Question

1. in an orchestra, the ratio of flutes to violas is 2:8. multiple orchestras are planning to combine to create a larger orchestra and they want to keep the ratio of instruments equivalent so the sound is balanced the same. complete the table to show how many flutes and violas would be needed.

flutes | violas
2 | 8
5 | ?
? | 72
? | 12
12 | ?

1. in an orchestra, the ratio of first violins to harps is 3:1. multiple orchestras are planning to combine to create a larger orchestra and they want to keep the ratio of instruments equivalent so the sound is balanced the same. complete the table to show how many first violins and harps would be needed.

first violins | harps
3 | 1
15 | ?
? | 8
? | 12
9 | ?

Explanation:

Problem 1 (Flutes to Violas, ratio 2:8)

The ratio of flutes to violas is \(2:8 = 1:4\) (simplified by dividing both by 2). So, violas = 4 × flutes, and flutes = violas ÷ 4.

Step 1: Flutes = 5

To find violas when flutes = 5:
Violas = \(5 \times 4 = 20\)

Step 2: Violas = 72

To find flutes when violas = 72:
Flutes = \(72 \div 4 = 18\)

Step 3: Violas = 12

To find flutes when violas = 12:
Flutes = \(12 \div 4 = 3\)

Step 4: Flutes = 12

To find violas when flutes = 12:
Violas = \(12 \times 4 = 48\)

Problem 2 (First Violins to Harps, ratio 3:1)

The ratio of first violins to harps is \(3:1\), so harps = first violins ÷ 3, and first violins = 3 × harps.

Step 1: First Violins = 15

To find harps when first violins = 15:
Harps = \(15 \div 3 = 5\)

Step 2: Harps = 8

To find first violins when harps = 8:
First Violins = \(8 \times 3 = 24\)

Step 3: Harps = 12

To find first violins when harps = 12:
First Violins = \(12 \times 3 = 36\)

Step 4: First Violins = 9

To find harps when first violins = 9:
Harps = \(9 \div 3 = 3\)

Final Tables:

Problem 1 Table (Flutes : Violas)

Flutes	Violas
5	20
18	72
3	12
12	48

Problem 2 Table (First Violins : Harps)

First Violins	Harps
15	5
24	8
36	12
9	3

Answer:

The ratio of first violins to harps is \(3:1\), so harps = first violins ÷ 3, and first violins = 3 × harps.

Step 1: First Violins = 15

To find harps when first violins = 15:
Harps = \(15 \div 3 = 5\)

Step 2: Harps = 8

To find first violins when harps = 8:
First Violins = \(8 \times 3 = 24\)

Step 3: Harps = 12

To find first violins when harps = 12:
First Violins = \(12 \times 3 = 36\)

Step 4: First Violins = 9

To find harps when first violins = 9:
Harps = \(9 \div 3 = 3\)

Final Tables:

Problem 1 Table (Flutes : Violas)

Flutes	Violas
5	20
18	72
3	12
12	48

Problem 2 Table (First Violins : Harps)

First Violins	Harps
15	5
24	8
36	12
9	3