CBSE Class 8 Mathematics A Story of Numbers MCQs Set 11

Answer: (b) numerals

Explanation:
1. The symbols a written number system uses to show numbers are called numerals, e.g. 0, 1, 5, 36 and 193 in the Hindu system.
2. Landmark numbers are numbers (not symbols), notches are tally marks, and a base is the fixed multiplier of a system.

Teacher's Note:
1. Number system = the sequence; numerals = its written symbols.
2. Keep the words 'number' and 'numeral' apart.

Answer: (c) 500

Explanation:
1. Roman landmark symbols: I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1000.
2. So D stands for 500.

Teacher's Note:
1. Learn the seven symbols in order: I, V, X, L, C, D, M.
2. Each step alternates between ×5 and ×2.

Answer: (a) base-10 system

Explanation:
1. Each Egyptian landmark number is 10 times the one before it: 1, 10, 10², 10³, …
2. These are the powers of 10, so it is a base-10 (decimal) system.

Teacher's Note:
1. The base is the number you multiply by to reach the next landmark number.
2. Base-60 belongs to Mesopotamia, not Egypt.

Answer: (d) 60

Explanation:
1. The later Mesopotamian (Babylonian) system was a base-60, or sexagesimal, system.
2. We still see it in 1 hour = 60 minutes and 1 minute = 60 seconds. So the answer is 60.

Teacher's Note:
1. Link 'Babylonian' with 60 through clock time.
2. Sexagesimal is the name for base 60.

Answer: (c) 1 and 5

Explanation:
1. The Mayans used a dot for 1 and a bar for 5.
2. Combining them they wrote every number from 1 to 19. So the answer is 1 and 5.

Teacher's Note:
1. Read 'respectively' carefully: dot first, then bar.
2. Three bars and four dots make 19, the largest in one position.

Answer: (b) may have had common ancestors who used this number system

Explanation:
1. All three groups counted in 2s and formed number names in the same way, yet there is no sign of contact between them.
2. One theory is that they had common ancestors who used this system, whose descendants later moved to these far-off places.

Teacher's Note:
1. The key clue is 'no contact', which rules out trade.
2. Counting in 2s has nothing to do with Roman or Mesopotamian systems.

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

Explanation:
1. A is true: the Hindu system is a place value system.
2. R is true, and it is exactly the definition of a place value system, so R correctly explains A.

Teacher's Note:
1. When R is the definition of the idea in A, R explains A.
2. In 375, the 3 counts hundreds only because of its position.

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

Explanation:
1. A is false: the Roman system has no symbol for zero at all, and it is not a place value system.
2. R is true: an empty position in a place value system must be marked, otherwise the numeral can be misread.
3. So A is false but R is true.

Teacher's Note:
1. Roman numerals never needed a zero because position does not decide value there.
2. Check each statement on its own first.

Answer: (a) Only Statement I is true.

Explanation:
1. Statement I is true: the Lebombo bone has 29 notches.
2. Statement II is false: the Ishango bone was found in the Democratic Republic of Congo; it is the Lebombo bone that comes from South Africa.
3. So only Statement I is true.

Teacher's Note:
1. Lebombo: South Africa, 29 notches, about 44,000 years old.
2. Ishango: Democratic Republic of Congo, notches in columns.

Answer: (a) LXVIII

Explanation:
1. Group 68 into landmark numbers: 68 = 50 + 10 + 5 + 1 + 1 + 1.
2. That is L X V I I I = LXVIII.
3. Check the others: XLVIII = 48, LXVII = 67, LXXVIII = 78.

Teacher's Note:
1. Always take the largest landmark number first.
2. XL before a number means 40, not 60.

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

Explanation:
1. Roman landmark numbers 1, 5, 10, 50, 100, … are not powers of one number: (i)-(r).
2. Egyptian is base-10 but needs a new symbol for every higher power of 10: (ii)-(s).
3. Mesopotamian is base-60 and positional: (iii)-(p).
4. Hindu is base-10 with ten digits including 0: (iv)-(q). So the answer is (c).

