NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Integer and rational-number calculations

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Required skills (click to reveal)

A craft designer in Melbourne is creating a symmetrical tile pattern for a public art installation. The total area of the central tile is determined by the expression: (-60/7) × (3/5) + (23/4) ÷ (-2/3).

Use efficient strategies such as simplifying fractions and applying the rules for operations with rational numbers to calculate the exact value of this expression. Show all steps and provide the final answer as a simplified fraction.

Concept illustrations

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Common denominators as equal strips. The first strip selects 4/12 = 1/3; the second selects 3/12 = 1/4. The result selects 7/12. Every strip represents the same whole.1/31/47/124/12 + 3/12 = 7/12
The first strip selects 4/12 = 1/3; the second selects 3/12 = 1/4. The result selects 7/12. Every strip represents the same whole.
Adding five to negative three. Start at −3. Move five units right to reach 2.-5-4-3-2-1012345+5
Start at −3. Move five units right to reach 2.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The expression to evaluate is (-60/7) × (3/5) + (23/4) ÷ (-2/3). This involves multiplication and division of rational numbers, including negative fractions.
  2. First, compute (-60/7) × (3/5). Multiply numerators: (-60) × 3 = -180. Multiply denominators: 7 × 5 = 35. So the result is -180/35.
  3. Simplify -180/35 by dividing numerator and denominator by their greatest common divisor, 5. -180 ÷ 5 = -36, 35 ÷ 5 = 7. So -180/35 = -36/7.
  4. Next, compute (23/4) ÷ (-2/3). Division of fractions is equivalent to multiplying by the reciprocal: (23/4) × (-3/2).
  5. Multiply numerators: 23 × (-3) = -69. Multiply denominators: 4 × 2 = 8. So the result is -69/8.
  6. Now add the two results: (-36/7) + (-69/8). This requires a common denominator. The least common multiple of 7 and 8 is 56.
  7. Convert -36/7 to a fraction with denominator 56: (-36 × 8)/(7 × 8) = -288/56.
  8. Convert -69/8 to a fraction with denominator 56: (-69 × 7)/(8 × 7) = -483/56.
  9. Add the two fractions: (-288/56) + (-483/56) = (-288 - 483)/56 = -771/56.
  10. Check if -771/56 can be simplified. The GCD of 771 and 56 is 1 (771 = 3 × 257, 56 = 2^3 × 7), so the fraction is already in simplest form.
  11. The exact value of the expression is -771/56. This is a negative rational number, and no further simplification is possible.
  12. Final answer: -771/56 square units (as area is a positive quantity, but the expression yields a negative value, which may indicate a directional or signed interpretation in context, but mathematically the value is correct as computed).

Reference answer: -771/56

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