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**Ncert exemplar Solutions Class 6 Mathematics Ratio and Proportion**

Welcome to NCTB Solutions. Here with this post we are going to help 6th class students for the Solutions of NCERT Class 6 Mathematics Book, Unit 8, Ratio and Proportion. Here students can easily find step by step solutions of all the problems for Ratio and Proportion, Also here our mathematics teacher’s are solved all the problems with easily understandable methods with proper guidance so that all the students can understand easily. Here in this post students will get Unit 8 solutions.

**Ratio and Proportion Unit 8 Solution :**

**Multiple Choice Questions : **

**Question no – (1) **

**The ratio of 8 books to 20 books is**

**Solution : **

The ratio of 8 books to 20 books = 8 / 20

We divide both by 4 we get, 2/5

The ratio of 8 books to 20 books = 2/5

Hence, correct answer is option – (A) 2 : 5

**Question no – (2) **

**The ratio of the number of sides of a square to the number of edges of a cube is**

**Solution : **

Number of sides of a square is 4.

Number of edges of a cube is 12.

The ratio of the number of sides of a square to the number of edges of a cube = 4 / 12

We divide both by 4 we get,

The ratio of the number of sides of a square to the number of edges of a cube = 1/ 3

Thus, the correct answer is option – (D) 1 : 3

**Question no – (3) **

**A picture is 60cm wide and 1.8 m long. The ratio of its width to its perimeter in lowest form is**

**Solution : **

A picture is 60 cm wide and 1.8 m long.

Picture is in rectangular form.

We convert 60 cm into m.

100 cm = 1 m

60 cm = 0.6 m

Perimeter of rectangular form of picture = 2 (Length + breadth)

Perimeter of rectangular form of picture = 2(1.8 + 0.6)

Perimeter of rectangular form of picture = 2(2.4)

Perimeter of rectangular form of picture = 4.8 m

The ratio of its width to its perimeter in lowest form = 0.6 m / 4.8 m

The ratio of its width to its perimeter in lowest form = 1/8

Therefore, the correct answer is option – (D) 1 : 8

**Question no – (4) **

**Neelam’s annual income is Rs. 288000. Her annual savings amount to Rs. 36000. The ratio of her savings to her expenditure is**

**Solution : **

Neelam’s annual income is Rs. 2, 88,000.

Annual savings amount to Rs. 36,000.

The ratio of her savings to her expenditure =

Expenditure = income – savings

Expenditure = Rs. 2, 88,000 – Rs. 36,000.

Expenditure = Rs. 2, 42,000

The ratio of her savings to her expenditure = Rs. 36,000 / Rs. 2, 42,000

We divide both by 36,000 we get,

The ratio of her savings to her expenditure = 1 / 7.

Hence, correct answer is option – **(B)** 1: 7

**Alternative Solution, **

Savings = 3600

Expenditure,

= 288000 – 3600

= 252000

**∴** The ratio of her savings and expenditure,

= 3600 : 252000

= 36 : 252

= 1 : 7

The correct option is = (B) 1 : 7

**Question no – (5) **

**Mathematics textbook for Class VI has 320 pages. The chapter ‘symmetry’ runs from page 261 to page 272. The ratio of the number of pages of this chapter to the total number of pages of the book is**

**Solution : **

Mathematics textbook for Class VI has 320 pages.

The chapter ‘symmetry’ runs from page 261 to page 272.

Total pages of chapter ‘symmetry’ = 272 – 261 + 1

Total pages of chapter ‘symmetry’ = 11 + 1

Total pages of chapter ‘symmetry’ = 12

The ratio of the number of pages of this chapter to the total number of pages of the book

= 12 / 320

We divide both by 4 we get, 3 / 80

Therefore, the correct answer is option – (C) 3 : 80

**Alternative Solution, **

The pages of Symmetry chapter,

= (272 – 261)

= 11

**∴ **11 : 320

The correct option is – (A) 11 : 320

**Question no – (6) **

**In a box, the ratio of red marbles to blue marbles is 7 : 4. Which of the following could be the total number of marbles in the box?**

**Solution : **

In a box, the ratio of red marbles to blue marbles is 7:4.

We adding ratio of red marbles to blue marbles = 7 + 4 = 11

The numbers given in option only 22 is divisible by 11.

22 / 11 = 2

No. of red marbles = 7 x 2 = 14

No. of blue marbles = 4 x 2 = 8

So, the correct answer is option – (D) 22

**Question no – (7) **

**On a shelf, books with green cover and that with brown cover are in the ratio 2 : 3. If there are 18 books with green cover, then the number of books with brown cover is**

**Solution : **

On a shelf, books with green cover and that with brown cover are in the ratio 2:3.

There are 18 books with green cover.

Books with green cover has ratio is 2.

We divide 18 books with green cover by its ratio 2.

18/2 = 9

1 ratio is 9.

Books with brown cover has ratio is 3.

Books with brown cover = we multiply 9 with 3.

Books with brown cover = 9 x 3

Books with brown cover = 27

Hence, correct answer is option – (C) 27

**Question no – (8)**

** The greatest ratio among the ratios 2 : 3, 5 : 8, 75 : 121 and 40 : 25 is**

**Solution : **

If the ratio is greatest ratio then its Numerator is greater than its denominator.

2 : 3 = 2 / 3 Numerator is smaller than its denominator.

5 : 8 = 5 / 8 Numerator is smaller than its denominator.

