Hi,
Neither of these is a typo, though both explanations could certainly contain more discussion of the approach taken.
148.
In this case, they go from 10x + 20y in the numerator to 10(x + y) + 10y in the numerator. If we multiply out the second expression, we will see that 10(x + y) + 10y = 10x + 10y + 10y = 10x + 20y, so it is in fact equal to the first expression.
In doing so, they skip a step in the explanation and do not explain their motivation. It would have been more clear if they had written 10x + 10y + 10y first, then factored to 10(x + y) + 10y.
They are doing this because the denominator contains (x + y), and they are seeking an opportunity to cancel out an (x + y) term from the numerator and denominator.
Another approach to this question is to plug in values for k based on the answer choices:
It is best to start out by multiplying everything through by (x + y). Most students are less likely to make mistakes when fractions are eliminated from a question; for this reason, it is helpful to eliminate denominators.
Once we do this, we are left with 10x + 20y = k(x + y), or 10x + 20y = kx + ky.
One thing you might notice about this is that plugging in k = 10 gives us 10x + 20y = 10x + 10y, or 10y = 0. Similarly, plugging in k = 20 gives us 10x + 20y = 20x + 20y, or 10x = 0. Since x and y are positive, neither of these is possible.
It is not immediately clear at this point whether k should be between 10 and 20 (as in B, C, D) or above 20 (as in E). There are more answers between 10 and 20, so we can start there.
We can start with k = 15, which is in the middle of the three choices B-D. This gives us 10x + 20y = 15x + 15y, or 5y = 5x, or x = y. Since we are told that x < y, this cannot be true.
It may not be totally clear at this point whether you should move to k = 12 or k = 18. Since we know that x < y, we need to end up in a situation in which we have an equation ax = by where a > b.
It turns out this is the case with k = 18, though if you tried k = 12 first, you would quickly realize that this is wrong and try k = 18.
k = 12 gives us 10x + 20y = 12x + 12y, or 8y = 2x, or x = 4y. Since we are told that x < y, and since they are both positive, this cannot be true.
k = 18 gives us 10x + 20y = 18x + 18y, or 2y = 8x, or y = 4x. This meets the x < y condition, so it is the correct answer.
The algebraic approach taken in the guide is not intuitive, and rather difficult. Even if you just plugged in answers A-E in order for k, you would be more likely to get to the correct answer faster and with less confusion.
Of course, if you started with B and D, as is typically advised when plugging in answer choices, you would have gotten there even faster, since the answer happens to be D.
230.
Here, they start with x * 2^-17 = (...)
As they move from step 1 to step 2, they DIVIDE by 2^-17.
This puts 2^-17 in the denominator.
In the next step, they simplify the expression. First, they use the definition of negative exponents, which says that 2^-17 = [1/(2^17)].
In general, when we divide by 1/n, it is the same as multiplying by n. In this case, dividing by [1/(2^17)] is the same as multiplying by (2^17), so having 2^-17 in the denominator is the same as having 2^17 in the numerator.
Therefore, when we simplify, we get (...) * 2^17. From here, they distribute 2^17 across the four terms in the numerator, and then apply exponent rules to simplify each term.
