NCERT Exemplar Solutions Fractions & Decimals - Maths Olympiad Class 6

Ans: (d)
(D) 9/15
All the options given in the question are further simplified as,
(A) 40/50 = 4/5
(B) 12/15
Divide both numerator and denominator by 3.
= 4/5
(C) 16/20
Divide both numerator and denominator by 4.
= 4/5
(D) 9/15
Divide both numerator and denominator by 3.
= 3/5
Therefore, 3/5 ≠ 4/5

Ans: (b)
A fraction whose numerator is less than the denominator is called a proper fraction.
So, 5/7 = 0.715
Therefore, 5/7 lies between 0 and 1.

Ans: (a)
(1 × 3)/(4 × 3) = 3/12
Consider, 3/12
Divide both numerator and denominator by 3.
= 1/4

Ans: (b)
Divide both numerator and denominator by 5.
= 3/4

Ans: (c)
Consider the given fraction, (5/8) = (20/P)
P = 20 × (8/5)
P= 4 × 8
P = 32

Ans: (c)
(C) 15/25
All the options given in the question are further simplified as,
(A) 6/8
Divide both numerator and denominator by 2.
= 3/4
(B) 12/16
Divide both numerator and denominator by 4.
= 3/4
(C) 15/25
Divide both numerator and denominator by 5.
= 3/5
(D) 18/24
Divide both numerator and denominator by 6.
= ¾
Comparing all results, (¾ = ¾ = ¾) ≠ 3/5
Therefore, (6/8 = 12/16 = 18/24) ≠ 15/25

Ans: (b)
We know that, among all fractions with same numerator, the one having smaller denominator will be the highest fraction.
5/9 < 5/8 < 5/7 < 5/6
Therefore, among four options, (b) 5/6 has small denominator. So, it is the greatest fraction.

Ans: (c)
We know that, among all fractions with same denominator, the one having smaller numerator will be the smallest fraction.
3/8 < 5/8 < 7/8 < 9/8
Therefore, among four options, (c) 3/8 has small numerator. So, it is the smallest fraction.

Ans: (a)
If denominators of the given fractions are same, we can add both fractions.
So, (4/17) + (15/17)
= (4 + 15)/17
= 19/17

Ans: (b)
If denominators of the given fractions are same, we can subtract both fractions.
So, (19/9) – (5/9)
= (19 – 5)/9
= 14/9

Ans: (c)
0.7499 lies between 0.749 and 0.75

Ans: (b)
0. 023 lies between 0.02 and 0.03

Ans: (c)
To convert an improper fraction 11/7 into a mixed fraction:

Divide the numerator (11) by the denominator (7): 11 ÷ 7 = 1 remainder 4

Write the quotient as the whole number, and the remainder as the numerator over the original denominator: 1(4/7)

So, 11/7 = 1(4/7), making option (c) the correct answer. 

Ans: (b)
5(4/7) can be expressed as = 5 + (4/7)
= (35 + 4)/7
= 39/7

Ans: (c)
First we have to convert given decimals into like decimals = 0.070 + 0.008
So, sum of 0.070 and 0.008 = 0.070 + 0.008
= 0.078

Ans: (c)
First we have to convert given decimals into like decimals = 0.1820, 0.0925, 0.2900, 0.0380
Now, by comparing 4 decimal numbers, 0.2900 is the greatest.

Ans: (c)
First we have to convert given decimals into like decimals = 0.270, 1.500, 0.082, 0.103
Now, by comparing 4 decimal numbers, 0.082 is the smallest.

Ans: (d)
Place value of the place immediately after the decimal point (i.e. tenth place) is 1/10, that of next place (i.e. hundredths place) is 1/100 and so on.
13.572 correct to the tenths place is 13.6

Ans: (b)
First we have to convert given decimals into like decimals = 15.80
Now, 15.80 – 6.73 = 9.07

Ans: (a)
Decimals can be converted into fractions by removing their decimal points and writing 10,100, etc in the denominators, depending upon the number of decimal places in the decimals.
So, 0.238 = 238/1000
Divide both numerator and denominator by 2
= 119/500

A number representing a part of a whole is called a fraction.
Example: ¼, ¾, 1/5, 3/6 etc.

