### Find basic information about a stem-and_leaf diagram and deduce two newly added data from increased mean and mode

Question Sample Titled 'Find basic information about a stem-and_leaf diagram and deduce two newly added data from increased mean and mode'

The stem-and-leaf diagram below shows the distribution of the weight (in kg) of the students in a swimming club.

Stem (tens)Leaf (units)
 ${4}$ ${5}$ ${6}$ ${7}$ ${8}$
 ${3}$ ${5}$ ${6}$ ${6}$ ${6}$ ${6}$ ${2}$ ${4}$ ${4}$ ${5}$ ${5}$ ${6}$ ${0}$ ${2}$ ${6}$ ${8}$ ${1}$ ${2}$ ${1}$ ${2}$

 (a) Find the mean, the median and the range of the above distribution. (3 marks) (b) Two more students now join the club. It is found that both the mean and the range of the distribution of the weights are increased by ${1}$ kg. Find the weight of each of these two students. (4 marks)

 (a) The mean $={58}$ $\text{kg}$ 1A The median $={55}$ $\text{kg}$ 1A The range $={82}-{43}$ $={39}$ $\text{kg}$ 1A  (b) Let ${a}$ and ${b}$ be the weights of the two students resepectively, where ${a}\le{b}$ . $\dfrac{{{58}\cdot{20}+{\left({a}+{b}\right)}}}{{22}}$ $={58}+{1}$ 1M ${a}+{b}$ $={138}$ $\ldots{\left(\ast\right)}$ Since the range is increased by ${1}$ $\text{kg}$ , the new range is ${40}$ $\text{kg}$ . There are only two cases.

 Case ${1}$: Assume ${a}={42}$ . 1M for either case When ${a}={42}$ , ${b}={96}$ . from ${\left(\ast\right)}$ However, the new range $={96}-{42}={54}\ne{40}$ $\text{kg}$ Thus, this case is impossible. Case ${2}$: Assume ${b}={83}$ . When ${b}={83}$ , ${a}={55}$ . from ${\left(\ast\right)}$ The new range $={83}-{43}={40}$ $\text{kg}$ Thus, this case is the only possible case.

 Therefore, the weights of these two students are ${55}$ $\text{kg}$ and ${83}$ $\text{kg}$ . 2A 1M for ${55}$ and 1M for ${83}$

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