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Difficulty: MediumFractions and Rational Numbers

An artist sold 13\frac{1}{3} of her paintings on the first day of an exhibition. On the second day, she sold 25\frac{2}{5} of the paintings that remained after the first day. On the third day, she sold 14\frac{1}{4} of the paintings that remained after the second day. If 1818 paintings remained unsold at the end of the third day, how many paintings did the artist have originally?

  1. A
    45
  2. 60Answer
  3. C
    72
  4. D
    90
  5. E
    108

Answer

The artist originally had 60 paintings.
To find the original total, determine the remaining fraction of paintings after each day sequentially. After day one, 23\frac{2}{3} remain. After day two, 35\frac{3}{5} of 23\frac{2}{3} remain, which is 25\frac{2}{5}. After day three, 34\frac{3}{4} of 25\frac{2}{5} remain, which simplifies to 310\frac{3}{10} of the original total. Given that 310\frac{3}{10} of the total equals 1818, dividing 1818 by 310\frac{3}{10} gives 6060.

Step-by-Step Solution

1
Calculate the fraction of paintings remaining after the first day.
Remaining fraction after Day 1 = 1−13=231 - \frac{1}{3} = \frac{2}{3}.
Selling 13\frac{1}{3} of the total leaves 23\frac{2}{3} of the original total.
2
Calculate the fraction of paintings remaining after the second day.
Remaining fraction after Day 2 = 23×(1−25)=23×35=25\frac{2}{3} \times \left(1 - \frac{2}{5}\right) = \frac{2}{3} \times \frac{3}{5} = \frac{2}{5}.
Selling 25\frac{2}{5} of the remaining paintings leaves 35\frac{3}{5} of that remaining portion.
3
Calculate the fraction of paintings remaining after the third day.
Remaining fraction after Day 3 = 25×(1−14)=25×34=620=310\frac{2}{5} \times \left(1 - \frac{1}{4}\right) = \frac{2}{5} \times \frac{3}{4} = \frac{6}{20} = \frac{3}{10}.
Selling 14\frac{1}{4} of the remaining paintings leaves 34\frac{3}{4} of the second day's remaining portion.
4
Set up the equation with the given remaining quantity and solve for the total number of paintings NN.
310N=18  ⟹  N=18×103=60\frac{3}{10} N = 18 \implies N = 18 \times \frac{10}{3} = 60.
The final remaining fraction represents 1818 paintings.

Key Concept

Sequential Fraction of Remainder Problems
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Topics

Fractions and Rational Numbers

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