Christabel A.

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Pre-Calculus

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Question:

The product of two consecutive even integers is 16 more than 8 times the smaller integer. Determine the integers.

Christabel A.

Answer:

This question seems confusing at first, but once you break it down you'll see that it's not so bad. The first thing we do is gather information. The question mentions "the product of two consecutive even integers". Therefore, we know that we have 2 variables: $$x$$ $$y$$ We also know that x and y are *consecutive*, *even* integers. For example, 6 and 8. This means that whatever the first number is, the second one will be two larger. Therefore: $$y = x+2$$ For example, if $$x$$ was $$6$$ then $$y$$ would be $$6+2 = 8$$. Now, the product of these two integers is 16 more than 8 times the smaller integer. Whew! Let's try to decompose this sentence. First, "the product of these 2 integers is" means $$x \times y=$$ "16 more" means we're adding 16 to one side $$x \times y= 16$$ And finally 8 times the smaller integer. Which one is smaller? When we defined our variables we said that $$y = 2x$$. If $$x=6$$ then $$y=6+2=8$$. Since we need to *add* to $$x$$ in order to get $$y$$, $$x$$ is the smaller integer. (Note: if we had defined our variables as $$x=y+2$$ or $$y=x-2$$ then $$y$$ would be the smaller integer but it wouldn't affect our final answer. What matters is that we're consistent with our variable definition throughout the entire question). Our equation now looks like: $$x\times y = 16 + 8x$$ Let's go ahead and sub in $$x+2$$ for $$y$$ so we can solve this as a quadratic equation. $$x(x+2) = 16 + 8x$$ This is starting to look familiar, right? Let's expand the left side $$x^2 + 2x = 16 + 8x$$ Then we move everything over to the left side by subtracting from both sides $$x^2 + 2x - 8x - 16 = 16 + 8x - 16 - 8x$$ And combine all the 'like' terms $$x^2 -6x - 16 = 0$$ Woohoo! A quadratic equation in standard form! We can solve this by factoring or using a graphing calculator. If we were to factor, we need 2 numbers that when multiplied together equal $$-16$$ and when added together equal $$-6$$. In this case, $$-8$$ and $$+2$$. Our factored equation is $$(x-8)(x+2)=0$$ So our potential values of $$x$$ are $$8$$ or $$-2$$. Going back to the original question, it mentions "integers" but doesn't specify if they're positive integers, so those are both valid values. The last step is to figure out $$y$$ $$y=x+2$$ $$x=8, y=8+2=10$$ $$x=-2, y=-2+2=0$$ And our final solution is: $$(-2,0)$$ or $$(8,10)$$ Tada!

Pre-Algebra

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Question:

Which is smaller, $$\frac{1}{3}$$ or $$\frac{1}{4}$$?

Christabel A.

Answer:

A fraction is made of 2 parts, the numerator (the number above the line) and the denominator (the number below the line). "Denominator" starts with "d" so it might help to tell yourself "the denominator is *down*". Fractions represent parts of a whole. For example, one pizza might be cut up into 6 slices. A fraction tells us how many parts we are talking about. In the example of our pizza, we are talking about *1* slice out of a total of *6* slices that make up a *whole* pizza. In this case, each slice is $$\frac{1}{6}$$ of the entire pizza. In our question, we're asking if $$\frac{1}{3}$$ is smaller or $$\frac{1}{4}$$ is. If you have two pizzas that are the same size but one pizza needs to be shared between 3 people and one pizza needs to be shared with 4 people, which pizza has smaller slices? The one being shared between 3 people, of course! The more people we have to share with, the smaller each slice has to be and the less amount of pizza each person gets for themselves. Therefore, when we're comparing fractions with the same *numerator* (in this case everyone gets 1 slice of pizza so the numerator is 1), the one with the bigger denominator is smaller (because we're cutting the pizza into smaller slices).

Basic Math

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Question:

Tanya won some money in a competition. She has two choices as to how she gets paid. Choice 1: $20 per week for one year Choice 2: $400 cash now plus $12 per week for one year Which method would pay Tanya more money? For what reasons might Tanya choose each method of payment?

Christabel A.

Answer:

The question asks us which method would pay Tanya more money. Since both options pay her over the course of a year, we need to find out how much money Tanya will have at the end of one year. Choice 1: $20 per week for one year There are 52 weeks in a year. Tanya's winnings look like this: Week 1: 20 Week 2: 20 + 20 = 40 Week 3: 40 + 20 = 60 ... all the way until week 52. We know that when we're adding the same number multiple times we can do it faster using multiplication. In this case: 52 weeks per year * $20 per week = $1040 per year Therefore, choice 1 gives Tanya $1040 by the end of the year Choice 2: $400 cash now plus $12 per week for one year This option gives Tanya some money right away and then a smaller amount of money over the course of the year. We can use the same method as above to understand how much money Tanya will get from her weekly payout. 52 weeks per year * $12 per week = $624 per year. She also gets $400 right away, so $624 + $400 = 1024 Therefore, choice 2 gives Tanya $1024 at the end of the year. Which method would pay Tanya more money? Choice 1 gives Tanya $1024 at the end of the year while choice 2 gives her $1024. Therefore, choice 1 gives her more money overall. For what reasons might Tanya choose each method of payment? Choice 1 gives Tanya more money overall, but choice 2 gives Tanya a large amount of money right away. Therefore, if Tanya needs money sooner, choice 2 is a better deal. If she doesn't need money right away and can wait until the end of the year then choice 1 is better because she gets more money overall.

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