# Tutor profile: Noah M.

## Questions

### Subject: Pre-Algebra

The top three countries in oil consumption in a certain year are as follows: the United States, China, and Japan. In millions of barrels per day, the three top countries consumed 32.2 million barrels of the world’s oil. The United States consumes 12 million more barrels a day than China. China consumes 3.2 million barrels a day than Japan. How many barrels of world oil consumption did the United States, Japan, and China consume?

For this question, I'm going to ignore the fact that all these numbers are in millions of barrels per day. For now, I'm just going to look at the numbers they give us: 1) United States, Japan, and China consume 32.2 total 2) The United States consumes 12 more than China 3) China consumes 3.2 more than Japan I'm going to assign a variable to represent the quantity that each country consumes: $$U$$ = United States, $$C$$ = China, $$J$$ = Japan. In order to figure out how much the United States consumes, we need to know how much China consumes. In order to figure out how much China consumes, we need to know how much Japan consumes. However, they don't tell us how much Japan consumes. If we can find $$J$$, we can find the other two. To solve for $$J$$ we need to turn our information into equations: 1) $$U + J + C = 32.2$$ 2) $$U = C + 12$$ 3) $$C = J + 3.2$$ If $$C = J + 3.2$$, then we can just replace $$J + 3.2$$ with $$C$$ in the second equation to get $$U = (J + 3.2) + 12$$. Once again, we can do the same thing with $$U$$ and $$C$$ in the first equation. This will give us $$([J + 3.2] + 12) + J + (J + 3.2) = 32.2$$. If we simplify this equation we can solve for $$J$$: $$\qquad ([J + 3.2] + 12) + J + (J + 3.2) = 32.2 \implies (J + 15.2) + J + (J + 3.2) = 32.2 \implies 3J + 18.4 = 32.2 \implies 3J = 13.8 \implies J = 4.6$$ Now that we know that $$J = 4.6$$, we can replace 4.6 with $$J$$ in our first and second equations to find the value of $$C$$ and then $$U$$: $$\qquad C = J + 3.2 \implies C = (4.6) + 3.2 \implies C = 7.8$$ $$\qquad U = (7.8) + 12 \implies U = 19.8$$ Now we have our values for $$U, C$$, and $$J$$. Don't forget that these numbers represent millions of barrels per day. So our final answer is: The United States consumes 19.8 million barrels of oil per day, Japan consumes 4.6 million barrels of oil per day, and China consumes 7.8 million barrels of oil per day.

### Subject: Basic Math

On a random Monday in the spring, 95% of students are present at school. If there are 589 students present, then how many students are absent on that Monday?

This is a tricky question because there are multiple steps involved to find the correct answer. The first thing you should do with word problems like this is list everything you know: 1) 95% of all the students are present 2) 589 students are present 3) It's a Monday in the spring. Now we have to use this information to find how many students are absent. But we don't know the total number of students that should be there. So before we can find out how many are absent, we need to find out how many students would be there if everyone was present. Remember that percentages are just fancy fractions: $$95\% = \frac{95}{100} = 0.95$$. We're trying to find how many students there are if 100% are present: $$100\% = \frac{100}{100} = 1$$. We know that $$95\% = 589$$ students. Therefore, if we can use this equation to turn the 95% into 100%, then we can find out how many students 100% is equal to. $$\qquad 95\% = 589 \implies 0.95 = 589 \implies \frac{0.95}{0.95} = \frac{589}{0.95} \implies 1 = 620 \implies 100\% = 620$$ Now that we know the total number of students that should be there are 620 students. It's a simple subtraction problem to find out how many students are absent on Monday. $$620 - 589 = 31$$ students absent.

### Subject: Algebra

Unleaded gas and supreme gas sell for different prices per liter. Suppose that 2 liters of unleaded gas and 3 liters of supreme gas cost a total of $2.52. Five liters of unleaded gas and 4 liters of supreme gas cost a total of $4.48. $$\qquad$$ 1) What is the price per liter of each kind of gas? $$\qquad$$ 2) If Ted filled his pickup with 30 liters of unleaded gas and also filled his car with 25 liters of supreme gas, how much would he have to pay for the gas? Assume prices have not changed.

This is a two-part question. So let's focus on the first part: "What is the price per liter of each kind of gas?" There are two types of gas in this problem: unleaded gas and supreme gas. Since we don't currently know the price for the two gases, I'm going to assign a variable to each one: $$U$$ = price of unleaded, $$S$$ = price of supreme. Now let's see if we can rewrite the information we're given into equations: "Suppose that 2 liters of unleaded gas and 3 liters of supreme gas cost a total of $2.52." $$\qquad 2 \cdot U + 3 \cdot S = 2.52$$ "Five liters of unleaded gas and 4 liters of supreme gas cost a total of $4.48" $$\qquad 5 \cdot U + 4 \cdot S = 4.48$$ Now we have two equations and two unknown variables. There are many methods you can use to solve this. I'm going to use substitution: $$\qquad 2 \cdot U + 3 \cdot S = 2.52 \implies 2 \cdot U = 2.52 - 3 \cdot S \implies U = 1.26 - \frac{3 \cdot S}{2}$$ $$\qquad 5 \cdot U + 4 \cdot S = 4.48 \implies 5 \cdot (1.26 - \frac{3 \cdot S}{2}) + 4 \cdot S = 4.48 \implies (6.30 - \frac{15 \cdot S}{2}) + 4 \cdot S = 4.48 \implies 6.30 - \frac{15 \cdot S}{2} + \frac{8 \cdot S}{2} = 4.48 \implies 6.30 - \frac{7 \cdot S}{2} = 4.48 \implies -\frac{7 \cdot S}{2} = -1.82 \implies S = 0.52$$ $$\qquad U = 1.26 - \frac{3 \cdot S}{2} \implies U = 1.26 - \frac{3}{2} \cdot (0.52) \implies U = 1.26 - (0.78) \implies U = 0.48$$ Now we have found the value for our (previously) unknown variables. Remember those variables represented the price of the two gases. So the answer to the first part is Unleaded gas = $0.48 per liter, Supreme gas = $0.52 per liter. On to the second part: "If Ted filled his pickup with 30 liters of unleaded gas and also filled his car with 25 liters of supreme gas, how much would he have to pay for the gas? Assume prices have not changed." Once again, I'm going to make an equation out of this sentence using the same variables as before: $$\qquad 30 \cdot U + 25 \cdot S = ?$$ Since the prices have not changed, we can use our answer from the first part $$U = 0.48, S = 0.52$$. If we plug those numbers into the equation it should tell us how much Ted has to pay for the gas: $$\qquad 30 \cdot U + 25 \cdot S = 30 \cdot (0.48) + 25 \cdot (0.52) = (14.40) + (13.00) = 27.40$$ And there we go, problem solved. Ted has to pay $27.40 for the gas.