Problem 20.
The expression 5 to the minus 8 times 7 to the minus 9 is
equal to which of the following?
And here we see a bunch of answers.
Let's see, where they have a 35 there.
So I suspect that they want to have us somehow multiply the 5
times the 7.
So let me write this in terms that I have an easier time
understanding it.
So, 5 to the minus 8 times 7 to the minus 9.
That's the same thing as 1 over 5 to the 8 times 1
over 7 to the 9.
The negative in the exponent just means one over the base
to the positive exponent, that's all the negative means.
That's equal to 1 over 5 to the 8 times 7 to the 9.
And that's not one of our answers.
Somehow we have to multiply the 5 and the 7.
And the way we can do that is by having the same exponent.
So let me rewrite 7 to the 9.
I can rewrite that as 7 to the eighth power times 7.
So then, our whole expression becomes 1 over 5 to the 8
times 7 to the 8 times 7.
All right, I just rewrote this.
I rewrote 7 to the ninth as 7 to the eighth times 7.
But now if I group this, just put parentheses around it, I
can do that because multiplication is associative.
It doesn't matter what order I do it.
This is the same thing as 5 times 7 to the
eighth power, right?
a to the x times b to the x is the same thing as ab to the x,
so the same thing as 5 times 7 to the eighth power.
So this is equal 1 over-- 5 times 7 to the eighth power is
35 to the eighth power times 7.
This piece, right here, is 35 to the eighth power, and then
you have your 7.
And then that is choice B, right there.
They just switched the order of the 7 and the 35 to the
eighth power.
21.
Use the diagram below to answer the
question that follows.
OK, they just have numbers listed in order and
have five per row.
Which of the following algebraic equations could best
be used to explain why, for any 3 by 3 cross, like the one
shown above, the sum of the numbers in the vertical
rectangle-- so this is the vertical rectangle, right
there-- is equal to the sum of the numbers in
the horizontal rectangle.
So the horizontal rectangle is that right there.
So let's look at this a little bit.
What's the sum in the vertical rectangle?
If we call this number x, let me just draw it generally.
Let me draw as best I can.
If that middle entry is x, what's the
entry right above that?
Well no matter where you go, it's going to be 5 less,
that's because we have 5 per row, right?
14 minus 5 is 9.
12 minus 5 is 7.
8 minus 5 is 3.
So this is going to be x minus 5.
By the same logic, if you go down a row, you're adding 5.
14 plus 5 is 19.
15 plus 5 is 20.
So this is going to be x plus 5.
So the sum of the entries in the vertical column, the sum
of the numbers in the vertical rectangle are x minus 5 plus x
plus x plus 5.
So that's the sum in the vertical column.
And then in the horizontal column, if this is x, what's
this going to be?
It's just one below that.
It's x minus 1, and then to the right of it
is just x plus 1.
So the sum in the horizontal column is going to be x minus
1 plus x plus x plus 1.
Now let's see if any of these choices look like what I just
wrote down.
A bunch of 9's.
Maybe, so that's simplified.
Here you go.
This looks very similar.
Actually this is exactly what I wrote up here if you just
switch the left- and the right-hand side.
And you can do that because there's equality.
They wrote this here, and they wrote this here.
So this is what I could use to prove that the sum of the
vertical rectangle is equal to the sum of the horizontal.
If I wanted to keep going, you'd notice that these 5's
would cancel out, these 1's would cancel out, you'd get 3x
is equal to 3x.
And that would be, that would complete your proof.
Next problem.
A landscaper bought some decorative cement blocks from
a landscaping store.
The supplier charged 5% sales tax, and the
total came to $315.
All right, 315 after 5%.
Without the tax, the landscaper could have bought
six more blocks for the same total cost. How many blocks
did the landscaper buy?
If p represents the price of one block.
OK, p is the price one block.
In which of the following equations does x represent the
answer to the problem.
But remember, the answer to the problem is, how many
blocks did the landscaper buy?
So x equals blocks.
x is equal to blocks bought.
Now, this first equation says that after paying 5% sales
tax-- so before tax, he bought x blocks, he payed p for each
of those blocks.
That's what they tell us right here.
So this is the pretax price.
p per blocks times x blocks is how much he pays, but he has
to pay sales tax on that.
He has to pay 5% of that on top of that as sales tax.
So plus 0.05-- that's the same thing as 5%-- times whatever
the pretax value is.
And that's going to be equal to $315.
And then this, we could just add the coefficients.
This is a 1 out here, right?
1 plus 0.05 is 1.05 times price times
quantity is equal to 315.
And let's see if we have anything like that over here
in our answers.
Which equation does x represent the answer to the
problem above?
0.95 px, price, he bought six more without the tax.
So another way you can think about it, if you didn't have
tax, $315 would have been the price without tax.
So let's see if that simplifies, because I don't
see exactly what I wrote here from that first equation.
So without the tax, he could have bought x plus 6, if there
was no tax.
So he would have bought x plus 6 for p per block, and then
that is going to be equal to 315.
This is the situation with no tax.
If there was no tax, for $315 he could have bought six more
than he originally bought.
So this is what the second, this-- let me graph it.
Without the tax.
So this first equation came from the first sentence.
The second equation comes from this red sentence, right here.
Let's see if we see this anywhere.
p times x plus 6 is equal to 315.
I don't see that anywhere here.
But we can see-- we can do something interesting.
We can use all of the information from the problem.
Remember, the first problem, it said, the supplier charged
5% sales tax, and the total came to 315.
That gave us this equation, right here.
The second one said, without the tax, for $315 he could
have bought six more blocks.
So we get this one, right here.
Notice, both of these equations equal 315.
So we could set this equal to that.
So we could say p times x plus 6 is equal to 1.05
times price times x.
And let's see if that looks like anything we have here.
So we have, let's see, p times x plus 6 is equal to 1.05px.
So the answer is C, right there.
We're just saying, both of these things equal 315, so we
set them equal to each other.
Problem 23.
A store that sells handcrafted items-- let me pick a nice
highlighting color-- a store that sells handcrafted items
takes $3 per item plus 40% of the sale price
for each item sold.
OK, the rest of the money-- takes $3 per item plus 40% of
the sale price for each item sold-- the rest of the money
from the item sales goes to the craft person.
That's nice of them.
All items sold cost $5 or more.
OK, if p represents the sale price of one item, which of
the following expressions represents the amount of money
the craft person gets for each item sold?
So you start off with p.
That's how much the customer gives.
And then the store takes $3.
So p minus 3, plus 40% of the sale price.
So not 40% of after the $3, it says plus
40% of the sale price.
So the store is going to take another 40%, right?
0.4, or 0.40, times the sales price.
So the store takes $3 and then 40% of the sales price.
And this is what the craft person gets.
So this is what, you know, this is our answer.
This is what craft person gets.
So what is this?
This is p-- switch colors-- p minus 0.4p is 0.6p, then you
have the minus 3.
That's what the craft person gets.
And if you look here, you don't see that answer.
They wrote it all as fractions.
0.6 is the same thing as 6/10 p minus 3.
6/10 is the same thing as 3/5 p minus 3.
So it's 3/5 p minus 3.
Choice D.