CBSE Class 8 Mathematics Tales by Dots and Lines MCQs Set 11

Answer: (a) dividing the sum of all the values by the number of values

Explanation:
1. Mean = (sum of all the values) ÷ (number of values).
2. Option (b) describes the median, not the mean.
3. Options (c) and (d) do not give an average at all.

Teacher's Note:
1. Mean means 'add all, then share equally'.
2. Keep the words mean and median separate in your mind.

Answer: (c) the middle value when the data is arranged in order

Explanation:
1. First arrange the data in order.
2. The value in the middle position is the median.
3. For an even number of values, the median is the mean of the two middle values.

Teacher's Note:
1. Always sort the data before picking the middle.
2. Median depends on position, not on the size of the extreme values.

Answer: (b) 10

Explanation:
1. Sum = 6 + 11 + 9 + 14 = 40.
2. There are 4 values.
3. Mean = 40 ÷ 4 = 10.

Teacher's Note:
1. Count the values carefully before dividing.
2. A quick check: the mean must lie between the smallest and largest value.

Answer: (d) the total of the distances of the values below it equals the total of the distances of the values above it

Explanation:
1. Add up how far each value below the mean is from it, and do the same for the values above it.
2. These two totals are always equal, so the data 'balances' at the mean, like a see-saw.
3. The mean need not be the midpoint of the extremes, need not be a data value, and need not have equal numbers of values on each side.

Teacher's Note:
1. Think of the values as equal weights on a see-saw.
2. The mean is the point where the see-saw balances.

Answer: (a) how a quantity changes over time

Explanation:
1. A line graph plots points and joins them with line segments in time order.
2. The rise and fall of the line shows how the quantity changes as time passes.

Teacher's Note:
1. Time usually goes along the horizontal axis.
2. Line graphs are best for trends over days, months or years.

Answer: (b) Only Statement I is true.

Explanation:
1. Statement I is the correct rule for an even number of values.
2. Statement II is false: for 8, 11 the median is \( \frac{8 + 11}{2} \) = 9.5, which is not in the data.
3. So only Statement I is true.

Teacher's Note:
1. With an even count, the median is often a value not in the list.
2. One counter-example is enough to show a statement is false.

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. R is the balance property of the mean, which is true.
2. With only two numbers, the point at equal distance from both is the halfway point.
3. So by R, the mean of two numbers is exactly halfway between them. A is true and R explains it.

Teacher's Note:
1. For two numbers, mean = midpoint.
2. For three or more numbers this is usually not true.

Answer: (c) 6

Explanation:
1. 'More than 3 hours' means 4 hours or 5 hours.
2. There are 4 dots above 4 and 2 dots above 5.
3. Number of students = 4 + 2 = 6.

Teacher's Note:
1. 'More than 3' does not include 3 itself.
2. Count the dots in each column carefully.

Answer: (c) =AVERAGE(B2:B9)

Explanation:
1. The range of cells from B2 to B9 is written B2:B9.
2. The AVERAGE function gives the mean of the cells in a range.
3. =AVERAGE(B2, B9) would average only the two cells B2 and B9.

Teacher's Note:
1. A colon means 'from … to'; a comma lists separate cells.
2. SUM adds; AVERAGE finds the mean.

Answer: (d) (i)-(r), (ii)-(p), (iii)-(s), (iv)-(q)

Explanation:
1. Line graph: points joined by segments to show change over time, so (i)-(r).
2. Dot plot: a dot for each value above a number line, so (ii)-(p).
3. Clustered-column graph: bars of different groups side by side, so (iii)-(s).
4. Activity strip: a day in equal coloured time boxes, so (iv)-(q).

Teacher's Note:
1. Match the easiest pair first and cross out options.
2. Here (i)-(r) and (ii)-(p) already narrow it to (b) or (d).

Answer: (d) 7

Explanation:
1. In order: 3, 5, 7, 9, 12.
2. There are 5 values, so the middle one is the 3rd value.
3. Median = 7.