Teacher's Note:
1. Match the easiest pair first (base-60 is Mesopotamian).
2. Then eliminate options that disagree with it.

Answer: (b) 23

Explanation:
1. The notches are in four groups of five and one group of three.
2. Total = 4 × 5 + 3 = 20 + 3 = 23.

Teacher's Note:
1. Grouping marks in fives makes tallies quick to count.
2. Count the last, smaller group carefully.

Answer: (c) the Bakhshali manuscript

Explanation:
1. The Bakhshali manuscript (about the 3rd century CE) has the first known use of the ten digits, with 0 as a dot.
2. The Aryabhatiya (499 CE) and Brahmasphutasiddhanta (628 CE) are later; the Yajurveda Samhita gives number names, not written digits.

Teacher's Note:
1. Bakhshali = earliest written zero (as a dot).
2. Arrange the four texts by date to eliminate options.

Answer: (a) Brahmagupta

Explanation:
1. Brahmagupta codified 0 as a number in his Brahmasphutasiddhanta in 628 CE.
2. Aryabhata used 0 earlier (499 CE) but did not codify its rules in this way.

Teacher's Note:
1. 628 CE and Brahmagupta go together.
2. Al-Khwarizmi and Fibonacci spread the numerals; they did not codify zero.

Answer: (d) 1, 7, 49, 343, …

Explanation:
1. Landmark numbers of a base-n system are the powers of n, starting from n⁰ = 1.
2. For base 7: 1, 7, 7² = 49, 7³ = 343, … So the answer is 1, 7, 49, 343, …

Teacher's Note:
1. The list must start with 1.
2. Each landmark number is multiplied by 7, not increased by 7.

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

Explanation:
1. A is true: 10 copies of any Egyptian symbol are replaced by one copy of the next symbol.
2. R is false: the Egyptian system has no place value; a symbol means the same wherever it is written.
3. So A is true but R is false.

Teacher's Note:
1. Egyptian = base-10 but not positional.
2. The regrouping rule, not position, explains A.

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

Explanation:
1. In a true base-20 system the third landmark number would be 20² = 400.
2. The Mayans used 360 instead, so A is false and R is true.
3. So A is false but R is true.

Teacher's Note:
1. For a true base, every landmark number must be a power of the base.
2. 360 is 18 × 20, not 20 × 20.

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

Explanation:
1. Both statements are true.
2. The Roman system spread with the Roman empire and was handy for recording numbers; its difficulty with multiplication is a drawback, not the reason it was used.
3. So R is not the correct explanation of A.

Teacher's Note:
1. Ask: does R give a reason for A? Here it works against A.
2. A true drawback cannot explain popularity.

Answer: (b) Only Statement II is true.

Explanation:
1. Statement I is false: the next landmark number is got by multiplying by n, not adding n.
2. Statement II is true: the landmark numbers are the powers of n.
3. So only Statement II is true.

Teacher's Note:
1. Adding gives multiples (n, 2n, 3n); multiplying gives powers.
2. Powers grow much faster, which is what a base needs.

Answer: (c) Both statements are true.

Explanation:
1. Statement I is true: hours, minutes and seconds come from the Mesopotamian base-60 system.
2. Statement II is true: the Chinese rod numerals left a blank for a skipped place, as the Mesopotamians did.
3. So both statements are true.

Teacher's Note:
1. Base 60 survives in time and in angle measure.
2. Leaving a blank was the step before a zero symbol.

Answer: (b) three 25s, two 5s and two 1s

Explanation:
1. 125 > 87, so start with 25: 87 = 3 × 25 + 12.
2. 12 = 2 × 5 + 2.
3. So 87 = three 25s, two 5s and two 1s (75 + 10 + 2 = 87).

Teacher's Note:
1. Option (a) also totals 87, but six 5s are not allowed in base 5.
2. Never use a landmark number 5 or more times in base 5.

