Question

The top three prices for works of art sold at auction in 2013 totaled $307.5 million. These three works of art were

a sculpture, a painting, and a photograph. The selling price of the painting was $47.8 million more than that of the
photograph. Together, the painting and the photograph sold for $17.7 million more than the sculpture. What was the
selling price of each work?

Answer 1

price of sculpture = 144,500,000

price of painting = 105,400,000

price of photograph = 57,600,000

Explanation:

Total price: $307,500,000

price of painting + price of sculpture + price of photograph = $307,500,000

Price of painting = 47,800,000 + price of photograph

Price of painting + price of photograph = $17,700,000 + price of sculpture

$17,700,700 + price of sculpture + price of sculpture = $307,500,000

2 price of sculpture = $289,000,000

price of sculpture = $144,500,000

price of painting + price of photograph + $144,500,000 = $307,500,000

price of painting + price of photograph = $163,000,000

Recall that price of painting is 47,800,000 + price of photograph so then

plug it back in to say the price of painting + price of painting + 47,800,000 + price of photograph = 163,000,000.

2(price of photograph) = price of painting = 115,200,000. The price of the photograph is 57,600,000. Then, we add 57,600,000 + 47,800,000 = 105,400,000 to get the price of the painting

Now, we know that the prices:

price of sculpture = 144,500,000

price of painting = 105,400,000

price of photograph = 57,600,000

To check, we add 144,500,000 + 105,400,000 + 57,600,100 = 307,500,000.

Answer 2

Explanation:

Let's denote the selling price of the photograph as P, the painting as P + $47.8 million, and the sculpture as S. According to the details provided, we can create the following equations:

1) The total price for all three works of art is $307.5 million:

\[ P + (P + 47.8) + S = 307.5 \]

2) The combined selling price of the painting and the photograph is $17.7 million more than the sculpture:

\[ (P + (P + 47.8)) = S + 17.7 \]

\[ 2P + 47.8 = S + 17.7 \]

Let's simplify the equations. First, combine like terms in the first equation:

\[ 2P + S + 47.8 = 307.5 \]

Now, let's express S from the second equation in terms of P:

\[ S = 2P + 47.8 - 17.7 \]

\[ S = 2P + 30.1 \]

We can substitute S from the second equation into the first one:

\[ 2P + (2P + 30.1) + 47.8 = 307.5 \]

\[ 4P + 77.9 = 307.5 \]

Now solve for P:

\[ 4P = 307.5 - 77.9 \]

\[ 4P = 229.6 \]

\[ P = \frac{229.6}{4} \]

\[ P = 57.4 \] (the selling price of the photograph in millions of dollars)

Now we can find the selling price of the painting, which is $47.8 million more than the photograph:

\[ P_{\text{painting}} = P + 47.8 \]

\[ P_{\text{painting}} = 57.4 + 47.8 \]

\[ P_{\text{painting}} = 105.2 \] (the selling price of the painting in millions of dollars)

And lastly, we can solve for the selling price of the sculpture using the price of the photograph:

\[ S = 2P + 30.1 \]

\[ S = 2(57.4) + 30.1 \]

\[ S = 114.8 + 30.1 \]

\[ S = 144.9 \] (the selling price of the sculpture in millions of dollars)

In summary, the selling prices were:

- Photograph: $57.4 million

- Painting: $105.2 million

- Sculpture: $144.9 million