Teacher's Note:
1. Never pick the middle of the unsorted list.
2. For 5 values the median is the 3rd value.

Answer: (b) 30 minutes

Explanation:
1. One day = 24 hours = 24 × 60 = 1440 minutes.
2. 1440 ÷ 48 = 30.
3. So each box stands for 30 minutes.

Teacher's Note:
1. Convert to minutes before dividing.
2. Two boxes make one hour.

Answer: (b) all the cells from B3 to G3 lying in row 3

Explanation:
1. A block of cells is written as Start:End.
2. B3 and G3 are both in row 3.
3. So B3:G3 means every cell of row 3 from column B to column G.

Teacher's Note:
1. The letter is the column; the number is the row.
2. A range is a block of cells, not a calculation.

Answer: (c) the quantity changed by a greater amount in that interval

Explanation:
1. Consecutive points are the same time-gap apart.
2. A steeper segment covers a bigger vertical distance in that same gap.
3. So the quantity changed by a greater amount.

Teacher's Note:
1. Steep = big change; flat = little or no change.
2. Upward steep means a big rise, downward steep a big fall.

Answer: (a) dry months with low rainfall for all six cities

Explanation:
1. In all six graphs the values for January, February and March are very low.
2. The peak months differ from city to city and come later in the year.
3. So January to March are dry months for all six cities.

Teacher's Note:
1. Look for what is common to all the graphs.
2. Low values mean little rain, not missing data.

Answer: (d) 12

Explanation:
1. Sum of the five values = 12.4 × 5 = 62.
2. Sum of the known values = 7 + 11 + 18 + 14 = 50.
3. x = 62 − 50 = 12.

Teacher's Note:
1. Mean × number of values gives the total.
2. Subtract the known values to get the missing one.

Answer: (c) 29.5

Explanation:
1. Adding 6 to each of 12 numbers adds 6 × 12 = 72 to the total.
2. The mean increases by 72 ÷ 12 = 6.
3. New mean = 23.5 + 6 = 29.5.

Teacher's Note:
1. Adding a fixed number to every value adds the same number to the mean.
2. No need to find the total first.

Answer: (a) Saturday

Explanation:
1. Read the two lines day by day: Visiting 18, 15, 12, 20, 26, 34, 30; Purchasing 11, 9, 8, 13, 18, 22, 21.
2. Gaps (Mon to Sun): 7, 6, 4, 7, 8, 12, 9.
3. The greatest gap is 12, on Saturday (34 − 22).

Teacher's Note:
1. The gap is the vertical distance between the two lines.
2. Do not confuse the highest point with the biggest gap.

Answer: (b) Both A and R are true, but R is not the correct explanation of A.

Explanation:
1. Mean = (10 + 10 + 11 + 17) ÷ 4 = 48 ÷ 4 = 12, so A is true.
2. Median = \( \frac{10 + 11}{2} \) = 10.5, so R is true.
3. The median does not tell us how the mean was found, so R does not explain A.

Teacher's Note:
1. Check each statement separately first.
2. Then ask: does R give the reason for A?

Answer: (c) A is true, but R is false.

Explanation:
1. A clustered-column graph would need 13 × 4 = 52 bars and look crowded, so A is true.
2. A line graph is generally used for data over time, but 'only' makes R too strong, so R is false.

Teacher's Note:
1. Watch for strong words like 'only', 'always', 'never'.
2. Four lines are much easier to follow than 52 bars.

Answer: (d) A is false, but R is true.

Explanation:
1. For data with frequencies, Mean = \( \frac{\sum (\text{value} \times \text{frequency})}{\sum \text{frequency}} \).
2. Averaging only the distinct values ignores the frequencies, so A is false.
3. R states the correct principle, so R is true.

Teacher's Note:
1. Each value counts as many times as it occurs.
2. Divide by the total frequency, not by the number of distinct values.

Answer: (c) Both statements are true.

