Welcome to another
practice SAT math question from the
numbers and operations section.
If M is the set of positive multiples of two less than 150,
and N is the set of positive multiples of nine
less than 150, how many members are there in M and N?
Again, M is the set of positive multiples of two
less than 150, and N is the set of
positive multiples of nine less than 150.
To get a better understanding of what's happening here,
let's go ahead and start by listing out multiples of two,
and multiples of nine that are positive, and less than 150.
The set M will consist of all of the numbers,
or members listed here.
Notice how we have multiples of two, but we stop at 148,
because the next multiple of two would be 150,
which is not less than 150.
Then, for the multiples of nine, we stop at 144,
because the next multiple of nine
would not be less than 150.
The key to understanding this question is to recognize
that if we want the members that are in both sets, M and N,
which can also be written using this notation here,
since two and nine don't share any common factors
other than one, the members in both M and N
would have to be multiples of two times nine,
or multiples of 18.
To recognize this, if you take a look at set M and N,
notice how the first member in common is 18,
So it comes down to identifying multiples of 18
that are positive, and less than 150.
There are a couple ways of doing this.
One way would be to just list them,
as we see here on the bottom;
18 times one is 18,
18 times two is 36,
18 times three is 54,
18 times four is 72,
all the way out to 18 times eight, which is equal to 144,
and the next multiple of 18 would not be less than 150.
If we list them all like this,
which is a bit time consuming,
we would just count these members, or these numbers,
one, two, three, four,
five, six, seven, and eight,
so our answer is B, eight.
Because listing all of these out is fairly time consuming,
let's take a look at some other strategies
to solve this problem.
Again, the key to recognize is that
if we want members that are in M and N,
and since two and nine don't share any common factors,
we're looking of multiples of two times nine, or 18,
that are positive, and less than 150.
Because we can use a calculator,
another way to find there are eight of 18 that are positive,
and less than 150,
would be to take 149,
which is the largest whole number that's less than 150,
and just divide by 18.
When we do this, we'll get a decimal,
but if we ignore the decimal,
notice how we would just have eight,
meaning there are eight multiples of 18
that are less than 150, and positive.
The other option would be to use
the table feature of the calculator.
If we press Y equals,
and then because we're looking for
multiples of 18 that are positive, and less than 150,
we'll enter 18 X,
and now we can use the table feature
to identify the multiples of 18.
Before we do this, let's check our table setting.
We'll press second window,
and make sure that the
independent feature is on automatic.
If it's not, we just highlight it, and press enter,
and now we are going to press second, graph,
and we'll start at 18,
since that's the first positive multiple of 18,
and we'll determine how many multiples of 18
there are that are less than 150.
We have one, two, three, four,
five, six, seven, eight.
Notice the next multiple of 18 is not less than 150,
so there are eight multiples of 18
that are positive, and less than 150.
Using the table feature here,
it would be a lot less time consuming
than completing the list of multiples of 18 by hand,
which we did here at the beginning.
Either way, there are eight members in sets M and N.
I hope you found this helpful.