Natalie earns $2.50 for each CD she sells and $3.50 for each DVD she sells. Natalie sold 45 DVDs last year. She earned a total of $780 last year selling CDs and DVDs.

Write an equation that can be used to determine the number of CDs (c) Natalie sold last year. How many CDs did Natalie sell last year? Show all work.

Natalie claims that she can earn more money this year by selling only 1/3 the number of CDs that she sold last year and by tripling the amount of DVDs she sold last year. Is Natalie's claim correct? Use words, numbers, and/or symbols to support your answer.

number of CD's --- x

number of DVDs --- 45

2.5x + 3.5(45) = 780
times 10
25x + 35(45) = 7800
25x + 1575 = 7800
25x = 6225
x = 249

she sold 249 CDs last year

her new plan:
number of DVDs she plans to sell = 135
number of CDs she plans to sell = (1/3)(249) = 83

income: 2.5(83) + 135(3.5) = 680

Looks like Natalie skipped a few math classes

To answer this question, let's start by writing the equation that can be used to determine the number of CDs (c) Natalie sold last year.

We know that Natalie earns $2.50 for each CD she sells and $3.50 for each DVD she sells.

Let's say the number of CDs Natalie sold last year is denoted by c.

The amount she earned from selling CDs would be $2.50 multiplied by the number of CDs sold, which gives us 2.50c.

The number of DVDs Natalie sold last year is already given as 45.

The amount she earned from selling DVDs would be $3.50 multiplied by the number of DVDs sold, which gives us 3.50 × 45 = $157.50.

We also know that Natalie earned a total of $780 last year selling CDs and DVDs.

So, the equation can be written as:

2.50c + 157.50 = 780.

Now, let's solve this equation to find the value of c, which represents the number of CDs Natalie sold last year.

Subtracting 157.50 from both sides of the equation, we have:

2.50c = 780 - 157.50.

Simplifying the right side of the equation, we get:

2.50c = 622.50.

Now, let's divide both sides of the equation by 2.50 to solve for c:

c = 622.50 ÷ 2.50.

Evaluating this division, we find:

c = 249.

Therefore, Natalie sold 249 CDs last year.

Now let's address Natalie's claim for this year:

Natalie claims that she can earn more money this year by selling only 1/3 of the CDs she sold last year (which would be 1/3 × 249) and tripling the amount of DVDs she sold last year (which would be 3 × 45).

Comparing the earnings from selling CDs in both years, last year she earned $2.50 for each of the 249 CDs sold. This year, if she sells only 1/3 of the CDs, she would earn $2.50 for each of 1/3 × 249 CDs sold.

Comparing the earnings from selling DVDs in both years, last year she earned $3.50 for each of the 45 DVDs sold. This year, if she triples the amount of DVDs sold, she would earn $3.50 for each of 3 × 45 DVDs sold.

To determine if her claim is correct, we need to compare the total earnings from CDs and DVDs in both years.

Last year's total earnings were $780. To calculate this year's total earnings, we sum the new earnings from CDs and DVDs:

New earnings = (earnings from CDs this year) + (earnings from DVDs this year).

Earnings from CDs this year = ($2.50/CD) × (1/3 × 249 CDs).

Earnings from DVDs this year = ($3.50/DVD) × (3 × 45 DVDs).

Calculating these values, we find:

Earnings from CDs this year = $2.50/CD × (1/3 × 249 CDs) = $2.50 × (83 CDs) = $207.50.

Earnings from DVDs this year = $3.50/DVD × (3 × 45 DVDs) = $3.50 × (135 DVDs) = $472.50.

Adding the earnings from CDs and DVDs, we have:

New earnings = $207.50 + $472.50 = $680.

Since $680 is less than the $780 Natalie earned last year, her claim that she can earn more money this year is not correct.

In conclusion, Natalie sold 249 CDs last year, and her claim to earn more money this year by selling 1/3 the CDs and tripling the DVDs is not correct.