Explanation:
1. Punjab goes from about 19°C in January up to about 38°C in June, a large change.
2. Kerala stays almost flat; its lowest value, about 29°C, comes in July.
3. So both statements agree with the graph.

Teacher's Note:
1. A flat line means small change; a hilly line means large change.
2. Read the lowest point of each line carefully.

Answer: (b) Only Statement II is true.

Explanation:
1. The colour scale shows the difference between per capita rice and wheat consumption.
2. +100 means this difference is the largest, in favour of rice, so Statement II is true.
3. It does not mean that no wheat is eaten, so Statement I is false.

Teacher's Note:
1. Always check what the scale actually measures.
2. A difference scale compares two things; it does not give either amount.

Answer: (a) (i)-(r), (ii)-(p), (iii)-(s), (iv)-(q)

Explanation:
1. (i) 5 values: the 3rd value is 9, so (r).
2. (ii) \( \frac{5 + 8}{2} \) = 6.5, so (p).
3. (iii) 5 values: the 3rd value is 10, so (s).
4. (iv) \( \frac{6 + 6}{2} \) = 6, so (q).

Teacher's Note:
1. Odd count: take the middle value; even count: average the two middle values.
2. All four lists are already in order here.

Answer: (d) Only Statement I is true.

Explanation:
1. Taking away a value that pulled the balance point down lets the mean rise, so Statement I is true.
2. Removing a value below the median moves the middle position towards larger values, so the median rises or stays the same; it never falls.
3. So Statement II is false; only Statement I is true.

Teacher's Note:
1. Removing a small value is like adding a large one, for the mean.
2. For the median, think about which value now sits in the middle.

Answer: (a) 0.225

Explanation:
1. Number of students = 2 + 5 + 9 + 12 + 8 + 4 = 40.
2. Total arrows = 6 + 20 + 45 + 72 + 56 + 32 = 231, so mean = 231 ÷ 40 = 5.775.
3. Running totals 2, 7, 16, 28: the 20th and 21st values are both 6, so median = 6.
4. Difference = 6 − 5.775 = 0.225.

Teacher's Note:
1. Multiply each value by its frequency before adding.
2. Use running totals to locate the middle positions.

Answer: (d) A is false, but R is true.

Explanation:
1. R is the correct rule, so R is true.
2. There are 12 values, so the middle values are the 6th (29) and the 7th (33).
3. Median = \( \frac{29 + 33}{2} \) = 31, not 33. So A is false.

Teacher's Note:
1. Count the values before choosing the middle.
2. With an even count, average the two middle values.

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. Let the total of the n values be S, so the mean is \( \frac{S}{n} \).
2. After doubling, the total is 2S and the new mean is \( \frac{2S}{n} \) = 2 × \( \frac{S}{n} \).
3. So A is true and R gives exactly the reason.

Teacher's Note:
1. Multiplying every value by k multiplies the mean by k.
2. R is the algebra behind A.

Answer: (b) (i)-(q), (ii)-(r), (iii)-(p), (iv)-(s)

Explanation:
1. Including a value equal to the mean keeps the balance, so the mean stays the same: (i)-(q).
2. Adding 4 to every value adds 4n to the total, so the mean increases by 4: (ii)-(r).
3. Multiplying every value by 4 multiplies the total and the mean by 4: (iii)-(p).
4. Removing a value below the mean raises the mean: (iv)-(s).

Teacher's Note:
1. 'Increases by 4' and '4 times as large' are different.
2. Adding shifts the mean; multiplying scales it.

Answer: (c) Cities X and Z both have their peak rainfall in July, while City Y peaks in November.

Explanation:
1. The highest point of X (about 920 mm) and of Z (about 260 mm) are both over July.
2. The highest point of Y (about 350 mm) is over November. So (c) is valid.
3. (a) fails: Y's line stays above zero; (b) fails: January to March are the driest months; (d) fails: X's line is far above the others for most of the year.

Teacher's Note:
1. Find the highest marker on each line.
2. Reject any option the graph contradicts.