Often when solving word problems, we are required to extract information and then use that information to solve some sort of equation. I like to use the following steps to make sure I never make a mistake when reading tricky problems. It is especially useful to figure out what information each phrase of the word problem gives us.
1) Identify Variables
The problem states that Amanda has two different tutoring rates. A rate is a unit divided by some unit of time. For example, miles per hour is a rate. The question asks "what is the least number of hours she must tutor to make more money..." which tells us that the unit in our rate is money and the unit of time is hours. We can represent these two variables as m and h, respectively.
2) Identify Equations
In word problems, the equations that we need will often be presented as tricky sentences. In this case, the first equation is given to us as "Rate A is a flat fee of $10 plus $10 per hour". The " flat fee" phrase indicates that this value will stand alone. The "$10 per hour" phrase indicates that this value must be multiplied by the total number of hours. We can write this equation as:
m_A = 10 + 10*h
where m_A and h are our variables for money and time, respectively, for rate A. The second equation is given to us as "Rate B is $12 per hour". This tells us that the value of $12 is multiplied by the total number of hours. Note that there is no "flat fee" in this equation. We can write this equation as:
m_B = 12*h
where m_B and h are our variables for money and time, respectively, for rate B. We have different variables for the money earned under each rate. This is because for the same hours worked, each rate will earn Amanda a different amount of money. However, we do not have different variables for the time under each rate. This is because regardless of which rate Amanda is using, one hour is still one hour.
3 Identify What to Solve For
Now that we have our variables and equations, we need to figure out what the problem wants us to solve for. This can be found in the question "what is the least number of hours she must tutor to make more money using rate B?". We want to know how many hours Amanda has to tutor for rate B to earn more money than rate A. The phrase 'least number' indicates that we probably want to use a 'greater than' inequality.
To solve for this value of time, we will need to set up the following inequality:
m_B > m_A
which can be rewritten as:
12*h > 10 + 10*h
This inequality allows us to solve for the minimum value of time at which rate B earns Amanda more money than rate A. Solving the inequality, we find the expression 2*h > 10, which leads us to the expression h > 5.
This solution tells us that in order for Amanda to earn more money using rate B, she must tutor at least 6 hours, since the number of hours must be greater than 5.
Hope this helps!