75 : 121 = 75 /121 Numerator is smaller than its denominator.

But, 40 : 25 = 40 / 25 Numerator is greater than its denominator.

The greatest ratio is 40: 25.

So, the correct answer is option – (D) 40: 25

**Question no – (9) **

**There are ‘b’ boys and ‘g’ girls in a class. The ratio of the number of boys to the total number of students in the class is **

**Solution : **

There are ‘b’ boys and ‘g’ girls in a class.

The ratio of the number of boys to the total number of students in the class,

Total number of students in the class = Sum of boys and girl

Total number of students in the class = b + g

The ratio of the number of boys to the total number of students in the class,

= b / b + g

Hence, correct answer is option – (A) b / b + g

**Question no – (10) **

**If a bus travels 160 km in 4 hours and a train travels 320 km in 5 hours at uniform speeds, then the ratio of the distances travelled by them in one hour is**

**Solution : **

If a bus travels 160 km in 4 hours and a train travels 320 km in 5 hours at uniform speeds

Distance travel by bus in 1 hour = 160 km / 4 hours

Distance travel by bus in 1 hour = 40 km.

Distance travel by train in 1 hour = 320 km / 5 hours

Distance travel by train in 1 hour = 64 km

The ratio of the distances travelled by them in one hour = Distance travel by bus in 1 hour / Distance travel by train in 1 hour

= 40 km / 64 km

Divide both by 8 we get,

= 5 / 8

Therefore, the correct answer is option – (C) 5 : 8

**Fill in the blanks to make equivalent ratios : **

**Question no – (11) **

**Solution : **

3/5 = __/20

= 3/5 = 15/20

**Question no – (12) **

**Solution : **

__/18 = 2/9

= 4/18 = 2/9

**Question no – (13) **

**Solution : **

8/__ = 32/4

= 8/10 = 32/4

**Question no – (14) **

**Solution : **

__/45 = 16/40 = 24/__

= 27/45 = 16/40 = 24/60

**Question no – (15) **

**Solution : **

16/36 = __/63 = 36/__ = __/117

= 16/36 = 28/63 = 36/81 = 52/117

**State True or False : **

**Question no – (16) **

**Solution : **

3/8 = 15/40 – This statement is **True.**

Because,

3 /8 = 15 /40

By using cross multiplication method,

3 x 40 = 8 x 15

120 = 120

**∴ **LHS = RHS

**Question no – (17) **

**Solution : **

4 : 7 = 20 : 35 – This statement is **True.**

Because,

4/ 7 = 20 /35

By using cross multiplication method,

4 x 35 = 7 x 20

140 = 140

**∴ **LHS = RHS.

**Question no – (18) **

**Solution : **

0.2 : 5 = 2 : 0.5 – This statement **False.**

Because,

0.2/ 5 = 2/ 0.5

By using cross multiplication method,

0.2 x 0. 5 = 5 x 2

1 = 10

**∴** LHS is not equal to RHS.

**Question no – (19) **

**Solution : **

3 : 33 = 33 : 333 -This statement is **False.**

Because,

3/ 33 = 33/ 333

By using cross multiplication method,

3 x 333 = 33 x 33

999 = 1089

**∴ ** LHS is not equal to RHS.

**Question no – (20) **

**Solution : **

15m : 40m = 35m : 65m – This statement is **False.**

Because,

15m / 40m = 35m / 65m

By using cross multiplication method,

15 x 65 = 40 x 35

975 = 1400

**∴ **LHS is not equal to RHS.

**Alternative Solution, **

15m **:** 40m = 15/40 – 3/8

= 35m 65m = 35/65 = 7/13

**∴** 15m : 40m ≠ 35 m : 65 m

Thus, the statement is False.

**Question no – (21) **

**Solution : **

27cm² : 57cm² = 18cm : 38cm – This statement is **True.**

Because,

Here, unit is different.

We take ratio of first two.

27cm² / 57cm²

= 27 / 57

Divide by 3

= 9 / 19

Now, we take ratio of 18cm: 38cm

18cm/ 38cm

Divide by 2,

= 9 / 19

The Ratio is same.

27 cm² : 57cm² = 18 cm : 38 cm

**Alternative Solution, **

= 27cm² : 75cm² = 27/57 = 9/19

= 18 cm : 38 cm

= 18/38

= 9/19

Hence, the statement is True.

**Question no – (22) **

**Solution : **

5 kg : 7.5 kg = Rs 7.50 : Rs 5 – This statement is **False.**

Because,

Here, unit is different.

We take ratio of first two.

5 kg/ 7.5 kg

We divide by 2.5

= 2 / 3

Now, we take ratio of Rs 7.50: Rs 5

Rs 7.50 / Rs 5

We divide by 2.5

= 3 / 2

The Ratio is not same.

**Question no – (23) **

**Solution : **

20 g : 100 g = 1 metre : 500 cm

This statement is **True.**

Because,

Here, unit is different.

We take ratio of first two.

20 g: 100 g

20 g/ 100 g

We divide by 20

= 1 / 5

Now, we take ratio of 1 metre : 500 cm

Here unit is different.

We make unit same.

We know, 100 cm = 1 m

100 cm: 500 cm

100 cm/ 500 cm

We divide by 100

= 1 / 5

The Ratio is same.