Answer: (d) (1) × 3600 + (22) × 60 + 5

Explanation:
1. 4925 = 1 × 3600 + 1325.
2. 1325 = 22 × 60 + 5 (since 22 × 60 = 1320).
3. So 4925 = (1) × 3600 + (22) × 60 + 5. Option (c) gives 4985 and option (b) gives 4865.

Teacher's Note:
1. Divide by 3600 first, then divide the remainder by 60.
2. Check by expanding back: 3600 + 1320 + 5 = 4925.

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

Explanation:
1. Aryabhata (c. 499 CE) first fully explained and computed with the ten symbols: (i)-(q).
2. Brahmagupta codified 0 as a number in 628 CE: (ii)-(s).
3. Al-Khwarizmi wrote On the Calculation with Hindu Numerals around 825: (iii)-(r).
4. Fibonacci urged Europe to adopt the numerals around 1200: (iv)-(p). Answer (a).

Teacher's Note:
1. Use the dates: 499, 628, 825, 1200 run in the same order as the names.
2. The word 'algorithm' comes from Al-Khwarizmi.

Answer: (a) 1267

Explanation:
1. Thousands line: 1 counter = 1000.
2. Hundreds line: 2 counters = 2 × 100 = 200.
3. Tens: 1 counter just above the line (50) and 1 on it (10) = 60.
4. Ones: 1 counter just above the line (5) and 2 on it (2) = 7.
5. Total = 1000 + 200 + 60 + 7 = 1267.

Teacher's Note:
1. A counter between two lines belongs to the line just below it and counts 5 of that line.
2. Work line by line from the top.

Answer: (c) 101

Explanation:
1. In base 7 the places are 49, 7 and 1.
2. 203 means 2 × 49 + 0 × 7 + 3 × 1 = 98 + 0 + 3 = 101.
3. Option (a) comes from using base 8 by mistake; option (d) ignores the empty 7s place.

Teacher's Note:
1. Write the place values above the digits before multiplying.
2. A 0 digit still holds its place.

Answer: (b) 23

Explanation:
1. 1101 in base 2 = 1 × 8 + 1 × 4 + 0 × 2 + 1 × 1 = 13.
2. Group 13 into powers of 5: 13 = 2 × 5 + 3.
3. So in base 5 it is written as 23.

Teacher's Note:
1. Convert to base 10 first, then to the new base.
2. Option (a) is the base-10 value, not the base-5 numeral.

Answer: (d) Neither statement is true.

Explanation:
1. Statement I is false: a dot stood for 1 and a bar for 5, not the other way round.
2. Statement II is false: the symbols were written one below the other and the lowest group gave the 1s.
3. So neither statement is true.

Teacher's Note:
1. Mayan numerals are read from the bottom up.
2. Dot = 1, bar = 5.

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

Explanation:
1. Both statements are true.
2. The Mesopotamians showed an empty place by a blank; since spacing was uneven, the same numeral could be read as different numbers.
3. That is exactly why a placeholder is needed, so R explains A.

Teacher's Note:
1. Think of 25 and 205: without 0 they would look alike.
2. The placeholder removes this ambiguity.

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

Explanation:
1. One-to-one mapping pairs each object with one item of a standard sequence: (i)-(r).
2. A landmark number is given its own basic symbol: (ii)-(s).
3. In a base-n system the landmark numbers are powers of n: (iii)-(q).
4. In a place value system position decides the landmark number: (iv)-(p). Answer (d).

Teacher's Note:
1. These four terms are the key vocabulary of the chapter.
2. Match one sure pair, then check the options.

Answer: (c) 367

Explanation:
1. Lowest group: one bar and two dots = 5 + 2 = 7 ones.
2. Middle group: the shell, so 0 twenties.
3. Top group: one dot = 1 three-hundred-sixty.
4. Number = 1 × 360 + 0 × 20 + 7 = 367. (407 would be right only if the top place stood for 400.)

Teacher's Note:
1. Read Mayan numerals from the bottom up.
2. Remember the third place is 360, not 400.