- Hello guys!
Welcome to Tariq's class.
You all know that SHSAT is
a very challenging test.
The biggest challenge is the time.
You get three hours to finish the test.
However, there are 114
questions to finish.
You get approximately 95
seconds to solve each questions.
Now think about it.
If you can finish each
question within 50 seconds,
you can save a lot of time.
You can use this time to review
your test towards the end
and you can do better at the test.
Now, the only way to save time is
if you know the shortcut techniques.
Today I am going
to show you five amazing
math shortcut techniques
that you can use on your exam.
My first shortcut is
about Finding Average.
This technique works when
there is a common difference
between the numbers.
For example, my first question is,
find the average of the following numbers.
The numbers are two, four, six, eight, 10,
and it goes all the way to 1000.
There is a common difference of two.
If you can see between
each and every number,
there is a common difference of two.
Like the difference between
two and four is two,
the difference between
four and six is two,
and between six and
eight is two, and so on.
Now, the traditional way
of finding average is
we'll have to add all
these numbers and divide it
by the total amount of
numbers which is 500.
There are a total of 500 even
numbers from two to 1000.
So that's so much work,
but the shortcut is,
if you have common difference
between the numbers,
then, in order to find the average,
you could just add the first number,
add the last number, and divide it by two.
Add the first number with the last number,
and divide it by two.
So, 1000 plus two is 1002
divided by two, you get 501.
and that's your average.
Is that easy guys?
Now, let's solve another question.
The second question says find the average
of the the following numbers.
This time, the numbers are 15, 20, 25, 30,
and to goes all the way to 50.
Is there any common difference guys?
What is the difference between 15 and 20?
It's five.
What is the difference between 20 and 25?
Also five.
So there you have common
difference between the numbers.
So we can use our shortcut technique.
So you can add the first number
with the last number and divide it by two.
So 15 plus 50 divided by two,
that's 65 over two, which is 32.5.
That's it.
Remember, the first plus the last,
divide by two.
That's all you gotta do.
Shortcut number two is
about finding the area
of the square, given the diagonal.
Now what is a diagonal, guys?
Now the diagonal is a straight line
that connects the two
opposite corner of a square.
For example, A and H, they
are two opposite corners.
The straight line
that is connecting them
is called diagonal.
Now, my question is, find the area
of the following square
math, means M-A-T-H
given that the diagonal A H is four.
Now, the traditional way of
finding the area of square is
using the formula.
The formula is area is
equal to length times width
or area is equal to any
of the sides power of two,
means side squares.
These are the two traditional
ways to find the area.
Now, where is the side length?
We don't know any side length.
So how are we gonna find the area?
Now we have the shortcut.
The shortcut is, if you know the diagonal,
in order to find the area,
you could just square the diagonal,
means diagonal power two
and divide it by two.
Square the diagonal, divide it by two.
Again, square the
diagonal, divide it by two.
What is the diagonal?
Four.
So square of four means four power of two,
divided by two.
Now, four power of two means 16.
Now, 16 divided by two is eight.
Now this is your area.
Now let's try another question.
What is the area of the square A B C D,
given the diagonal D B is three?
Again guys, what is the
formula to find the area
of the square knowing the diagonal?
You are correct.
Square the diagonal and divide it by two.
Square the diagonal and divide it by two.
Remember this.
So the diagonal of this square is three.
So three square divided by two.
So that's nine divided
by two which is 4.5.
Remember guys.
Square the diagonal and divide it by two.
That's all.
The shortcut number three is
about the divisibility rule for three.
There is a divisibility rule
for almost every number.
Now, the question is,
is the number 45,942 divisible by three?
Now, if you want to divide
it the traditional way
and then check if it
is divisible by three,
it is gonna take you a lot of time
and later you might find out
that it is actually
not divisible by three.
So you kinda wasted your time.
So let's check if it
is divisible by three.
Now the number is 45,942.
The technique is...
the shortcut is you add the digits,
four, five, nine, four,
and two from 45,942.
Your gonna get 24.
Now check.
The sum 24.
Is it divisible by three?
Let's check.
24 divided by three, yes
it is divisible by three
because you get eight when you divide.
