# Tutor profile: Hanh L.

## Questions

### Subject: Geometry

a) Given a triangle has a total interior angle of 180$$^{\circ}$$, what is the total interior angle of a pentagon? b) An irregular pentagon has 3 corners with an angle of 100$$^{\circ}$$ each and corner A and corner B with an angle of $$x$$ and $$2x$$ respectively. What are the interior angles of corner A and corner B?

Answer: a) $$540^{\circ}$$ b) Corner A = $$80^{\circ}$$ Corner B = $$160^{\circ}$$ -------------------------------------------------------------------- Short explanation: a) Using the formula for the total interior angle of any 2d polygon, $$(n - 2) * 180^{\circ}$$, we can work this out. As it's a pentagon, it has five sides so $$n = 5$$. With this, we substitute in $$n = 5$$ into the formula and we get, $$(n - 2) * 180^{\circ}$$ $$(5 - 2) * 180^{\circ}$$ $$3 * 180^{\circ}$$ $$540^{\circ}$$ b) The sum of the three known angles in the pentagon is $$300^{\circ}$$ ($$100^{\circ} * 3$$), this leaves $$240^{\circ}$$ for the remaining two angles. The sum of these two angles is $$x + 2x$$ which gives us $$3x$$. Using the information we currently have, we can set up an equation relating the two to solve for $$x$$. $$3x = 240^{\circ}$$ $$x = 80^{\circ}$$ As corner A is just $$x$$, corner A is simply $$80^{\circ}$$. Corner B however is $$2x$$, so corner B is $$2 * 80^{\circ}$$ which is $$160^{\circ}$$. Long explanation: a) As you might know, a triangle has a total interior angle of $$180^{\circ}$$ and the total interior angle of a square (or any four-sided 2d shape) is $$360^{\circ}$$. If you cut a square in half (corner to corner), you'll see that a square is simply two triangles. And if a single triangle is $$180^{\circ}$$, then it would be right to assume that two triangles are just $$180^{\circ} + 180^{\circ}$$ which is $$360^{\circ}$$. Using this idea, very people clever generalized it into the formula $$(n - 2) * 180^{\circ}$$. And whilst it is fine to use the formula, you can also do it intuitively by drawing lines corner to corner on shapes (as long as they don't overlap) and count the number of triangles each shape has. If you do this to a pentagon, you'll see it has three triangles inside it, using the logic from two paragraphs ago, you simply add the angles of the three triangles together to get $$540^{\circ}$$. b) We know that 3 of the angles are $$100^{\circ}$$ each which means that we have $$300^{\circ}$$ of the total $$540^{\circ}$$ accounted for. The sum of the remaining two angles, therefore, is $$240^{\circ}$$. As corner A has an angle of $$x$$ and corner B has an angle of $$2x$$, they have a combined angle of $$3x$$. With this information, we can set up an equation relating $$3x$$ and $$240^{\circ}$$ together in the following form: $$3x = 240^{\circ}$$ $$x = 80^{\circ}$$ However we are not done with this, corner B has an angle of $$2x$$, and as we just solved what $$x$$ was, we can work out that corner B is just $$2 * x$$ or in other words $$2 * 80^{\circ}$$ which is $$160^{\circ}$$. Corner A has an angle of just $$x$$ so we can just use the answer we initially worked out for $$x$$ which is just $$80^{\circ}$$.

### Subject: Basic Math

Alex is playing a board game with her friend Timothy. She is currently 2 spaces away from the finish line and is about to roll a six-sided die. What are the chances of her winning as a fraction?

Answer: $$\frac{5}{6}$$ ------------------------------------------------- Short explanation: If Alex rolls anything but a 1, she wins and as there are five other numbers she could possibly roll for, she has a 5 in 6 chance of winning or a $$\frac{5}{6}$$ chance to win. Long explanation: As there are only 2 spaces left until Alex wins, Alex just needs to roll a 2 or higher on the die (the same as not rolling a 1). Now there are five possible numbers Alex could roll on the die that would let her win, these being 2, 3, 4, 5, and 6. However, there are six possible numbers Alex could roll on the six-sided die, these being 1, 2, 3, 4, 5, and 6. The questions require us to give the answer as a fraction. There are two parts to a fraction, these being the numerator (the number on the top) and the denominator (the number that goes on the bottom). So how do we know which number to use and in which half they go? The number that goes into the denominator is the total number of outcomes that exist. In this scenario, we do not count winning and losing as the two outcomes because whilst there exists only one way to lose, there are five ways to win (rolling a 2 to win, 3 to win, etc.). This means there are six different outcomes so the denominator becomes a 6. The number that goes into the numerator is the total number of the desired outcomes that exist, the desired outcome was stated in the question, this being Alex's chances of winning. As stated before, there are five outcomes in which Alex wins (rolling a 2 to win, 3 to win, etc.). So the numerator becomes a 5. Put together, the answer is $$\frac{5}{6}$$.

