NCERT Exemplar Solutions
Class 6 - Mathematics

Unit 1: Number System

In Roman numeration, a symbol is not repeated more than three times.

In Roman numeration, if a symbol is repeated, its value is multiplied as many times as it occurs.

\(5555=5\times1000+5\times100+5\times10+5\times1\).

\(39746=3\times10000+9\times1000+7\times100+4\times10+6\).

\(82546=8\times1000+2\times1000+5\times100+4\times10+6\).

\(532235=5\times100000+3\times10000+2\times1000+2\times100+3\times10+5\).

\(\text{XXIX}=31\).

\(\text{LXXIV}=74\).

The number LIV is greater than LVI.

4578, 4587, 5478, 5487 are in descending order.

85764 rounded off to nearest hundreds is 85700.

Estimated sum of 7826 and 12469 rounded off to hundreds is 20,000.

The largest six-digit number using digits 5,3,4,7,0,8 once each is 875403.

81652318 is read as eighty one crore six lakh fifty two thousand three hundred eighteen.

The largest 4-digit number formed by 6,7,0,9 using each once is 9760.

Among kilo, milli and centi, the smallest is centi.

Successor of a one digit number is always a one digit number.

Successor of a 3-digit number is always a 3-digit number.

Predecessor of a two digit number is always a two digit number.

Every whole number has its successor.

Every whole number has its predecessor.

Between any two natural numbers, there is one natural number.

The smallest 4-digit number is the successor of the largest 3-digit number.

Of two natural numbers, the one having more digits is greater.

Natural numbers are closed under addition.

Natural numbers are not closed under multiplication.

Natural numbers are closed under subtraction.

Addition is commutative for natural numbers.

1 is the identity for addition of whole numbers.

1 is the identity for multiplication of whole numbers.

There is a whole number which when added to a whole number, gives the number itself.

There is a natural number which when added to a natural number, gives the number itself.

If a whole number is divided by another whole number greater than the first one, the quotient is not equal to 0.

Any non-zero whole number divided by itself gives the quotient 1.

The product of two whole numbers need not be a whole number.

A whole number divided by another whole number greater than 1 never gives the quotient equal to the former.

Every multiple of a number is greater than or equal to the number.

The number of multiples of a given number is finite.

Every number is a multiple of itself.

Sum of two consecutive odd numbers is always divisible by 4.

If a number divides three numbers exactly, it must divide their sum exactly.

If a number exactly divides the sum of three numbers, it must exactly divide the numbers separately.

If a number is divisible both by 2 and 3, then it is divisible by 12.

A number (≥3 digits) is divisible by 6 if its last two digits form a number divisible by 6.

A number with 4 or more digits is divisible by 8, if the last three digits form a number divisible by 8.

If the sum of the digits of a number is divisible by 3, then the number itself is divisible by 9.

All numbers which are divisible by 4 may not be divisible by 8.

The HCF of two or more numbers is greater than their LCM.

LCM of two or more numbers is divisible by their HCF.

LCM of two numbers is 28 and their HCF is 8.

LCM of two or more numbers may be one of the numbers.

HCF of two or more numbers may be one of the numbers.

Every whole number is the successor of another whole number.

Sum of two whole numbers is always less than their product.

If the sum of two distinct whole numbers is odd, then their difference also must be odd.

Any two consecutive numbers are coprime.

If the HCF of two numbers is one of the numbers, then their LCM is the other number.

The HCF of two numbers is smaller than the smaller of the numbers.

The LCM of two numbers is greater than the larger of the numbers.

The LCM of two coprime numbers is equal to the product of the numbers.

99.

(a) 10 million = ____ crore.

(b) 10 lakh = ____ million.

100.

(a) 1 metre = ____ millimetres.

(b) 1 centimetre = ____ millimetres.

(c) 1 kilometre = ____ millimetres.

101.

(a) 1 gram = ____ milligrams.

(b) 1 litre = ____ millilitres.

(c) 1 kilogram = ____ milligrams.

100 thousands = ____ lakh.

Height of a person is 1m 65cm. His height in millimetres is ____.