A fraction with denominator greater than the numerator is called a proper fraction.
Example: 2/5, 3/8, 10/11 etc. are proper fractions.

Fractions with the same denominator are called like fractions.
Example: ½, 3/2, 5/2, 7/2 etc.

Mixed fraction.

18/5 is an improper fraction.
A fraction whose numerator is greater than the denominator is called an improper fraction.

7/19 is a proper fraction.
A fraction whose numerator is less than the denominator is called a proper fraction.

5/8 and 3/8 are like proper fraction.
Fractions with same denominators are called like fractions.

6/11 and 6/13 are unlike proper fractions.
If the denominators are different, then they are called unlike fractions.

The fraction 6/15 in simplest form is 2/5.
The given fraction 6/15, is further simplified by dividing both numerator and denominator by 3.

The fraction 17/34 in simplest form is ½.
The given fraction 17/34, is further simplified by dividing both numerator and denominator by 17.

18/135 and 90/675 are proper, unlike and equivalent fractions.
Consider the two given fractions, 18/135 and 90/675
So, (18/135) = (90/675)
By cross multiplication, we get
(18 × 675) = (90 × 135)
12,150 = 12,150
Therefore, 18/135 and 90/675 are proper, unlike and equivalent fractions.

8(2/7) is equal to the improper fraction 58/7.
Given mixed fraction is converted into improper fraction as = ((7 × 8) + 2)/7
= (56 + 2)/7
= 58/7

87/7 is equal to the mixed fraction
12(3/7)
We know that, mixed fraction = Quotient Remainder/Divisor

Therefore, 87/7 is equal to the mixed fraction
12(3/7)

9 + (2/10) + (6/100) is equal to the decimal number 9.26.
Fractions with denominators 10,100, etc. can be written in a form, using a decimal point, called decimal numbers or decimals.
9 + (2/10) + (6/100) = 9 + 0.2 + 0.06
= 9.26

Decimal 16.25 is equal to the fraction 16¼ or 65/4.
Decimals can be converted into fractions by removing their decimal points and writing 10,100, etc in the denominators, depending upon the number of decimal places in the decimals.
16.25 = 1625/100
Divide both numerator and denominator by 25.
= 65/4
= 16¼

Fraction 7/25 is equal to the decimal number 0.28.
Multiply numerator and denominator by 4 to get denominator 100.
(7/25) = (7 × 4)/(25 × 4)
= 28/100
We know that, fractions with denominators 10,100, etc. can be written in a form, using a decimal point, called decimal numbers or decimals.
= 0.28

(17/9) + (41/9) = 58/9.
Fractions with same denominators are called like fractions.
Sum of two like fractions = (17 + 41)/9
= 58/9

(67/14) – (24/14) = 43/14.
Fractions with same denominators are called like fractions.
Difference of two fractions = (67 – 24)/14
= 43/14

17/2 + 3½ = 12.
First we have to convert mixed fraction into improper fraction = 3½ = 7/2
Fractions with same denominators are called like fractions.
Sum of two like fractions = (17/2) + (7/2)
= (17 + 7)/2
= 24/2
= 12

9 ¼ – 5/4 = 37/4 – 5/4 = (37 – 5)/4 = 32/4 = 8.

4.55 + 9.73 = 14.28.

8.76 – 2.68 = 6.08.

The value of 50 coins of 50 paisa = ₹25.
We know that, ₹ 1 = 100 paisa
So, 50 coins of 50 paisa = 50 × 50
= 2500 paisa.
Then,
= 2500/100
= ₹ 25

3 Hundredths + 3 tenths = 0.33.
Place value of the place immediately after the decimal point (i.e. tenth place) is 1/10, that of next place (i.e. hundredths place) is 1/100 and so on.
3 Hundredths is written as = 3 × (1/100)
= 0.03
3 tenths is written as = 3 × (1/10)
= 0.3
Then sum of 3 Hundredths, 3 tenths = 0.03 + 0.3
= 0.33

False.
Fractions with same denominators are called like fractions.