20 g : 100 g = 1 metre : 500 cm

**Alternative Solution, **

20 g** :** 100 g = 20/100 = 1/5

= 1 meter : 500 cm

**∴** 100 cm : 500 cm

= 100/500

= 1/5

Thus, the statement is True.

**Question no – (24) **

**Solution : **

12 hours : 30 hours = 8 km : 20 km

This statement is **True.**

Because,

Here, unit is different.

We take ratio of first two.

12 hours : 30 hours

12 hours / 30 hours

We divide by 6

= 2 / 5

Now, we take ratio of 8 km: 20 km

8 km/ 20 km

We divide by 4

= 2 / 5

The Ratio is same.

12 hours: 30 hours = 8 km: 20 km

**Alternative Solution, **

= 12 hours : 30 hours

= 12/30

= 2/5

= 8 km : 20 km

= 8/20

= 2/5

Thus, the statement is **True.**

**Question no – (25) **

**Solution : **

The ratio of 10 kg to 100 kg is 1:10

This statement is **True.**

Because,

10 kg: 100 kg

We divide by 10

= 1 / 10

The ratio of 10 kg to 100 kg is 1:10.

**Question no – (26) **

**Solution : **

The ratio of 150 cm to 1 metre is 1:1.5

This statement is **False.**

Because,

We know,

1 metre = 100 cm

The ratio of 150 cm to 1 metre (100 cm)

= 150 / 100

We divide by 50

= 3 / 2

= 1.5 / 1

The ratio of 150 cm to 1 metre is 1.5 : 1

**Alternative Solution, **

The ratio of 150 cm to 1 meter = 150 cm : 1 meter

= 150 cm : 100 cm

= 3 : 2

The statement is False.

**Question no – (27) **

**Solution : **

25kg : 20g = 50kg : 40g

This statement is **True.**

Because,

We know, 1 kg = 1000 gm.

25 kg = 25000 gm.

25kg : 20g

25000 gm. / 20 gm.

We divide by 20

= 1500 / 1

Now, we take ratio of 50kg: 40g

We know, 1 kg = 1000 gm.

50 kg = 50000 gm.

50kg : 40g

50000 gm. / 40 gm.

We divide by 40

1500 / 1

The Ratio is same.

25kg: 20g = 50kg: 40g

**Alternative Solution, **

25 kg : 20g = 25000g : 20g

= 25000/20

= 2500 : 2

= 50 kg : 40 kg = 50000 g : 40g

= 50000/40

= 5200/2

= 2500 : 2

∴ The statement is True.

**Question no – (28) **

**Solution : **

The ratio of 1 hour to one day is 1 : 1

This statement is **False.**

Because,

In day there are 24 hours.

The ratio of 1 hour to one day = 1 / 24

The ratio of 1 hour to one day is 1: 24

**Alternative Solution, **

The ratio of hour to one day is

= 1 hour : 1 day

= 1 hour : 24 hour

= 1 : 24

**∴** The statement is False.

**Question no – (29) **

**Solution : **

The ratio 4 : 16 is in its lowest form

This statement is **False.**

Because,

The ratio 4:16 is not in its lowest form.

We divide both by 4.

We get 1 / 4

1 : 4

**Alternative Solution, **

= 4 : 16 = 1 : 4

∴ The lowest from is 1 : 4

∴ The statement is False.

**Question no – (30) **

**Solution : **

The ratio 5 : 4 is different from the ratio 4 : 5

This statement is **True.**

Because,

The ratio 5: 4 is different from the ratio 4: 5.

Value of 5/4 is different.

Value of 4/5 is different.

**Alternative Solution, **

= 5 : 4 = 5/4

= 4 : 5 = 4/5

∴ 5 : 4 ≠ 4 : 5

Thus, the statement is True.

**Question no – (31) **

**Solution : **

A ratio will always be more than 1

This statement is **False.**

Because, A ratio will be less than 1, Equal to 1 or more than 1.

**Question no – (32) **

**Solution : **

A ratio will always be more than 1

This statement is **False.**

Because, a ratio will be less than 1, Equal to 1 or more than 1.

**Question no – (33) **

**Solution : **

If b : a = c : d, then a, b, c, d are in proportion

This statement is **False.**

Because, b : a = c: d, then a, b, c, d are in proportion.

We are not knowing the value of a, b, c and d.

We cannot say they are in proportion

**Alternative Solution, **

If a, b, c, d are in proportion,

= a/b = c/d

Here is b : a = c : d

= b/a = c/d

= a/b ≠ b/a

Therefore the a, b, c, d are not in proportion.

Thus, the statement is False.

**Question no – (34) **

**Solution : **

The two terms of a ratio can be in two different units

This statement is **False.**

Because, The two terms of a ratio can be in same units.

**Fill in the blanks :**

**Question no – (35) **

**Solution : **

A ratio is a form of comparison by **division.**

**Question no – (36) **

**Solution : **

20 m: 70 m = Rs 8: **Rs 28.**

Explanation :

We first take ratio of 20 m : 70 m

20 m/ 70 m

= 2/ 7

This ratio is also same for Rs. Value.

2 / 7 = Rs. 8 /?

By cross multiplication method,

2 x? = 7 x 8

? = 7 x 8 / 2

? = 7 x 4(Divide by 2)

? = Rs.28

**Question no – (38) **

**Solution : **

If two ratios are equal, then they are in **Proportion.**

**Question no – (39) **

**Solution : **

Perimeter is the sum of all exterior sides.

perimeter of the boundary of the shaded portion = 6 unit

Perimeter of the whole figure = 14 units.