Since the sum of the digits
is divisible by three,
it means the number itself
is divisible by three.
It means you can divide 45,942 by three.
Now you can go ahead and divide.
Now let's check another one.
Now the second question.
Let's see if this is divisible by three.
What is the technique guys again?
Add all the digits.
So four plus three plus
seven plus four plus five,
what is the sum?
four plus three seven,
seven plus seven 14, 18, 23.
The sum is 23.
Now guys, is the sum
23 divisible by three?
No it is not.
It means 43,745,
this number cannot be divided by three.
The shortcut number four is
about fraction simplification.
Question number one.
Simplify the fraction multiplication.
So the traditional way
would be multiplying
all the numerators of this
fraction and divide the product
by the product of all the denominators.
Right here.
The question that should
come in your head is,
why don't you do it?
Now, I'm not gonna do
it in traditional way.
I'm gonna use my shortcut.
Before learning the shortcut, let's learn
about fraction simplification.
Let's say you have a fraction.
In the fraction, you have a
numerator and a denominator.
We can always simplify a
numerator and a denominator
if they are both divisible
by the same number.
Now, if another fraction is multiplied
with the previous fraction,
then we can simplify the
numerator of the previous fraction
with the denominator of the next fraction
and this simplification is
called cross simplification,
means we are simplifying cross.
Now let's use this cross
simplification technique
into our fraction multiplication.
Now let's look at it.
4,020 and 8,040, we can see
there in one to two ratio,
means if I divide 8,040
by 4,020, then I get two.
It means both 4,020 and
8,040 are divisible by 4,020.
Now, 4,020 is a numerator
and 8,040 is a denominator,
and there is a
multiplication in the middle
so I'm allowed to use
cross simplification.
So if I divide 4,020 by 4,020 itself,
the quotient will be one.
What do I get when I
divide 8,040 by 4,020?
I'm gonna get two.
Now that was a cross simplification.
Now let's do another cross simplification
between three and 15.
Is there a number by which you can divide
both three and 15?
Yeah, we can divide both
three and 15 by three.
Let's divide three by three.
Then the three would become one
because three divided by three is one.
Now let's divide 15 by three,
15 divided by three is what, guys?
Five.
Okay.
Now, can we simplify the six and nine?
Yes, as I said, every fraction
you can always divide,
you can always simplify the
numerator and the denominator.
If you want to simplify, you
can simplify six and nine
or you could simplify two and six.
That would be a cross simplification.
What do you like guys?
Cross simplification or
the numerator divided
by denominator?
Let's use another cross simplification.
So I can divide this
two by two, I get one,
and divide this six by two as well.
What do I get, guys?
Three.
Now I can simplify this
three and this nine.
As I said, you can always
simplify one numerator
with another denominator
if it is divisible.
So divide three by three.
You get one.
Divide this nine by three.
What do you get, guys?
Three.
Okay.
Now we have this five surviving,
this three surviving,
and that 60 surviving.
Can we simplify this five and three guys?
No.
Two reasons.
First of all, they are both denominators
and the second reason is they
are not divisible by any...
They don't have any common factor
means you cannot divide both of them
by any numbers except for one.
Hey, wait a minute, I
can divide three by three
and I can divide 60 by three.
That will be a cross simplification.
Let's divide this three by three,
you get one.
Divide this 60 by three,
what do you get, guys?
20.
Can we do anymore cross simplification?
Well, we have a five
surviving and we have a 20.
20 is a numerator and
five is a denominator.
Can we simplify cross?
Yes.
So divide five by five we get one.
Divide 20 by five, we get four.
Finally, as you can
see, all the survivors,
we have one times one,
times one, times four,
which is four in the numerator.
At the bottom we have one times one,
times one, times one, which is one.
So we just forward the numerator,
one is the denominator, so answer is four.
Much easier.
Isn't it guys?
Now let's try one more.
Well, let's see if we can
do cross simplification
between four and eight.
Look.
Four, We can divide both
four and eight by four.
four divided by four is what, guys?
One.
Eight divided by four is two.
Well, let's see.
Hey, we can divide nine
by nine, we get one,
nine by nine we get one.
Can we cross simplify anything else?
Yes, 10 and 10.
