Sophia finds two landscape gardeners online. The first charges a fixed fee of 20 dollars per job plus 15 dollars per hour for labor, while the second charges a fixed fee of 90 dollars but only five dollars per hour for labor. After how many hours will the second gardener be cheaper than the first?
To solve this problem, we can actually think about it as two different equations and actually solving them simultaneously. Well, we can say that, actually, the first equation which is for our first gardener would be 𝐶, which we’re gonna have as the total cost, is equal to 20, because that’s the fixed fee, plus 15ℎ, where ℎ is the number of hours worked.
We can then set up a second equation where 𝐶 is equal to 90, because that’s our fixed fee for the second gardener, plus five ℎ, cause we know it’s five dollars per hour for labor for the second gardener. Okay, great! We’ve now got two equations. Okay, well we want to find out when the second gardener is gonna be cheaper than the first gardener. Well, in order to do this, we first wanna work out when they’ll actually be the same cost for this first gardener and the second gardener.
And to do that, we’re gonna make our equations equal to each other. So we get that 20 plus 15ℎ is equal to 90 plus five ℎ. So then we’re gonna subtract five ℎ from each side, which gives us 20 plus 10ℎ is equal to 90. Then we’re next gonna subtract 20 from each side, which gives us 10ℎ is equal to 70. And then our final step to find ℎ, we divide by 10 to give us ℎ is equal to seven. So that means that, after seven hours, they’re actually gonna have the same cost.
So therefore, we can say that, at any time after seven hours, the second gardener will be cheaper. And now we can quickly check that by substituting ℎ equals seven back into our equations. So for the first equation, we have 20 plus 15 multiplied by seven is gonna give us 𝐶, our total cost, which gives us 20 plus 105 equals 𝐶. So therefore, we can say that 𝐶 is equal to 125 dollars.
Then we can substitute it back into the second equation. So we get 90 plus five multiplied by seven equals 𝐶, which gives us 90 plus 35 equals 𝐶. So therefore, we also have that 𝐶 is equal to 125 dollars. So therefore, it’s checked and correct. We can say that yes seven hours is the correct answer.