48:2x is equivalent to 144:600

[jpv17]

Aloha,

I am using the 5th edition Fraction, Decimal, & Percents guide. This problem is in the problem set of Ch. 4, #1, page 69.

48:2x is equivalent to 144:600. What is x?

So I know that I can translate this to:
48/2x = 144:600

But I guess, I don't quite fully understand the properties and rules behind equations with an equal sign in between.

for an example, the solution states:
a simplified version of problem above is...
24/x = 12/50 (Since you can divide the left side by 2 and divide the right side by 12)

then next process that the solution gives is:
4/x=1/25 (how do you get this?)

Then the book proceeds to the the answer which is x=100 (cross multiply).

Can anyone help me understand this problem and guide me to a source that I can learn some rules and properties of how to tackle problems with a equal sign in between?

Looking forward to moving ahead,
Jon

[jpv17]

Can anyone help me understand this problem and guide me to a source that I can learn some rules and properties of how to tackle problems with a equal sign in between?
[/quote]

before I get hackled by anyone for this ;) ... Let me clarify, that I need help with proper rules for fractions or rations that equal each other. Like in the problem mentioned above.

[RonPurewal]

In the first two steps shown here, two very different things are happening. Here's a general presentation of those two things.

1/
There are manipulations of individual numbers.
* These manipulations work on numbers always, regardless of whether those numbers are in equations. However, they must preserve the value of the number"”you can't end up with a different numerical value.
E.g., 10/2 is the same as 5. If you ever see 10/2, anywhere, you can change it to 5, no matter where you find it. (You could also change 5 to 10/2 anywhere, too, though there would be little call for that unless you were making a common denominator.)

And...
2/
There are manipulations of equations.
Here, you just have to do the same thing to both sides.
The numerical values will normally change when you do so"”but that's fine, because, in an equation, you don't really care about the value of each side, as long as the equation is true.
E.g.,
R = M (Ron is as tall as Matt.)
You can add 4 to each side:
R + 4 = M + 4 (If Ron and Matt are both wearing dress shoes with a 4-centimeter heel lift, they will have the same height.)
Obviously the numerical values are not the same anymore"”both are 4 centimeters higher"”but the equation is still true. If you start with identical heights and make the same change to both of them, you'll still have identical heights. Most likely different from the original ones... but still the same as each other. That's how equations work.

[RonPurewal]

The first step"”reducing 144/600 to 12/50"”is type #1 here.
Those fractions are exactly the same number. The left side is not involved at all here. (This is a weird way to simplify, though; I'll explain later.)

In the second step, both sides are divided by 6. So, the numerical values are no longer the same, but the equation is still true (type #2).
If we tried to do this to just one side, it wouldn't work. (If Matt puts the shoes on but I don't, we won't be the same height anymore.)

Hope that makes sense.

[RonPurewal]

This comment:

(This is a weird way to simplify, though; I'll explain later.)

If I were to simplify 144:600 here, I'd do one of two things.

1/
I'd make it match the other side.
If the other side were still as originally written (48:2x), then I'd simplify it to 48:200, and then stop. So we'd have 48:2x = 48:200, so that 2x = 200.
If the other side had already been simplified to 24:x, I'd simplify 144:600 down to 24:100. That would give 24:x = 24:100, making x = 100 for sure.

Or...

2/
I'd simplify to lowest terms"”not for any particularly good reason, but just because that's the iconic thing to do with fractions if you don't have anything better in mind.
The lowest-terms form of 144:600 is 6:25.

Weirdly, they simplified to 12:50"”which doesn't do either of these things.
In the answer key, is there an explanation of why they stopped there? Simplified too far to match the other side, but not far enough to reach lowest terms?

Hmm.