The ratio of the perimeter of the boundary of the shaded portion to the perimeter of the whole figure

= 6 / 14 (Divide by 2)

= 3 / 7

The ratio of the perimeter of the boundary of the shaded portion to the perimeter of the whole figure is 3: 7.

**Question no – (40) **

**Solution : **

The ratio of the area of the shaded portion to that of the whole figure is** 1 : 6**.

Area of the shaded portion = Length x Breadth

Length = 2 units.

Breadth = 1 units.

Area of the shaded portion = 2 x 1 = 2 units 2

Area of whole figure = Length x Breadth

Length = 4 units.

Breadth = 3 units.

Area of whole figure

= 4 x 3

= 12 units

The ratio of the area of the shaded portion to that of the whole figure = 2 / 12

The ratio of the area of the shaded portion to that of the whole figure = 1 / 6

**Alternative Solution, **

The area at should portion,

= 2 × 1

= 2

The area of whole figure,

= 4 × 3

= 12

**∴** 2 : 12

= 1 : 6

**Question no – (41) **

**Solution : **

The ratio of sleeping time to awaking time is **3 : 1**

The awaking time,

= 24 – 18

= 6 hour

Sleeping time = 18 hours.

**∴** The ratio of sleeping time to awaking time is,

= 18 : 6

= 3 : 1

**Question no – (42) **

**Solution : **

A ratio expressed in lowest form has no common factor other than **One** in its terms.

**Question no – (43) **

**Solution : **

To find the ratio of two quantities, they must be expressed in **Same** units.

**Question no – (44) **

**Solution : **

Ratio of 5 paise to 25 paise is the same as the ratio of 20 paise to **100 paise.**

Explanation :

Ratio of 5 paise to 25 paise

5 paise / 25 paise = 20 paise /?

By cross multiplication method,

5 x ? = 25 x 20

? = 25 x 20 / 5(Divide by 5)

? = 5 x 20

? = 100 paise

**Question no – (45) **

**Solution : **

Saturn and Jupiter take 9 hours 56 minutes and 10 hours 40 minutes, respectively for one spin on their axes. The ratio of the time taken by Saturn and Jupiter in lowest form is** 149 / 160.**

Explanation :

Saturn and Jupiter take 9 hours 56 minutes and 10 hours 40 minutes, respectively for one spin on their axes.

Saturn taken 9 hours 56 minutes.

We know, 1 hour = 60 minute

9 hour = 60 x 9 = 540 minute.

Saturn taken 9 hours 56 minutes. = 540 + 56 = 596 minutes.

Jupiter taken 10 hours 40 minutes.

We know, 1 hour = 60 minute

10 hour = 60 x 10 = 600 minute.

Jupiter taken 10 hours 40 minutes.

= 600 + 40

= 640 minutes.

The ratio of the time taken by Saturn and Jupiter in lowest form = 596 / 640

We divide by 4. We get,

The ratio of the time taken by Saturn and Jupiter in lowest form = 149 / 160

**Alternative Solution, **

The ratio of the time taken by Saturn and Jupiter in lowest form is 149 : 160

Saturn : Jupiter = 9 hours 56 min : 10 hours 40 min

= (9 × 60 + 56) min : (10 × 60 + 40) min

= (540 + 56) min : (600 + 40) min

= 596 : 640

= 149 : 160

**Question no – (46) **

**Solution : **

10g of caustic soda dissolved in 100mL of water makes a solution of caustic soda. Amount of caustic soda needed for 1 litre of water to make the same type of solution is **100 gm.**

Explanation :

10g of caustic soda dissolved in 100 mL of water makes a solution of caustic soda.

Amount of caustic soda needed for 1 litre of water to make the same type of solution =

We know, 1 litre = 1000 ml

10g : 100 ml = ? : 1000 ml

Here unit is different.

We find first ratio of 10g: 100 ml

10g / 100 ml

= 1 / 10

1 / 10 =? / 1000 ml

By cross multiplication method,

1 x 1000 = 10 x ?

? = 1000 / 10

? = 100 gm. Of caustic soda is required.

**Alternative Solution : **

Amount of caustic soda needed for 1 litre of water to make the same type of solution is **100.**

= 1 Lt = 1000 ml

= 10g : 100 = 100 : 1000

= 100

**Question no – (47) **

**Solution : **

The ratio of the sale price to the marked price = **4 / 5**

Explanation :

The marked price of a table is Rs 625.

Sale price is Rs 500.

The ratio of the sale price to the marked price = Rs 500 / Rs 625

We divide both by 125, we get,

**Alternative Solution :**

So, the ratio of the sale price to the marked price is **4 : 5**

In the given question,

Price of a table is = Rs 625

Its sale price is = Rs 500

**∴** Ratio the sale price to market price

= 500 : 625

= 100 : 125

= 20 : 25

= 4 : 5

**Question no – (48) **

**Solution : **

**(i) 2/3, 4/6**

Two ratios are equivalent, if the fractions corresponding to them are equivalent.

2 /3 we multiply this fraction by 2 we get,

2 x 2 /3 x 2 = 4 / 6

Hence, 2 /3 and 4/ 6 are equal or equivalent ratio.

**(ii) 8/4, 2/1**

Two ratios are equivalent, if the fractions corresponding to them are equivalent.

