andy had 20% fewer marbles than henry. when andy gave 80 marbles to henry, the number of marbles he had left was half of what henry had. how many marbles did andy have in the beginning?

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- mathstudent55

Let's say originally Henry had x marbles.
Then Andy originally has 20% fewer marbles than x.
How do you write mathematically 20% fewer than x?

- EXOEXOEXO

.8x?

- mathstudent55

Correct.
We have this:
Originally After Giving
Henry x
Andy 0.8x

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- mathstudent55

Now we need to figure out what the numbers are after giving.

- EXOEXOEXO

1/2x for andy?

- mathstudent55

Correct.
Originally After Giving
Henry x x + 80
Andy 0.8x 0.8x - 80
After giving the marbles, Andy had 0.8x - 80 marbles.
From the given, we know Andy had 0.5x left, so
0.8x - 80 = 0.5x

- mathstudent55

Remember that we started above by letting x = number of marbles Henry had originally.
If you solve the last equation above for x, that will tell you how many marbles Henry had originally.
Then 0.8x is how many marbles Andy had originally.

- EXOEXOEXO

about 213 marbles?

- mathstudent55

Ok. we see there is a problem.
I get 266.6666...

- mathstudent55

The number of marbles must be a whole number.
When I read the problem I noticed there is a sentence that can be interpreted two ways.
We interpreted one way and got 266.666... marbles.
Let's interpret the sentence the other way and see what we get.
I'll show you which sentence I am referring to below.

- Mertsj

I believe that .8x-80=1/2(x+80)

- mathstudent55

This is how we interpreted the question. I am adding words in square brackets to show our original interpretation.
[Originally] Andy had 20% fewer marbles than Henry.
When Andy gave 80 marbles to Henry, the number of marbles he had left was half of what Henry had [before Henry was given the extra marbles].
How many marbles did Andy have in the beginning?

- mathstudent55

This is the other way to interpret the question. I am adding words in square brackets to show our original interpretation.
[Originally] Andy had 20% fewer marbles than Henry.
When Andy gave 80 marbles to Henry, the number of marbles he had left was half of what Henry had [***after*** Henry was given the extra marbles].
How many marbles did Andy have in the beginning?

- mathstudent55

Since with the second interpretation, we need to deal with Henry's marbles after he got the extra 80 marbles from Andy, then the number of marbles Andy has in the end is equal to 0.5(x + 80)
That gives us the new equation:
0.8x - 80 = 0.5(x + 80)

- Mertsj

Bingo!!!

- mathstudent55

If we solve the new equation, we get x = 400, which is a whole number and makes more sense.

- EXOEXOEXO

.8x400=320

- mathstudent55

If Henry had 400 marbles originally, then Andy had 0.8(400) marbles originally.

- mathstudent55

Yes, you got it.
Good job!

- EXOEXOEXO

thank you!

- mathstudent55

You're welcome.

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