Simplify this 10 by 10
and that by 10 as well
so we get the same thing.
321 and 642.
They look like one to two ratio.
So I can divide 321 by
321, I'm gonna get one.
If I divide 642 by 312, we get two
and finally we can see that this two
and this two can be
simplified to one and one,
means we divide both of them by two,
and the final answer would be just one
because that's all is surviving.
One.
So if you know the simplification,
your multiplication will be much easier
unless you want to do
it the traditional way.
The shortcut number five
is about finding LCM.
The least common multiple.
Now the questions is,
find the LCM of 15 and 18.
The traditional way would
be using the tree method
like the 15 can be factored,
the two factors of 15
is three and five, 18 can be factored,
the two factors are three and six,
six can be further prime
factor to two and three.
My question is, then what?
It's pretty messy.
Now the shortcut is and the
fun method is the cake method.
C-A-K-E, cake method.
So we can draw a cake, means draw a box.
Let's start with the box and let's put the
both numbers 15 and 18
in the box using a comma.
15 comma 18.
Now let's divide by any
prime numbers in there,
the smallest and the only
even prime number is two.
So let's divide by two.
Wait a minute.
We cannot divide 15 by two.
But can we divide 18 by two?
Yes.
As long as you can divide one of them,
you are allowed to divide by two.
So what we do with the
15, we keep it as it is.
We cannot divide 15 by two
so we keep it as it is.
But 18 divided by two,
the answer is nine.
Now, you make another box on top of it
like we are building up a cake.
Now can I divide both of them
or either one of them by two?
No.
So let's use another prime number.
Three.
I can divide both of
them by three actually.
15 divided by three is five
and nine divided by three is three.
Now, let's make another cake,
another layer of the cake.
Now can I divide by three again?
Yes. Let's divide by three.
Can I divide five by three?
No.
so I keep the five as it is.
Remember, if you cannot
divide, you keep it as it is.
Can I divide the three by three?
Yes.
Then you get a one.
Now, can I divide five and one by five?
Yes, I could divide this five by five.
Remember, you can only
divide by prime numbers.
So, five is a prime number
so I can divide by five.
So I get a one and this
one stays as it is.
This is my candle.
Look at this.
My candle.
Now cut the cake, Happy
Birthday, and this is you LCM.
So your LCM
is equal to five times three, times three,
and the last one is two.
So five times three is 15.
15 times three, 45 times two, 90.
So that's your LCM.
So let's do another one.
Find the LCM of 20, 25, and 15.
So let's draw a cake.
Put all the three numbers, 20, 25, and 15.
Now think about a prime number.
It could divide either one of them
or two of them or all three of them.
So if you can divide all three of them,
it will be faster.
So let's divide by five.
Five is a prime number so I get four,
25 divided by five is five,
15 divided by five is three.
Now let's make another layer of the cake.
Now can I divide by four?
Yes you can divide this four
by four but you're not allowed
to do it because four
is not a prime number.
So what number can we use now, guys?
Let's use two.
Then I get four divided by two is two
but can I divide five by two?
No.
So I will keep it as it is.
Three is not divisible by two as well.
Keep three as it is.
Now another layer of the cake.
Remember, you keep building the layers
until you get your candles.
Now let's divide by two again.
You get one, five stays
as it is, three as it is.
Now let's divide by three.
You get one, stays as it is.
Five stays as it is.
Three divided by three is one.
Now what is the last number
you want to pick, guys?
Now divide by five.
You get one, one, and one,
these other three candles,
Happy Birthday, and this is your LCM.
So the LCM is equal to the
product of all these numbers.
I have two twos, so two times two,
I have one three and two fives.
So what do you get, guys?
That's 25, 50, 100, 300.
And that's your LCM.
Hope you guys enjoyed and understood
all my shortcut techniques, guys.
These shortcut techniques can help you go
through many exams.
You can save a lot of time
unless you say traditional way for life.
For more videos, guys trust me.
This is just a drop of the ocean.
There are a lot more to learn,
lot more shortcut techniques
that I want to share with you guys.
For more videos, subscribe
to Bobby-Tariq Tutoring
Center, my channel,
or leave a comment underneath the video.
If you want me to make a
video on a particular topic.