8 / 4 we divide this fraction by 4. We get,

8 / 4 = 2 / 1

Hence, 8 / 4 and 2 / 1 are equal or equivalent ratio.

**(iii) 4/5, 12/20**

Two ratios are equivalent, if the fractions corresponding to them are equivalent.

4 / 5 we multiply this fraction by 3 we get,

4 x 3 / 5 x 3 = 12 / 15

Which is not equal.

4 / 5 and 12 / 20 are not equal.

**Alternative Solution :**

**(i) 2/3, 4/6**

= 2/3 = 2 : 3

= 4/6 = 2 : 3

**(ii) 8/4, 2/1**

= 8/4 = 8 : 4 = 2 : 1

= 2/1 = 2 : 1

**(iii) 4/5, 12/20**

= 4/5 = 4 : 5

= 12 : 20 = 3 : 5

Therefore, (i) and (ii) are same.

**Question no – (49) **

**Solution : **

For deciding which ratio is larger, we are doing cross multiplication.

10/ 21 = 21/ 93

10 x 93 = 21 x 21

930 > 441

Hence, Ratio 1 is larger.

10 : 21 is larger ratio.

**Question no – (50) **

**Solution : **

Reshma prepared 18 kg of Burfi by mixing Khoya with sugar in the ratio of 7: 2.

We are adding ratio of Khoya and sugar

= 7 + 2

= 9

Reshma prepared 18 kg.

For getting value of 1 ratio, we are dividing 18 kg with 9.

Value of 1 ratio

= 18 / 9

= 2

Ratio of Khoya = 7

Khoya she use = we are multiplying 7 with value of 1 ratio.

Khoya she use,

= 7 x 2

= 14 kg.

**Alternative Solution : **

As per the question,

Reshma prepared = 18 kg of Burfi

By mixing Khoya with sugar in ratio = 7 : 2

**∴** 18 × 7/7+2

= 18 × 7/9

= 14 kg

**∴** She used 14 kg khoya.

**Question no – (51) **

**Solution : **

A line segment 56 cm long is to be divided into two parts in the ratio of 2 : 5.

We are adding ratio of two parts = 2 + 5 = 7

A line segment 56 cm long.

For getting value of 1 ratio, we are dividing 56 cm with 7.

Value of 1 ratio = 56 / 7

Value of 1 ratio = 8

Two parts in the ratio of 2: 5

Length of 1st part,

= 8 x 2

= 16 cm.

Length of 2st part,

= 8 x 5

= 40 cm.

**Alternative Solution : **

A line segment = 56 cm long

Divided into two parts in ratio = 2 : 5

**∴** 56 × 2/2+5

= 56 × 2/7

= 16 cm

= (56 – 16) = 40 cm

**∴** The length of each par is 16 cm and 40 cm.

**Question no – (52) **

**Solution : **

The number of milk teeth in human beings is 20.

The number of permanent teeth is 32.

The ratio of the number of milk teeth to the number of permanent teeth

= 20 / 32

We are dividing both by 4 we get,

The ratio of the number of milk teeth to the number of permanent teeth

= 5 /8

**Alternative Solution : **

Number of milk teeth = 20

Number of permanent teeth is = 32

**∴** The ratio of the number of milk teeth to the number of permanent teeth,

= 20 : 32

= 5 : 8

**Question no – (53) **

**Solution : **

There are 3732 females per 4000 males in a town.

**∴** Females : Male

= 3732 : 4000

= 933 : 1000

**∴** The number of females is 933 per 1000 meals in the population.

**Question no – (54) **

**Solution :**

**(a) income to income tax.**

Ravi earns Rs 3, 60,000 and paid Rs 24000 as income tax.

The ratio of his income to income tax = Rs 3, 60,000 / Rs 24000

We divide both by 12,000

We get, The ratio of his income to income tax = 300 /2

The ratio of his income to income tax = 15 / 1

**(b) Income tax to income after paying income tax.**

After paying income tax

Money left with Ravi = Rs 3, 60,000 – Rs 24000

Money left with Ravi = Rs. 3, 36,000

The ratio of his Income tax to income after paying income tax = Rs 24000 / Rs. 3, 36,000

The ratio of his Income tax to income after paying income tax =

We divide both by 12,000

We get, The ratio of his Income tax to income after paying income tax = 2 /28

The ratio of his Income tax to income after paying income tax = 1 / 14

**Question no – (55) **

**Solution : **

**(a) Ramesh’s earnings to their total earnings**

Income to income tax ration,

= 360000 : 24000

= 360 : 24

= 60 : 4

= 15 : 1

**(b) Rama’s earnings to their total earnings.**

Income after paying income tax,

= 360000 – 24000

= 336000

Ratio of his income tax to income after paying income tax,

= 24000 : 336000

= 24 : 336

= 1 : 14

**Question no – (56) **

**Solution : **

**(a)** Men to that of women.

Of the 288 persons working in a company, 112 are men and the remaining are women.

