Class 8 Maths, Chapter 3 (NCERT). 15 questions by concept. Open “Show answer” for the answer and why.
How to teach this chapter · Common misconceptions & worked visual for this chapter
In the extended Gumulgal system, 'ukasar-ukasar-ukasar-ukasar-urapon' and 'ukasar-ukasar-ukasar-urapon' are put together. What number is the total?
(a) 16
Ukasar means 2 and urapon means 1. Add the values within each name, then add the two totals: 9 + 7 = 16.
In the extended Gumulgal system, how many words does the name for 30 need?
(a) 15
Each ukasar accounts for 2. Use one word for each pair, and one urapon if 1 remains. This needs 15 words.
How is (2) x 3600 + (88) x 60 + (15) x 1 written in base 60?
(c) (3) x 3600 + (28) x 60 + (15) x 1
88 sixties are the same as 1 group of 3600 and 28 sixties. So carry 1 into the 3600s place and leave 28 sixties behind. The number itself does not change, and now every place holds less than 60. This gives (3) x 3600 + (28) x 60 + (15) x 1.
How many places are needed to write 754 in base 60, counting any empty places between the highest occupied place and the units?
(b) 2
The place values from the right are 1, 60, 3600 and 216000. Since , 754 needs 2 places. Count every place from the highest occupied place to the units, including any empty place.
Compare 11001 in base 2 and 31 in base 8. Which answer correctly describes their values?
(b) 25 and 25
Use the powers of each stated base as place values. Both numerals represent 25. Their different lengths do not imply different values.
The numeral 10 is read in base 5 and in base 2. Which reading gives the greater number?
(b) Base-5 reading
In base 5, the value is . In base 2, the value is . These totals are written in base ten. Since , the base-5 reading gives the greater number.
Use the seven Roman symbols I, V, X, L, C, D and M. Their values are 1, 5, 10, 50, 100, 500 and 1000, respectively.
and .
How many of these two products have a single symbol among the seven Roman symbols?
(c) 0
2500 is not among the seven symbol values. 25 is also not among the seven symbol values. Therefore, the number of products with a single symbol is 0.
While adding Roman numerals you collect a pile of Is. How many of them make one V?
(a) 5
Divide the value of V by the value of I. It takes 5 copies of I to make V.
A Roman numeral records a number as a sum of landmark numbers, taking as many of the largest as possible each time. Which sum does the numeral for 715 record?
(a) 500 + 100 + 100 + 10 + 5
Take the largest available landmark that fits, then repeat with the remainder. The resulting sum is 500 + 100 + 100 + 10 + 5.
A positional number system uses all the digits from 0 to 5 and no others. What is its base?
(a) 6
The base is one more than the largest digit: 5 + 1 = 6. When a place collects 6 units, they regroup into one unit of the next place.
Each shell in a collection gets a different letter, picked in no fixed order. The last letter used is 'u'. How many shells are there?
(a) Cannot tell
Labels tell how many only when they are used in their fixed order a, b, c, and so on. Then the last label reached is the count. Here the letters are picked in no fixed order, so 'u' is only the letter that happened to be used last. Its place in the alphabet is not a count of the shells. The alphabet has 26 letters, but that is how many labels exist, not how many shells there are. So the number of shells cannot be told.
The grouping 1 x 27 + 3 x 9 + 1 x 3 + 1 x 1 comes to 58, but one of its places holds 3. Written as a base-3 numeral, what is 58?
(c) 2011
Regroup 3 groups of 9 as one group of 27. This gives 2 x 27 + 0 x 9 + 1 x 3 + 1 x 1, and the base-3 numeral is 2011.
In Egyptian numerals, a stroke is 1, a heel bone is 10, a coiled rope is 100, a lotus flower is 1000, a bent finger is 10000. Numeral A: 7 coiled ropes, 8 heel bones and 4 strokes. Numeral B: 8 heel bones, 7 coiled ropes and 4 strokes. How do the values of A and B compare?
(a) A = B
An Egyptian numeral is the sum of its symbols, so the order in which they are written does not matter. A and B hold the same symbols in the same counts, so both stand for 784 and A = B. Reading the counts as digits in the order listed gives 784 and 874, but an Egyptian numeral has no place value.
A rod numeral shows 4 and then 7, both in upright zong rods, with exactly one empty place between the groups. The last group is in the units place. Zong rods stand in units, hundreds and ten-thousands places; heng rods in tens, thousands and hundred-thousands places. What number is it?
(b) 407
The last group, 7, is in the units place. The explicitly stated single empty intervening place is the tens place. The first group, 4, therefore stands two places above the last group. Keeping that empty place gives 407. The rod orientation alone fixes place parity; the stated gap fixes the actual spacing.
In Mesopotamian base-60 numerals, each wedge adds 1 within its group. A numeral contains 4 wedges, with no clear spacing between groups. Which of 4, 63 and 122 could this numeral represent?
(a) 4, 63 and 122
One group in the units place gives 4. One wedge in the 60s place gives , using the remaining wedges as units. Two wedges in the 60s place give . Without clear group spacing, each listed value is possible.
Wrong
IX is larger than X because IX has two symbols.
Why it feels right. In our usual whole-number notation, more digits mean a larger number when there are no leading zeros.
Right
Roman numerals use a different rule. I before X subtracts one.
Worked example
