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Zorluk: ZorFractions, Decimals, and Percents Arithmetic

At the start of a month, an online bookstore's inventory consisted of Fiction, Non-Fiction, and Textbook titles. Exactly 38\frac{3}{8} of the total inventory consisted of Fiction titles, and 0.400.40 of the remaining inventory consisted of Non-Fiction titles, with the balance consisting of Textbook titles. During the month, the number of Fiction titles increased by 20%20\%, the number of Non-Fiction titles decreased by 25%25\%, and the number of Textbook titles remained unchanged. If the total inventory increased by a net amount of 1515 titles at the end of the month, how many total book titles were in the inventory at the start of the month?

  1. A
    400
  2. B
    600
  3. C
    800
  4. 1,200Cevap
  5. E
    2,400

Cevap

1,200 total book titles
The initial inventory consists of 38T\frac{3}{8}T Fiction titles and 0.40×58T=14T0.40 \times \frac{5}{8}T = \frac{1}{4}T Non-Fiction titles. A 20%20\% increase in Fiction adds 340T\frac{3}{40}T, while a 25%25\% decrease in Non-Fiction subtracts 116T\frac{1}{16}T. Combining these gives a net change of 340T−116T=680T−580T=180T\frac{3}{40}T - \frac{1}{16}T = \frac{6}{80}T - \frac{5}{80}T = \frac{1}{80}T. Setting 180T=15\frac{1}{80}T = 15 gives T=1,200T = 1,200.

Adım Adım Çözüm

1
Determine the initial fractional breakdown of each book category relative to the total inventory T
Fiction =38T= \frac{3}{8}T. Remaining inventory =1−38=58T= 1 - \frac{3}{8} = \frac{5}{8}T. Non-Fiction =0.40×58T=25×58T=14T= 0.40 \times \frac{5}{8}T = \frac{2}{5} \times \frac{5}{8}T = \frac{1}{4}T.
The question specifies that Non-Fiction is 0.40 of the remaining inventory, not the total inventory.
2
Calculate the net change in titles as a fraction of total initial inventory T
Increase in Fiction =20%×38T=15×38T=+340T= 20\% \times \frac{3}{8}T = \frac{1}{5} \times \frac{3}{8}T = +\frac{3}{40}T.
Decrease in Non-Fiction =25%×14T=14×14T=−116T= 25\% \times \frac{1}{4}T = \frac{1}{4} \times \frac{1}{4}T = -\frac{1}{16}T.
Net fractional change =340T−116T=680T−580T=+180T= \frac{3}{40}T - \frac{1}{16}T = \frac{6}{80}T - \frac{5}{80}T = +\frac{1}{80}T.
Percentage changes must be applied to each category's specific share of the total inventory.
3
Equate the net fractional change to the numerical net increase and solve for T
\frac{1}{80}T = 15 \implies T = 15 \times 80 = 1,200.
The overall net increase is given as 15 titles.

Anahtar Kavram

Multi-step successive fraction and percentage change with changing base values
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Fractions, Decimals, and Percents Arithmetic

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