Twice as many? 3/4 as many? How to interpret?

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Main Question or Discussion Point

How do you write "twice as many x as z" algebraically?

Is this 2x=z or x=2z? How do you know?

How about "3/4 as many x as z" algebraically?

Is this (3/4)x=z or z=(3/4)x? How do you know?

An applied example:
There are twice as many apples as there are oranges, so does this mean for each orange there are 2 apples, so that if apples=x and oranges=y, then y=2x; 2y=4x, and etc?

Are these statements ambiguous to you?
 
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  • #2
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How do you write "twice as many x as z" algebraically?
Is this 2x=z or x=2z?
x = 2z

How do you know?
There's no trick to it; all of the information is right there in the expression. It says that you need 2 of z to equal 1 of x, hence: x = 2z.
 
  • #3
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How do you write "twice as many x as z" algebraically?

Is this 2x=z or x=2z? How do you know?
x = 2z

An equivalent problem is "x is twice as big as z." It might help to look at numbers. 50 is twice as big as 25. Would you write an equation expressing this relationship as
a) 50 = 2*25
or
b)2*50 = 25?
How about "3/4 as many x as z" algebraically?
x = (3/4)y
Is this (3/4)x=z or z=(3/4)x? How do you know?

An applied example:
There are twice as many apples as there are oranges, so does this mean for each orange there are 2 apples, so that if apples=x and oranges=y, then y=2x; 2y=4x, and etc?

Are these statements ambiguous to you?
I would not let apples = x and oranges = y. These variables should represent the number of apples or oranges.

As far as translating to equations, see what I said at the beginning of this post.
 
  • #4
Vay
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I don't know what has gotten to me, sometimes I just can't get something. It must be doing my math homework for 6 hours straight... I can't think straight anymore. Anyone else have similar experience? Or it must be something I ate...
 
  • #5
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How do you write "twice as many x as z" algebraically?
I usually try to think of this in terms of ratios. The phrase "twice as many x as z" gives you an equality of ratios, x:z is 2:1. Assuming z isn't zero, we can rewrite this using fractions as [itex]\frac{x}{z} = 2 [/itex] so x = 2z.
 

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