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Multiplication Makes It ... Smaller?!

Date: 07/21/2012 at 16:13:01
From: Justine
Subject: Multiplying with a decimal equals a smaller product

Hello!

My question is, when you multiply with a decimal that is smaller than 1,
why does it give a smaller product?

For example:

      2 x .5  = 1

And

    500 x .25 = 125
    
Isn't multiplying supposed to give a larger product?

My dad says 2 x .5 = 1 is the same as 2/2, which is also 1. Why is
multiplying by .5 the same as dividing by 2?

He tries to explain it to me, but I just can't seem to get it!

Related to my question, I think, is when someone says "25% off of 50."
That means you don't have to pay the .25. So you can do

   50 x .75 = 37.5 

I don't understand why, though.

When you get 25% off something, that means you get it for a smaller price
... but yet I multiplied? That is kind of strange to me.

Please reply back because this has been bugging me lots. Thanks!



Date: 07/21/2012 at 16:38:21
From: Doctor Peterson
Subject: Re: Multiplying with a decimal equals a smaller product

Hi, Justine.

When you multiply a number by 2, you get a bigger result. That's what
you're used to until now.

When you multiply by 1, what do you get? The same number, right? 

Multiplying by a number GREATER than 1 makes it bigger; multiplying by 1
leaves it the same; and, continuing the pattern, multiplying by a number
LESS than 1 makes it smaller.

For example, multiplying a number by 1/2 cuts it in half, making it
smaller. Multiplying by 0.5 is the same thing.

In general, you can think of multiplication as scaling, which can make
something either larger or smaller, scaling up or scaling down.

Multiplying by 3 scales up:

   0     1     2     3
   +-----+-----+-----+
   |      \      \      \
   |       \       \       \
   |        \        \        \
   |         \         \         \
   |          \          \          \
   |           \           \           \
   |            \            \            \
   |             \             \             \
   |              \              \              \
   |               \               \               \
   |                \                \                \
   +-----------------+-----------------+-----------------+
   0                 3                 6                 9

This enlargement turned 1 into 3.

Multiplying by 1/3 scales down:

   0     1     2     3     4     5     6     7     8     9
   +-----+-----+-----+-----+-----+-----+-----+-----+-----+
   |                /                /                /
   |               /               /               /
   |              /              /              /
   |             /             /             /
   |            /            /            /
   |           /           /           /
   |          /          /          /
   |         /         /         /
   |        /        /        /
   |       /       /       /
   |      /      /      /
   +-+-+-+-+-+-+-+-+-+
   0     1     2     3

This shrinking turned 1 into 1/3, and 3 into 1. Notice that multiplying by
1/3 undoes a multiplication by 3; it divides by 3.

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



Date: 07/21/2012 at 19:21:54
From: Justine
Subject: Thank you (Multiplying with a decimal equals a smaller product)

Thanks so much! Your answer really cleared up my confusion.

So now I won't get bugged by this question anymore.
Associated Topics:
Elementary Fractions
Elementary Multiplication

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