Women in company = 288 – 112

Women in company = 176

The ratio of the number of men to that of women

= 112 / 176

We divide both by 16

We get,

The ratio of the number of men to that of women

= 7 / 11

**(b)** Men to the total number of persons.

The ratio of the number of men to the total number of persons

= 112 / 288

We divide both by 16

We get,

The ratio of the number of men to the total number of persons

= 7 /18

**(c)** Women to the total number of persons.

The ratio of the number of Women to the total number of persons

= 176/ 288

We divide both by 16

We get,

The ratio of the number of Women to the total number of persons

= 11 / 18

**Alternative Solution : **

Women in that company,

= 288 – 112

= 176

Now, **(a)** ratio of men to that of women,

= 112 : 176

= 56 : 88

= 7 : 11

**(b)** ratio of men to the total no of persons

= 112 : 288

= 56 : 144

= 28 : 72

= 7 : 18

**(c)** ratio women to that total no of persons

= 176 : 288

= 22 : 36

= 11 : 18

**Question no – (57) **

**Solution : **

A rectangular sheet of paper is of length 1.2m and width 21cm

We know, 100 cm = 1 m

= 1.2 m = 120 cm

The ratio of width of the paper to its length

= 21 / 120

We divide both by 3,

We get,

The ratio of width of the paper to its length

= 7 / 40

**Alternative Solution : **

A rectangular sheet of paper is of Length = 1.2 m

A rectangular sheet of paper is of Width = 21 cm.

**∴** 1.2 m = 12 × 100 cm

= 120.0 cm

The ratio of width of the paper to its length,

= 21 : 120

= 7 : 40

**Question no – (58) **

**Solution : **

A scooter travels 120 km in 3 hours.

Speed of scooter = 120 / 3 = 40 kmph

Train travels 120 km in 2 hours

Speed of Train = 120 / 2 = 60 kmph

The ratio of their speeds = 40 kmph / 60 kmph

We divide both by 20,

We get, The ratio of their speeds = 2 / 3

**Alternative Solution : **

Scooter’s speed,

= 120/3

= 40 km/h

Train’s speed,

= 120/2

= 60 km/h

**∴** The ratio of sector and train’s speed,

= 40 : 60

= 2 : 3

**Question no – (59) **

**Solution : **

An office opens at 9 a.m. and closes at 5.30 p.m. with a lunch break of 30 minutes

Total period in the office = opens at 9 a.m. and closes at 5.30 p.m. = 8.5 hours

We know, 1 hour = 60 minute

8.5 hour = 60 x 8. 5 = 510 minutes

The ratio of lunch break to the total period in the office

= 30 minutes / 510 minutes

We divide both by 30,

We get, The ratio of lunch break to the total period in the office

= 1 / 17

**Alternative Solution : **

Total period,

= 8 hour 30 min

= 8 × 60 + 30

= 510 min

Lunch break = 30 min

**∴** The ratio of lunch break to the total period in the office,

= 30 : 510

= 1 : 17

**Question no – (60) **

**Solution : **

The shadow of a 3 m long stick is 4m long.

The shadow of a flagstaff is 24 m long,

Height of the flagstaff =?

We making ratio of 4 quantities.

3 m : 4 m =? : 24 m

By cross multiplication method,

3 x 24 = 4 x?

? = 3 x 24 / 4

? = 3 x 6

? = 18 m.

Height of the flagstaff is 18 m.

**Alternative Solution : **

The shadow of flagstaff is,

= 24 m

**∴** The flagstaff is,

= 24 × 3/4

= 18 m

So, the flagstaff is 18 m long.

**Question no – (61) **

**Solution : **

A recipe calls for 1 cup of milk for every 2 x ½ cups of flour to make a cake that would feed 6 persons.

Ratio of milk and floor to make cake of 6 persons = 1: 2.5

We adding this ratio = 1 + 2.5 = 3.5

3.5 cups cake feed 6 persons.

Then How many cups feed to 8 peoples =

3.5 / 6 = ? / 8

By cross multiplication method,

3.5 x 8 = 6 X?

? = 3.5 x 8 / 6

? = 3.5 x 4 / 3

? = 14 / 3

? = 4.66 cups

4.66 cups of both flour and milk will be needed to make a similar cake for 8 people.

**Alternative Solution : **

Milk : Flour

= 1 : 2 1/2

= 1 : 5/2

= 2 : 5

Total Cake = (2 + 5) = 7

= 6 persons cup 7 cup

= 1 person cup 7/6 cup

= 8 persons cup 8 × 7/63 = 28/3

**∴ Milk,**

= 28×/3 2/7

= 8/3

= 2 2/3

**∴ Flour,**

= 28/3 × 5/7

= 20/3

= 6 2/3

**Question no – (62) **

**Solution : **

In a school, the ratio of the number of large classrooms to small classrooms is 3:4.

The number of small rooms is 20.

The ratio of small classrooms is 4.

The ratio of 1 class room = we divide 20 by 4

The ratio of 1 class room = 20 / 4 = 5

The ratio of the number of large classrooms = 3

We multiply the ratio of 1 class room with 3.

The number of large rooms = 5 x 3

The number of large rooms = 15

**Alternative Solution : **

Given in the question,

The ratio of large class room to small class room is,

= 3 : 4

**∴ Small class room**

= 4 × 5

= 20

**∴ Large class room**

= 3 × 5

= 15

**Question no – (63) **

**Solution : **

**(a)** The number of English newspapers to the number of Hindi newspapers.

Samira sells newspapers at Jan path crossing daily.

On a particular day, she had 312 newspapers out of which 216 are in English and remaining in Hindi.

