at the beginning of the month a song on a streaming service had 500 listens and was getting 35 new listens each day. Meanwhile a different song had 700 listens and was getting 15 new listens each day. When will the 2 songs have the same number of listens?

Answer 1

Let's define the number of listens for each song with respect to the number of days (let's call it \(d\)) that have passed since the beginning of the month.

For the first song:

• Initial listens: 500
• New listens per day: 35

The equation for the number of listens after \(d\) days is: \[ L_1 = 500 + 35d \]

For the second song:

• Initial listens: 700
• New listens per day: 15

The equation for the number of listens after \(d\) days is: \[ L_2 = 700 + 15d \]

To find when the two songs have the same number of listens, we set the two equations equal to each other: \[ 500 + 35d = 700 + 15d \]

Now, we’ll solve for \(d\):

1. Subtract \(15d\) from both sides: \[ 500 + 20d = 700 \]

2. Subtract 500 from both sides: \[ 20d = 200 \]

3. Divide both sides by 20: \[ d = 10 \]

This means that the two songs will have the same number of listens in 10 days.

To verify, we can calculate the total listens for each song after 10 days:

For the first song: \[ L_1 = 500 + 35 \times 10 = 500 + 350 = 850 \]

For the second song: \[ L_2 = 700 + 15 \times 10 = 700 + 150 = 850 \]

Both songs will have 850 listens after 10 days.