False.
Lowest form of given fraction 18/39
Divide both numerator and denominator by 3,
= 6/13

True.
Consider the two given fractions, 15/39 and 45/117
So, (15/39) = (45/117)
By cross multiplication, we get
(15 × 117) = (45 × 39)
1,755 = 1,755

True.
For example: consider two fractions 10/5 and 15/5.
Sum of two fractions = (10 + 15)/5
= 25/5
= 5
= 5/1
A fraction in which there is no common factor, except 1, in its numerator and denominator is called a fraction in the simplest or lowest form.
When 2 fractions are added, the result in most cases will be a fraction p/q form, but in some case if it does happen to be just an integer, it can always be written with denominator 1 (hence p/q form).

False.
Not necessarily a fraction. But can be written in fraction.

True.
A fraction is a number representing a part of a whole. This whole may be a single object or a group of objects.
For example: consider a circle is divided into 4 equal parts. Out of four equal parts 3 of them are shaded.
So, it can be represented in the form of fraction = 3/4

False.
Let us assume a digit be ‘y’.
The place value of a digit at the tenths place = y × (1/10)
= y/10
Then,
The tenths place is 10 times the same digit at the ones place.
y/10 = 10y is not possible.

True
Let ‘a’ be the same digit at tens and hundreds place in a number.
Place value of digit at tens place = 10 × a = 10a
Place value of digit at hundreds place = 100 × a = 100a
Hence, the place value of a digit at the hundreds place is 10 times the same digit at the tens place.

False.
Consider the given decimal number, 3.725
The thousandths place has number 5.
Then, hundredths has number 2 it will be increased by 1 number to correct two decimal places.
Therefore, the decimal 3.725 is equal to 3.73 correct to two decimal places.

True.
25/8 can be further simplified by dividing both numerator and denominator by 8.
= 3.125

False.
The decimal 23.2 = 232/10
Dividing both denominator and numerator by 2, we get.
= 116/5 =
23(1/5)

True.
Circle is divided into 8 equal parts. Out of 8 equal parts 3 of them are shaded.

False.
Rectangle is divided into 9 equal parts. Out of 9 equal parts 4 of them are unshaded. So, fraction represented by the unshaded portion in the adjoining figure = 4/9.

False.
So, (25/19) + (6/19)
= (25 + 6)/19
= 31/19
Hence, 31/19 ≠ 31/38

False.
Consider Left Hand Side (LHS),
LCM of 18 and 15 = 90
Then,
(8/18) = (8 × 5)/(18 × 5) = 40/90
(8/15) = (8 × 6)/(15 × 6) = 48/90
Difference of two fractions (40/90) – (48/90)
= -8/90
Right Hand Side (RHS) = 8/3
By comparing LHS and RHS,
LHS ≠ RHS
-8/90 ≠ 8/3

True.
Consider Left Hand Side (LHS),
Sum of like fractions = (7/12) + (11/12)
= (7 + 11)/12
= 18/12
Divide both numerator and denominator by 6, we get,
= 3/2
Right Hand Side (RHS) = 3/2
By comparing LHS and RHS,
LHS = RHS
3/2 = 3/2

False
Firstly convert 3.03 and 0.016 into like fractions, writing the decimals in column form and finally by adding we get,

True

True
The given fractions are like fractions. On comparing the numerators, we get

False
Since, the digit at tenth place of 19.25 is 2 and the digit at tenth place of 19.053 is 0, where 2 > 0.
∴ 19.25 >19.053

True
Since, after converting the given decimals in like decimals we get, 13.730 = 13.73.
Directions: In each of the questions 66 to 71, fill in the blanks using ’<’, ‘>’ or ‘=’

