NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Algebraic expressions and equations

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Required skills (click to reveal)

A library has a collection of 96 books, and each bookshelf holds 59 books. If the library uses a new shelving system that allows exactly 59 books per shelf, how many shelves are needed to store all 96 books?

Express your answer as an exact fraction in simplest form.

Concept illustrations

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Combining like terms as equal tiles. Each tile represents x. Three x-tiles and two x-tiles make five x-tiles. A constant tile represents a different quantity.xxxxx3x2x3x + 2x = 5x
Each tile represents x. Three x-tiles and two x-tiles make five x-tiles. A constant tile represents a different quantity.
Distributive law as a rectangle. A rectangle of height 3 and total width x + 4 splits into areas 3x and 12. Therefore 3(x + 4) = 3x + 12. Algebraic lengths are schematic.x433x123(x + 4) = 3x + 12
A rectangle of height 3 and total width x + 4 splits into areas 3x and 12. Therefore 3(x + 4) = 3x + 12. Algebraic lengths are schematic.

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Formula

Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The library has 96 books to store, and each shelf holds exactly 59 books.
  2. To find the number of shelves needed, divide the total number of books by the capacity per shelf: 96 ÷ 59.
  3. The division 96 ÷ 59 results in a fraction: 96/59.
  4. Check if the fraction 96/59 can be simplified by finding the greatest common divisor (GCD) of 96 and 59.
  5. 59 is a prime number and does not divide 96 evenly (96 ÷ 59 ≈ 1.627), so the GCD is 1.
  6. Since the GCD is 1, the fraction 96/59 is already in its simplest form.
  7. The result must be expressed as an exact fraction in simplest form, as required.
  8. No further simplification is possible, and the fraction 96/59 is exact.
  9. Verify by multiplication: 59 × (96/59) = 96, confirming the fraction correctly represents the number of shelves needed.
  10. Since shelves must be whole units and 96 > 59, at least 2 shelves are needed, but the question asks for the exact fractional representation, not the smallest integer greater than or equal to it.
  11. The exact number of shelves required to store all 96 books, given 59 per shelf, is 96/59.
  12. The answer is confirmed as 96/59, which is already in simplest form and matches the required format.

Reference answer: 96/59

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