Length of river Narmada is about 1290 km. Its length in metres is ____.

The distance between Srinagar and Leh is 422 km. The same distance in metres is ____.

Writing of numbers from the greatest to the smallest is called an arrangement in ____ order.

By reversing the order of digits of the greatest number made by five different non-zero digits, the new number is the ____ number of five digits.

By adding 1 to the greatest ____ digit number, we get ten lakh.

The number five crore twenty three lakh seventy eight thousand four hundred one can be written, using commas, in the Indian System of Numeration as ____.

In Roman Numeration, the symbol X can be subtracted from ____, M and C only.

The number 66 in Roman numerals is ____.

The population of Pune was 2,538,473 in 2001. Rounded off to nearest thousands, the population was ____.

The smallest whole number is ____.

Successor of 106159 is ____.

Predecessor of 100000 is ____.

400 is the predecessor of ____.

____ is the successor of the largest 3 digit number.

If 0 is subtracted from a whole number, then the result is the ____ itself.

The smallest 6 digit natural number ending in 5 is ____.

Whole numbers are closed under ____ and under ____.

Natural numbers are closed under ____ and under ____.

Division of a whole number by ____ is not defined.

Multiplication is distributive over ____ for whole numbers.

2395 × ____ = 6195 × 2395

1001 × 2002 = 1001 × (1001 + ____)

10001 × 0 = ____

2916 × ____ = 0

9128 × ____ = 9128

125 + (68+17) = (125 + ____) + 17

8925 × 1 = ____

19 × 12 + 19 = 19 × (12 + ____)

24 × 35 = 24 × 18 + 24 × ____

32 × (27 × 19) = (32 × ____) × 19

786 × 3 + 786 × 7 = ____

24 × 25 = 24 × ____

A number is a ____ of each of its factor.

____ is a factor of every number.

The number of factors of a prime number is ____.

A number for which the sum of all its factors is equal to twice the number is called a ____ number.

The numbers having more than two factors are called ____ numbers.

2 is the only ____ number which is even.

Two numbers having only 1 as a common factor are called ____ numbers.

Number of primes between 1 to 100 is ____.

If a number has ____ in ones place, then it is divisible by 10.

A number is divisible by 5, if it has ____ or ____ in its ones place.

A number is divisible by ____ if it has any of the digits 0, 2, 4, 6, or 8 in its ones place.

If the sum of the digits in a number is a ____ of 3, then the number is divisible by 3.

If the difference between the sum of digits at odd places (from the right) and the sum of digits at even places (from the right) of a number is either 0 or divisible by ____, then the number is divisible by 11.

The LCM of two or more given numbers is the lowest of their common ____.

The HCF of two or more given numbers is the highest of their common ____.

Given below are two columns – Column I and Column II. Match each item of Column I with the corresponding item of Column II.

Arrange the numbers in descending order: 8435, 4835, 13584, 5348, 25843

Of the following numbers which is the greatest? Which is the smallest? 38051425, 30040700, 67205602

Write in expanded form:

(a) 74836

(b) 574021

(c) 8907010

As per the census of 2001, the population of four states are given. Arrange in ascending and descending order.

(a) Maharashtra: 96878627

(b) Andhra Pradesh: 76210007

(c) Bihar: 82998509

(d) Uttar Pradesh: 166197921

The diameter of Jupiter is 142800000 metres. Write it with commas suitably according to International System of Numeration.

India’s population increased from 439 million in 1961 to 1028 million in 2001. Find total increase and write in Indian System with commas.

Radius of Earth = 6400 km, Radius of Mars = 4300 km. Whose radius is bigger and by how much?

In 2001, populations of Tripura = 3,199,203 and Meghalaya = 2,318,822. Write in words.

In a city, polio drops were given to 2,12,583 children in March 2008 and to 2,16,813 children in April. Find the difference.

A person had Rs 10,00,000. He purchased TV = 16580, Motorcycle = 45890, Flat = 870000. How much money left?

Out of 1,80,000 Vitamin A tablets, 18,734 distributed. Find remaining.