The L.C.M. of 16 and 15 is 2 × 2 × 2 × 2 × 3 × 5 = 240
Thus, 

and

On comparing, we observe that

The L.C.M of 15 and 14 is
2 × 3 × 5 × 7 = 120

The L.C.M of 75 and 200 is 2 × 2 × 2 × 3 × 5 × 5 = 600

Converting the given decimals into like decimals, they become 3.25 and 3.40. The whole number part of these is same. On comparing, we get their tenths digits 2 < 4
∴ 3.25 < 3.4

1.3 = (13/10)
∵ The L.C.M. of 15 and 10 is 2 × 3 × 5 = 30
Now, 

Thus, 

∴ 6.25 = 25/4

In the given figure, total parts in which figure has been divided is 8 and out of which 7 parts are shaded.
∴ The required fraction is 7/8

In the given figure, total parts in which figure has been divided is 15 and out of which 4 parts are unshaded.
∴ The required fraction is 4/15.

Since, Ali has to divide one fruit cake equally among 6 persons
∴ Each person will get 1/6 part.

∵ The digits are already given in the form of like decimals.
Clearly,
12.104 < 12.124 < 12.142 < 12.214 < 12.401

The required number is 0.8531, which is the largest four digit decimal number less than 1.

The required number is 0.2345, which is the smallest four digit decimal number.

We have, 

We have, 

We have, 

Now,  

We have, 
0.041 = 41/1000

We have, 
6.03 = 603/100

We have, 
5201g = (5201/1000) kg
[∵ 1 kg = 1000 g]
= 5.201 kg

We have,

We have, 
1537 cm = (1537/100) m
[∵  1 m = 100 cm]
= 15.37 m

We have, 
2435m = (2435/1000) km
[ ∵ 1 km = 1000 m]
= 2.435 km
Firstly, convert the fraction (2435/1000) into the simplest form, for this dividing the numerator and denominator by 5, we get

i.e.,
2(87/200) km

We have given, 

Firstly find the L.C.M. of 3, 4, 2 and 6.

The L.C.M. of 3, 4, 2 and 6 is 2 × 2 × 3 = 12

We have given, 

Let 3/4 = ?/44
Then, we have to find the missing numeral.
To get 44 in the denominator, we multiply 4 by 11.
So, we multiply the numerator and denominator by 11.

Hence, 3/4 and 33/44are equivalent fractions.

Let 

Then, we have to find the missing numeral. To get 60 in the numerator, we multiply 5 by 12.
So, we multiply the numerator and denominator by 12.

Hence, 5/6 and 60/72 are equivalent fractions.

We have, 

The estimated value of 20.83 to the nearest tenths is 20.8

The estimated value of 75.195 to the nearest hundredths is 75.20

The estimated value of 27.981 to the nearest tenths is 28.0

L.C.M. of 8 and 3 is 2 × 2 × 2 × 3 = 24
Now,  

L.C.M of 8 and 4 is 2 × 2 × 2 = 8

The L.C.M of 6 and 2 = 6

The L.C.M of 3 and 9 = 9

The L.C.M of 4 and 2 = 4
Now, 

The L.C.M. of 4 and 2 = 4

Now,

The distance travelled by Katrina in the morning 

The distance travelled by Katrina in the evening 

∴ Total distance travelled by her

Let the number of parts in which the rectangle has been divided be x.
According to question, 

By cross-multiplication, x = 16 × 4 = 64
∴ The required number of parts is 64.

We have given, a grip size of a tennis racquet which is the required improper fraction.

We have given, 1/10 of the food eaten is turned into organism’s own body.
∴ The required fraction of the food eaten not available for the next level is 

[∵ L.C.M. of 1 and 10 is 10]

Mr. Rajan got a job at the age of 24 years.
He got retired at the age of 60 years.
He worked for (60 – 24) years = 36 years
∴ The required fraction 

[ ∵ H.C.F. of 36 and 60 is 12]

The food we eat remains in the stomach for a maximum of 4 hours.
Total number of hours in a day = 24 hours
∴ The required fraction  

[∵ H.C.F of 4 and 24 is 4]

Alok purchased,
Potatoes = 1 kg 200 g = 1.200 kg
Dhania = 250 g = 0.250 kg
Onion = 5 kg 300 g = 5.300 kg
Palak = 500 g = 0.500 kg
Tomatoes = 2 kg 600 g = 2.600 kg
∴ The total weight of the above purchases

Since, all the decimals are already given in like fractions, i.e., 0.011, 1.001, 0.101, 0.110
∴ Arranging them in ascending order, we get
0.011, 0.101, 0.110, 1.001

We have, 20.02 and 2.002
To add the above decimals, we must convert them into like decimals first.
Writing 20.020 and 2.002 in a column
So,

which is the required sum.

