What are two possible combinations of eighth notes and quarter notes that they could use to make a bar that's 4 beats long?

Steps to Solve

1. Understand the note values

A quarter note is worth 1 beat, while an eighth note is worth $\frac{1}{2}$ beat. The goal is to create a combination of these notes that totals 4 beats.

2. Set up the equation

Let ( x ) represent the number of quarter notes, and ( y ) represent the number of eighth notes. The total length in beats can be expressed as:

$$ 1x + \frac{1}{2}y = 4 $$

3. Find integer combinations

To simplify the equation, multiply everything by 2 to eliminate the fraction:

$$ 2x + y = 8 $$

Now, we can find integer combinations of ( x ) (quarter notes) and ( y ) (eighth notes).

4. Generate combinations

We can choose values for ( x ) (from 0 to 4) and calculate ( y ):

• If ( x = 0 ): ( y = 8 ) (8 eighth notes)
• If ( x = 1 ): ( y = 6 ) (1 quarter note + 6 eighth notes)
• If ( x = 2 ): ( y = 4 ) (2 quarter notes + 4 eighth notes)
• If ( x = 3 ): ( y = 2 ) (3 quarter notes + 2 eighth notes)
• If ( x = 4 ): ( y = 0 ) (4 quarter notes)

5. Select two combinations

From the above possible combinations:

• Combination 1: 0 quarter notes and 8 eighth notes
• Combination 2: 2 quarter notes and 4 eighth notes