NavGanitMateby NavGreen Compute, AustraliaBased on Victorian Curriculum Mathematics v2.0
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Modelling with linear relations

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Required skills (click to reveal)

A public library in Sydney is cataloguing new book acquisitions. The number of books added to the collection each week follows a linear pattern.

After 11 weeks, the total number of books added is 50 more than the number added in the first week. If the initial weekly addition was 12 books, how many books were added in week 5?

Express your answer as an exact number.

Concept illustrations

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Translate a situation into a model. With a fixed charge of $4 and $2 per kilometre, cost C = 4 + 2d. At d = 3 km the cost is $10. Assume this tariff applies for the trip.$4 + $2/kmDefine tariff→C = 4 + 2dBuild rule→d=3 → C=$10Interpret
With a fixed charge of $4 and $2 per kilometre, cost C = 4 + 2d. At d = 3 km the cost is $10. Assume this tariff applies for the trip.
A straight-line relationship. Illustration: y = x + 1. Each unit right adds one unit up; the y-intercept is 1.-2-2-1-111223344xy
Illustration: y = x + 1. Each unit right adds one unit up; the y-intercept is 1.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The number of books added each week follows a linear pattern, so the weekly additions form an arithmetic sequence. The first term a is given as 12 books.
  2. After 11 weeks, the total number of books added is 50 more than the number added in the first week. The first week's addition is 12 books, so the total after 11 weeks is 12 + 50 = 62 books.
  3. The sum of the first n terms of an arithmetic sequence is given by S_n = n/2 * (2a + (n-1)d), where a is the first term, d is the common difference, and n is the number of terms.
  4. Substitute known values into the sum formula: S_11 = 11/2 * (2*12 + (11-1)*d) = 62.
  5. Simplify the equation: 11/2 * (24 + 10d) = 62.
  6. Multiply both sides by 2 to eliminate the denominator: 11 * (24 + 10d) = 124.
  7. Expand the left side: 264 + 110d = 124.
  8. Subtract 264 from both sides: 110d = 124 - 264 = -140.
  9. Solve for d: d = -140 / 110 = -14/11 ≈ -1.27 books per week.
  10. Now find the number of books added in week 5 using the formula for the nth term: a_n = a + (n-1)d. Substitute a = 12, n = 5, d = -14/11.
  11. Calculate: a_5 = 12 + (5-1)*(-14/11) = 12 + 4*(-14/11) = 12 - 56/11 = (132 - 56)/11 = 76/11 ≈ 6.91 books.
  12. The number of books added in week 5 is 76/11, which is approximately 6.91, but since books must be whole numbers, and the model allows fractional values for intermediate steps, the exact value is 76/11. However, the context implies a real-world count, so we must check if the model allows non-integer weekly additions. The problem states a linear pattern and gives initial values, so fractional values are acceptable in the model. The question asks for the number added in week 5, and the model gives 76/11, which is not a whole number. But since the problem does not state that weekly additions must be integers, and the initial value is 12, which is integer, but the common difference is fractional, the model permits non-integer values. Therefore, the answer is 76/11 books, but since the question asks for an exact number and books are discrete, we must re-evaluate: the sum after 11 weeks is 62 books, and the first week is 12, so the total increase over 10 intervals is 50 books, so average weekly increase is 50/10 = 5 books over 10 weeks, but that's not correct because the sum is not linear in the increase. The correct interpretation is that the sum of 11 terms is 62, and the first term is 12. The sum formula gives d = -14/11. Then a_5 = 12 + 4*(-14/11) = 12 - 56/11 = 76/11. Since the problem says 'express as an exact number', and 76/11 is exact, but books are whole items, so the model must allow fractional books for the purpose of the linear relation. Therefore, the answer is 76/11, but this is not a whole number. However, the problem says 'how many books were added in week 5?' and books are countable, so the answer must be an integer. This suggests a contradiction. But the problem states a linear pattern, and the sum after 11 weeks is 50 more than the first week's addition. The first week is 12, so total is 62. The sum of 11 terms is 62, first term 12. The average per week is 62/11 ≈ 5.636. The common difference d must be such that the sum is 62. We solved d = -14/11. Then a_5 = 12 + 4*(-14/11) = 76/11 ≈ 6.91. But 76/11 is not an integer. The problem says 'express as an exact number', so it accepts fractions. Therefore, the answer is 76/11 books. But books are discrete, so this is a mathematical model, not a real count. The problem allows it. So final answer is 76/11.

Reference answer: 76/11

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