3/4 of _____ Is 90 in Three Ways

Date: 03/24/2010 at 22:36:23
From: Rachel
Subject: 3/4 of _____ is 90

Another one was 5/6 of 120 is ______

How do I find the answer and explain it to my son?

Not sure what to be dividing by -- I think you divide -- but then how to
explain it simply to my son in the 4th grade?

4 divided by 90 then multiply by 3?

I know that is wrong, but doesn't it have something to do with division
using the demoninator?

Date: 03/25/2010 at 09:17:17
From: Doctor Ian
Subject: Re: 3/4 of _____ is 90

Hi Rachel,

The most basic thing to know here is that a multiplication and a division
are two different ways of expressing the same relationship. For example,
these equalities ...

  3 * 4 = 12

      3 = 12 / 4

      4 = 12 / 3

... are just three ways of saying the same thing.

Sometimes, we know two factors, and need a product:

  1/2 * 6 = _____

But sometimes, we have a product and a factor, and need the other factor:

  3/4 * _____ = 90

In this case, we can either (1) use guess-and-improve to identify the
missing factor, or (2) rewrite the multiplication as a division, and
evaluate that.

The first approach might look like this:

  Could _____ be 12?

    3/4 * 12 = (3 * 12)/4 = 36/4 = 9

So 12 is too small!

Could _____ be 400?

    3/4 * 400 = (3 * 400)/4 = 1200/4 = 300

So 400 is too large. The value we're looking for must be somewhere between
12 and 400. How about 200?

    3/4 * 200 = (3 * 200)/4 = 600/4 = 150

Still too large. So we know it's between 12 and 150. How about 120?

    3/4 * 120 = (3 * 120)/4 = 360/4 = 90

So now we know what the answer is.

Note how I used each guess to improve my choice of the next guess. You can
often solve problems pretty quickly this way, if you pay attention to what
you're doing.

But let's look at the second approach. We can rewrite the multiplication
...

  3/4 * _____ = 90

... as the equivalent division ...

        _____ = 90 / (3/4)

This is good if you know the rule for dividing by a fraction, i.e., that
to divide by a/b, you multiply by b/a:

        _____ = 90 / (3/4)

              = 90 * (4/3)

              = (90 * 4)/3

              = 360/3

              = 120

So we get the answer a little more quickly this way. 

There is a third way to think about it, which is this. If I have an
equation that is true, like ...

  2 + 3 = 5

... and I multiply both sides by the same amount, e.g., ...

  4 * (2 + 3) = 4 * (5)

... then the equation must still be true. (Does that make sense? If not,
write back and we can talk about this, since it's one of the basic ideas
that makes everything else work.)

It's useful to know that the product of a fraction and its reciprocal is
1. For example,

  3/4 * 4/3 = (3*4)/(4*3) = 12/12 = 1

So given something like ...

  3/4 * _____ = 90

... we can multiply both sides by the reciprocal of 3/4, 

  4/3 * 3/4 * _____ = 4/3 * 90

          1 * _____ = 4/3 * 90

              _____ = 4/3 * 90

Remember, if the first equation was true, then this last one must also be
true! And note that it ends up at the same place as doing the equivalent
division: That is, we multiply 90 by 4/3, which is the same as dividing it
by 3/4. 

I know this seems like a lot, but the key thing to focus on here is that
even if you end up doing a division, you can always start by writing the
equivalent multiplication, which is often easier to understand. Once you
have it, you have all kinds of options available to you. 

Does this help? Let me know if this helps. If it doesn't, we can look for
some other way to approach it that will work better.

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 03/25/2010 at 18:07:12
From: Rachel
Subject: Thank you (3/4 of _____ is 90)

THANK YOU!!!!!!!
I look to Mr. Math website a lot to help my son with his homework, and
then be able to help him understand. Your team has never let me down! The
website is fantastic. Thank you again for your time!
God bless,
Rachel