By the end of year 2007,
The quantity of CNG saved 33000 tonnes,
The quantity of diesel saved 3300 tonnes and The quantity of petrol saved 21000 tonnes

Milk provides the least energy and rice provides the maximum energy.
∴ The required fraction 

A cup is 1/3 full of milk.
∴ The remaining part of the cup which is still to be filled by milk = 1 - (1/3)

[∵ L.C.M of 1 and 3 is 3]
= 2/3

Mary bought the lace

Lace used by Mary

∴ She is left with

[ ∵ L.C.M. of 2 and 4 is 4]

of lace

Sunita had gainedEarlier her weight was

∴  Her total weight on Monday

Sunil purchased juice on Monday

∴ Total quantity of juice Sunil purchased in two days 

Total quantity of juice 

Nazima gave to her friends =

∴ The required quantity of juice she is left

Total length of a wooden board
The carpenter sawed off a piece of length
The carpenter sawed off a piece of length
∴ The length of the remaining piece

Nasir travelled by a bus Nasir walked = ∴ Total distance travelled by him

The weight of fish caught by Neetu
The weight of fish caught by Narendra
∴ Neetu’s fish weigh more than that of Narendra by

Neelam’s father purchased the length of cloth for the skirt and for the scarf ∴ Total length he buys in all  

(a) Equal denominators are added.
(b) Numerators and denominators are added.

1 metre 40 centimetres + 60 centimetres = 1.40 metres + 0.60 metres
[ 100 centimetres = 1 metre]
= 2.00 metres
Since, 2.6 > 2.00
∴ 2.6 metres is greater.

(i) ➝ (D) ; (ii) ➝ (A) ; (iii) ➝ (E) ; (iv) ➝ (B)
Marked point in (A) = 6/10
Shaded fraction in (B)= 6/16
Shaded fraction in (C)  = 6/7
Shaded fraction in (D)  

Shaded fraction in (E)= 6/6

Out of 0, 1, 2, 3, 4, 5 ➝ 1, 2, 3, 4 and 5 are the natural numbers.
∴ The fraction that represents the number of natural numbers to the total numbers = 5/6 and the whole numbers are 0,1, 2, 3, 4 and 5.
∴ The fraction that represents the number of whole numbers to the total numbers = 6/6.

Out of -3, -2, -1, 0, 1, 2, 3 -> 1, 2 and 3 are the natural numbers, 0,1, 2 and 3 are the whole numbers and -3, -2, -1, 0, 1, 2, 3 are integers.
∴ The fraction representing the natural numbers to the total numbers = 3/7=37
The fraction representing the whole numbers to the total numbers = 4/7=47
And the required fraction representing the integers to the total numbers = 7/7

Let one fraction be x.
Another fraction be (7/11) - x711−x
Now, according to question,
Thus, one fraction is 9/22 and another fraction is

Since, we know that the measurement of a straight angle is 180° and a right angle is 90°.
∴ The required fraction is 90°/180° = 1/2.

We know that if numerator is less than the denominator, then the fraction is less than 1.
If numerator is equal to the denominator, then the fraction is equal to 1 and if numerator is greater than the denominator, then the fraction is greater than 1.
Cards in Bag I are
(i) 3/7,
(iv) 8/9
(v) 5/6
(vi) 6/11
(viii) 19/25
(ix) 2/3 and
(x) 13/17
Cards in Bag II are (ii) 4/4 , (vii) 18/18
And cards in Bag III are (iii) 9/8.