Question

Ethan makes candles and sells them at a holiday market. He uses the equation



P = 10.5n − (4n + 25)



to determine his total profit, in dollars, when he sells n candles.



If Ethan wants to earn a profit of $100, how many candles must he sell?

Answer 1

To determine how many candles Ethan must sell to earn a profit of $100, we can start by setting the profit equation equal to 100. The equation given is:

\[
P = 10.5n - (4n + 25)
\]

We can simplify the equation first:

\[
P = 10.5n - 4n - 25
\]

Combining like terms gives us:

\[
P = (10.5n - 4n) - 25 = 6.5n - 25
\]

Now we set this equal to 100 to find the number of candles, \( n \):

\[
100 = 6.5n - 25
\]

Next, we add 25 to both sides:

\[
100 + 25 = 6.5n
\]

This simplifies to:

\[
125 = 6.5n
\]

Now, we can solve for \( n \) by dividing both sides by 6.5:

\[
n = \frac{125}{6.5}
\]

Calculating the division:

\[
n \approx 19.230769
\]

Since Ethan can't sell a fraction of a candle, we round up to the nearest whole number, so \( n = 20 \).

To verify, we can calculate the profit when he sells 20 candles:

\[
P = 6.5(20) - 25 = 130 - 25 = 105
\]

And for 19 candles:

\[
P = 6.5(19) - 25 = 123.5 - 25 = 98.5
\]

Since selling 19 candles gives a profit less than $100, but selling 20 candles gives a profit greater than $100, the minimum number of candles Ethan must sell to earn at least $100 in profit is:

\[
\boxed{20}
\]