: $\frac{128}{16}=8$. Longest side: $7\times8=56$ yards

Answer:

1. $7:4$
2. $5:6$
3. $3:11$
4. $36:29$
5. $22:19$
6. $2:9$
7. $42^\circ$
8. $24^\circ$, $108^\circ$, $48^\circ$
9. $90^\circ$
10. $80^\circ$
11. 105 feet, 40 feet, 70 feet
12. 56 yards