Chinmay had Rs 6,10,000. He gave Rs 87,500 to Jyoti, Rs 1,26,380 to Javed and Rs 3,50,000 to John. How much money left?

Find the difference between the largest 7-digit number and the smallest 8-digit number.

Mobile number: first 4 digits = 9, 9, 8, 7; last 3 digits = 3, 5, 5. Find remaining distinct digits to make greatest possible number.

Mobile number: first 4 digits = 9,9,7,9. Make smallest number using one digit twice from 8,3,5,6,0.

Five-digit number: Ten’s place digit = 4, Unit digit = 1/4 of ten’s place = 1, Hundred’s = 0, Thousand’s = 5×unit = 5, Ten-thousand’s = 2×ten’s = 8. Write number.

Find sum of greatest and least 6-digit numbers formed by digits 2,0,4,7,6,5 (using each digit once).

A container has 35874 litres cold drink. How many bottles of 200 ml each?

Population of a town = 450772. One out of 14 illiterate. Find number of illiterate persons.

Find LCM of 80, 96, 125, 160.

Make greatest and smallest 5-digit numbers using different digits, with 5 at ten’s place.

How many grams should be added to 2kg 300g to make it 5kg 68g?

A box contains 50 biscuit packets each 120g. How many such boxes can be loaded in a van of 900kg?

How many lakhs make 5 billion?

How many millions make 3 crores?

Estimate each of the following by rounding off each number to nearest hundreds:

(a) 874 + 478

(b) 793 + 397

(c) 11244 + 3507

(d) 17677 + 13589

Estimate each of the following by rounding off each number to nearest tens:

(a) 11963 – 9369

(b) 76877 – 7783

(c) 10732 – 4354

(d) 78203 – 16407

Estimate each of the following products by rounding off each number to nearest tens:

(a) 87 × 32

(b) 311 × 113

(c) 3239 × 28

(d) 1385 × 789

The population of a town was 78787 in 1991 and 95833 in 2001. Estimate the increase in population by rounding off each population to nearest hundreds.

Estimate the product \(758 \times 6784\) using the general rule.

A garment factory produced 216315 shirts, 182736 trousers and 58704 jackets in a year. Find the total production.

Find the LCM of \(160, 170\) and \(90\).

A vessel has 13 litres 200 mL of juice. In how many glasses of capacity 60 mL can it be filled?

Determine the sum of the following: (a) successor of 32, (b) predecessor of 49, (c) predecessor of the predecessor of 56, (d) successor of the successor of 67.

A tempo carries 482 boxes of 15 kg each and a van carries 518 boxes of the same weight. Find the total weight.

Leela spent Rs 216766, Rs 122322, Rs 88234 and Rs 26780 on various items. Find the total amount spent.

A box has 5 strips, each with 12 capsules of 500 mg medicine. Find the total weight of medicine in grams in 32 such boxes.

Find the least number that leaves remainder 2 when divided by 3, 4 and 5.

Oil volumes are 120 L, 180 L and 240 L. He fills equal-capacity tins to use all oil. What is the greatest capacity of a tin?

Find a 4-digit odd number using digits 1, 2, 4 and 5 once each such that interchanging the first and last digits gives a number divisible by 4.

Using digits 1, 2, 3 and 4 only once, determine the smallest 4-digit number divisible by 4.

Postal charges are Rs 20, Rs 28 and Rs 36. If stamps of one denomination are to be used, what is the greatest denomination needed?

Biscuit packets contain 12, 15 and 21 biscuits. Find the minimum number of packets of each brand to buy an equal number of biscuits of each brand.

A room is 8 m 96 cm by 6 m 72 cm. Find the minimum number of square tiles of the same size needed to cover the floor.

A library has 780 English and 364 Science books. Each shelf must have the same number of books of each subject. What is the minimum number of books on each shelf?

A van stops every 6th block and a bus every 10th block in a colony of 1 to 100. At which block numbers will both stop?

Test divisibility by 11: (a) 5335 (b) 9020814

Using divisibility tests, determine which of the following are divisible by 4: (a) 4096 (b) 21084 (c) 31795012

Using divisibility tests, which of the following are divisible by 9? (a) 672 (b) 5652