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TMUA 2018 · Paper 1 · Question 17 of 20

TMUA 2018 Paper 1 Question 17

Algebra and functions — Means · deducing the sizes of two data sets. Try it first; the answer and a full worked solution are below.

TMUA 2018 · Paper 1Algebra and functionsMeans · deducing the sizes of two data sets5 options
There are two sets of data: the mean of the first set is 15, and the mean of the second set is 20.

One of the pieces of data from the first set is exchanged with one of the pieces of data from the second set.

As a result, the mean of the first set of data increases from 15 to 16, and the mean of the second set of data decreases from 20 to 17.

What is the mean of the set made by combining all the data?

  1. A1614
  2. B1613
  3. C1612
  4. D1623
  5. E1634
Show the answer and worked solution
answer · A
  1. A1614
  2. B1613
  3. C1612
  4. D1623
  5. E1634
The exchange does not change the sizes of the sets, so let them be n1 and n2, and let the swapped values be x from the first set and y from the second. The first set's total rises by y − x, raising its mean by 1, so y − x = n1; the second set's total falls by y − x, lowering its mean by 3, so y − x = 3n2. Hence n1 = 3n2. The combined mean is unaffected by the swap, so it equals 15n1 + 20n2n1 + n2 = 45n2 + 20n24n2 = 654 = 1614.