### Subject: Algebra

Seraa visits his friends and brings along with him a bag of snacks. He plans to keep 3 snacks for himself and give 2 to his friends. a) Express how many snacks Seraa would need to bring along with him as an equation. b) Seraa's friend told him that there will only be five people present (including Seraa), how many snacks does he need to bring along?

Answer: a) Let snacks be $$S$$ and total snacks be $$T$$ $$2S + 3 = T $$ b) Using the equation that we wrote in the previous question, $$2S + 3 = T $$ Substitute $$S = 4$$ $$(2 * 4) + 3 = T $$ $$8 + 3 = T $$ $$ 11 = T$$ Seraa would need to bring with him 11 snacks. ---------------------------------------------------------------------------------------- Short explanation: a) We don't know how snacks Serra's friends need or the total amount of snacks he needs to bring along so we assign both of these unknown a variable $$S$$ and $$T$$. $$S$$ will be snacks for his friends and $$T$$ is total snacks. If he has one friend there, he needs to bring 2 snacks, if he has two friends, then he needs to bring along 4 snacks. (Along with the additional 3 for himself). As the amount of snacks increases by 2 per friend, we can say $$2S$$ instead of $$S$$. As we stated earlier, $$T$$ is the total snacks needed and that's equal to $$2S + 3$$ which gives us the equation $$2S + 3 = T$$. b) In section a), we made an equation to help figure out how many snacks Seraa needs to bring along, The equation was $$2S + 3 = T$$ and the question gives us a number for the variable $$S$$, that being 4 (the number of people present is 5 but Seraa doesn't count because the variable $$S$$ is only for his friends and not him). So substitute 4 into the equation and we get the following: $$2S + 3 = T $$ Substitute $$S = 4$$ $$(2 * 4) + 3 = T $$ $$8 + 3 = T $$ $$ 11 = T$$ As $$T$$ is the total amount of snacks needed, and the equation states $$T = 11$$, that means Seraa needs to bring 11 snacks along with him. ---------------------------------------------------------------------------------------- Long explanation: a) In the question, we can see there are two unknown variables present; the number of snacks Seraa needs to bring for his friends, and the total number of snacks he needs to bring all together. Seraa currently only knows to bring 3 snacks for himself, he does not know how many snacks he needs to bring for his friend making it the first variable. As a result, he also does not know how many snacks he needs to bring along in total making this the second variable. Now we need to place these two variables into a question but the question does not state what to call each variable so we can make it up. So I chose $$S$$ to be snacks and $$T$$ to be total snacks however it is perfectly reasonable to choose $$z$$ to be snacks and $$Δ$$ to be total snacks as long as it remains consistent throughout answering the questions. As it remains, his friends currently receive $$S$$ amount of snacks each, but the question states his friends get 2 snacks each so instead of $$S$$, we need to put a 2 in front of it to give us $$2S$$. With the two unknown variables accounted for, we can include all of them (including the 3 snacks Seraa needs for himself) into one equation. The complication lies in which variable goes where however we can think of it intuitively. One of the unknown variables is the total amount of snacks required, to get the total you need to add Seraa's 3 snacks and the total amount of snacks his friends need. So we add Seraa's 3 snacks and the number of snacks his friends need (that being $$2S$$) to give us $$2S + 3$$. This is then equal to the total amount of snacks (that being $$T$$) which gives us the final equation $$2S + 3 = T$$. b) This question can be answered it two ways, the first requires you to use the equation we made in the previous question, this being $$2S + 3 = T$$ and the other is a more intuitive approach. To help answer the question, when reading the question, it states that 5 people will be present at Seraa's friend's house including Seraa himself. In the first method, we can simply plug that number 5 into the equation and get an answer; however, it will be an incorrect answer because we shouldn't be using the number 5. If we look back at our equation, the $$2S$$ is only meant for Seraa's friend, and Seraa himself gets 3 snacks rather than the 2 that his friends get. Knowing this, we can assert that Seraa only has 4 friends at his friend's house. Now using the equation $$2S + 3 = T$$, we replace the $$S$$ with 4 to get $$2(4) + 3 = T$$ Whenever a number is directly next to a bracketed number, it means we have to multiply the two, this equation would then look like the following $$(2 * 4) + 3 = T$$. Now we can solve the equation. $$8 + 3 = T$$ $$11 = T$$ As $$T$$ is the total number of snacks Seraa needs to bring along and the equation says $$T$$ is equal to 11, the total number of snacks Seraa needs to bring along is 11.