Hindi newspapers

= 312 – 216

= 96

The ratio of the number of English newspapers to the number of Hindi newspapers = 216/96

We divide both by 24,

We get,

The ratio of the number of English newspapers to the number of Hindi newspapers = 9/4

**(b)** The number of Hindi newspapers to the total number of newspapers.

The ratio of the number of Hindi newspapers to the total number of newspapers = 96/312

We divide both by 24,

We get,

The ratio of the number of Hindi newspapers to the total number of newspapers

= 4/13

**Alternative Solution : **

The number of Hindi newspaper,

= (312 – 216)

= 96

Now, **(a)** ratio of English newspaper to Hindi newspaper,

= 216 : 96

= 36 : 16

= 9 : 4

**(b)** ratio of Hindi newspaper to total no of newspaper is,

= 96 : 312

= 16 : 52

= 4 : 13

**Question no – (64) **

**Solution : **

**(a) The number of Hindu students to the number of Christian students.**

The number of Hindu students is 288.

The number of Christian students is 72.

The ratio of number of Hindu students to the number of Christian students = 288 / 72

We divide both by 72,

We get,

The ratio of number of Hindu students to the number of Christian students = 4 /1

**(b) The number of Muslim students to the total number of students.**

The number of Muslim students is 252.

Total number of students,

= 288 + 252 + 144 + 72

= 756 students.

The ratio of number of Muslim students to the total number of students = 252 / 756

We divide both by 3,

We get,

The ratio of number of Muslim students to the total number of students = 1/3

**Question no – (65) **

**Solution : **

The ratio of North Indian food stalls to South Indian food stalls is 5:4.

Total number of food stalls is 117.

We are adding ratio of both food

= 5 + 4

= 9

Ratio of 1 food = Total number of food stalls / ratio of addition of food stalls.

Ratio of 1 food = 117 / 9

Ratio of 1 food = 13.

The ratio of North Indian food stalls is 5.

Number of North Indian food stalls = we multiply 5 with 13.

Number of North Indian food stalls

= 13 x 5

= 65 stalls

The ratio of South Indian food stalls is 4.

Number of South Indian food stalls = we multiply 4 with 13.

Number of South Indian food stalls

= 4 x 13

= 52

Number of South Indian food stalls = 52 stalls

**Question no – (66) **

**Solution : **

Kartik counted that there are 115 cycles, 75 scooters and 45 bikes

Total number of vehicles,

= 115 cycles + 75 scooters + 45 bikes

= 235

The ratio of the number of cycles to the total number of vehicles

= 115 / 235

We divide both by 5,

We get, the ratio of the number of cycles to the total number of vehicles

= 23 / 47

**Question no – (67) **

**Solution : **

A train takes 2 hours to travel from Ajmer to Jaipur, which are 130 km apart.

Distance travelled in 1 hour

= 130 / 2

= 65 km

Much time will it take to travel from Delhi to Bhopal which are 780 km = ?

We have to divide 780 km by 65 km.

Time will it take to travel from Delhi to Bhopal

= 780 / 65

We divide both by 65,

We get, time will it take to travel from Delhi to Bhopal

= 12 hours

**Question no – (68) **

**Solution : **

The length and breadth of a school ground are 150 m and 90 m respectively.

The length and breadth of a mela ground are 210 m and 126 m, respectively.

We are finding first ratio of length and breadth of a school ground

Ratio of length and breadth of a school ground = 150 / 90

We divide both by 30,

We get, Ratio of length and breadth of a school ground

= 5 / 3.

Now we find first ratio of length and breadth of a mela ground.

Ratio of length and breadth of a mela ground

= 210 / 126

We divide both by 42,

We get, Ratio of length and breadth of a Mela ground

= 5 / 3

Both ratios are same.

These measurements in proportion.

**Question no – (69) **

**Solution : **

**(a) Africa to Europe**

Area of Africa = 26

Area of Europe = 10

The ratio of the areas of Africa to Europe

= 26 / 10

We divide both by 2,

We get, The ratio of the areas of Africa to Europe

= 13 / 5

**(b) Australia to Asia**

Area of Australia = 8

Area of Asia = 44

The ratio of the areas of Australia to Asia

= 8 / 44

We divide both by 4,

We get, the ratio of the areas of Australia to Asia

= 2 / 11

**(c)** Antarctica to Combined area of North America and South America.

Combined area of North America and South America

= 18 + 17

= 35

Area of Antarctica = 13

The ratio of the areas of Antarctica to Combined area of North America and South America = 13/35

**Question no – (70) **

**Solution : **

A tea merchant blends two varieties of tea costing her Rs 234 and Rs 130 per kg in the ratio of their costs.

The ratio of two varieties of tea = 234 : 130

We divide both by 26,

We get, The ratio of two varieties of tea = 9 : 5

We are adding ratio of two varieties of tea

= 9 + 5

= 14

Weight of the mixture is 84 kg.

Value of 1 ratio = we divide 84 kg / 14

Value of 1 ratio = 6

Ratio of 1st variety = 9

Weight of 1st variety

= 6 x 9

= 54 kg.

Ratio of 2st variety = 5

Weight of 1st variety

= 6 x 5

= 30 kg.

**Question no – (71) **

**Solution : **

An alloy contains only zinc and copper and they are in the ratio of 7 : 9.

The weight of the alloy is 8 kg.

We are adding ratio of zinc and copper

= 7 + 9

= 16.

Value of 1 ratio = we divide 8 kg by 16

Value of 1 ratio

= 8 / 16

= 1/2

Ratio of copper = 9

Weight of copper = 1/2 x 9

Weight of copper = 4.5 kg.

