Word problem help with explanations

Question

At an auction, a lucky wealthy collector paid $7,000,000 for three paintings: a Monet, a Picasso, and a van Gogh. The Monet cost $800,000 more than the Picasso. The price of the van Gogh was $200,000 more than twice the price of the Monet. What was the price of each painting?

 

This is what I have thus far, I was wondering if there was an easier way to simplify this problem and better explanations.

 

X+(x+800) +2(x+800)+200 =7000
X+x+800 + 2x+1600+200 = 7000

 

4x=7000-2600

4x=4400

 

P= Picaso
x=1,100,000 P

M= Monet
x+800=1,900,000 M

V= Van Gogh
2 * 1900000+200000 =4, 000, 000 V

Answer 1

Well here is the check step:

 

   The VanGogh is 4,000,000

   The Monet is     1,900,000

   The Picasso is   1,100,000

 

  The total is in fact 7 million. So that condition is satisfied.

 

  THe Monet is in fact 800k more than the picasso.

 

  2 x 1,900,000 + 200,000 = 3,800,000 + 200,000 = 4,000,000,

  So the Van Gogh is in fact 200k more than twice the price of 

  the MOnet.

 

 So all of the conditions stated in the problem are satisfied

with these results.

 

Therefore, your answer is correct.

 

Algebra equations CAN (AND MUST) be checked by

plugging in the alleged answer, and checking

that the conditions stated in the problem

are satisfied.

 

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The only bit of friendly critique and coaching is that 

when you state your answers, you are making

run-on equation statements.

(Yes, I am being very picky here)

 

For example, starting with the statement 4x=4400

So x = 1100.  The price of the Picasso is then P=1,100,000

 

However, in your statement of the answer, you

are equating the variable x with 1100000 which

it is not. You even defined variable P, but did not

use it for that purpose. You did it for all three of

the paintings.

 

Here's an quick rewrite:

   4x = 4400

   x = 1100.

 

 So the price of the Picasso is P = 1100000

 

 x+800 = 1900, so the price of the Monet

is M=1900000.

 

Finally, 2x + 200 = 2(1900) + 200 = 

                     3800 + 200 = 4000.

So the price of the Van Gogh isV= 4000000.

 

Again, this is being very picky.

However, on proficiency tests, and statewide

exams that you take at the end of the year,

penalty points are deducted from

your score for run-on statements,

even if the answer is correct.

 

KEEP UP THE EXCELLENT WORK!!!

YOU ARE DOING AN AWESOME JOB!!!