**Question no – (72) **

**Solution : **

**(i) AC : AF**

Length of AC = 2

Length of AF = 5

AC : AF = 2 / 5

**(ii) AG : AD**

Length of AG = 6

Length of AD = 3

AG: AD = 6 / 3 = 2 / 1

**(iii) BF : AI**

Length of BF = 4

Length of AI = 8

BF: AI = 4 / 8 = 1 / 2

**(iv) CE : DI**

Length of CE = 2

Length of DI = 5

CE: DI = 2 / 5

**Question no – (73) **

**Solution : **

We adding ratio of two numbers.

Ratio is 9 : 16. = 9 + 16 = 25

Two numbers whose sum is 100.

We are dividing sum by 25

Value of 1 ratio = 100 / 25

Value of 1 ratio = 4

Ratio of 1st number = 9

**1 St Number**

= 4 x 9

= 36

Ratio of 2st number = 16

**2 St Number, **

= 4 x 16

= 64

**Question no – (74) **

**Solution : **

**Figure – (i)**

Area of shaded portion of fig. 1 = 8

Area of whole figure. 1 = 16

The ratio of the area of the shaded portion to that of the whole figure.1 = 8 / 16

The ratio of the area of the shaded portion to that of the whole figure.1 = 1 / 2

**Figure – (ii)**

Area of shaded portion of fig. 2 = 8

Area of whole figure. 2 = 16

The ratio of the area of the shaded portion to that of the whole figure.2 = 8 / 16

The ratio of the area of the shaded portion to that of the whole figure.2 = 1 / 2

**Question no – (75) **

**Solution : **

A typist has to type a manuscript of 40 pages.

She has typed 30 pages of the manuscript.

Number of pages left to type

= 40 – 30

= 10 pages

The ratio of the number of pages typed to the number of pages left = 30 / 10

The ratio of the number of pages typed to the number of pages left = 3 / 1

**Question no – (76) **

**Solution : **

**(a) The perimeter of shaded portion to the perimeter of the whole design.**

In a floral design made from tiles each of dimensions 40 cm by 60 cm.

Perimeter of shaded portion = 2 (Length + Breadth)

Perimeter of shaded portion = 2(120 + 120)

Perimeter of shaded portion = 2 x 240 = 480 cm.

Perimeter of the whole design = 2 (Length + Breadth)

Perimeter of the whole design = 2(240 + 200)

Perimeter of the whole design = 2 x 440

Perimeter of the whole design = 880 cm.

The ratio of perimeter of shaded portion to the perimeter of the whole design

= 480 /880

We divide both by 40,

We get,

The ratio of perimeter of shaded portion to the perimeter of the whole design

= 12 / 22

The ratio of perimeter of shaded portion to the perimeter of the whole design

= 6/ 11

**(b) The area of the shaded portion to the area of the unshaded portion.**

The area of the shaded portion = Length x Breadth

The area of the shaded portion

= 120 x 120

= 14400 cm²

The area of unshaded portion = area of whole design – area of the shaded portion

Area of whole design = Length x Breadth

Area of whole design = 240 x 200 = 48000 cm²

The area of unshaded portion = 48000 cm² -14400 cm²

The area of unshaded portion = 23600 cm²

The ratio of area of the shaded portion to the area of the unshaded portion

= 14400 cm² /23600 cm²

We divide both by 100,

We get, the ratio of area of the shaded portion to the area of the unshaded portion

= 144 / 236

The ratio of area of the shaded portion to the area of the unshaded portion

= 72 /118

= 36 /59

**Question no – (77) **

**Solution : **

**(a) shaded portion I to shaded portion II ?**

Area of shaded portion I = Length x Breadth

Length of shaded portion I = 5

Breadth of shaded portion I = 5

Area of shaded portion I

= 5 x 5

= 25 uni²

Area of shaded portion II = Length x Breadth – (10)

Length of shaded portion II = 10

Breadth of shaded portion II = 5

Area of shaded portion II = Length x Breadth – (10)

Area of shaded portion II = 10 x 5 – 10

Area of shaded portion II = 40 uni.²

the ratio of the areas of shaded portion I to shaded portion II

= 25 / 40

we divide both by 5

we get, the ratio of the areas of shaded portion I to shaded portion II

= 5 / 8

**(b) Shaded portion II to shaded portion III?**

Area of shaded portion II = 40 uni.²

Area of shaded portion III = Length x Breadth

Length of shaded portion III = 7

Breadth of shaded portion III = 5

Area of shaded portion III = 7 x 5 = 35 uni²

the ratio of the areas of shaded portion II to shaded portion III = 40 / 35

we divide both by 5

we get, the ratio of the areas of shaded portion II to shaded portion III = 8 / 7

**(c) shaded portions I and II taken together and shaded portion III?**

Area of shaded portions I and II taken together

= 25 + 40

= 65 uni²

Area of shaded portion III

= 7 x 5

= 35 uni²

the ratio of the areas of shaded portions I and II taken together and shaded portion III

= 65 / 35 we divide both by 5

We get, the ratio of the areas of shaded portions I and II taken together and shaded portion III = 13 / 7

**Question no – (78) **

**Solution : **

A car can travel 240 km in 15 litres of petrol.

Distance travel by car in 1 litre

= 240/15

= 16 km

Distance it travel in 25 litres of petrol = we multiply 16 km with 25 litres.

Distance it travel in 25 litres of petrol = 16 x 25

Distance it travel in 25 litres of petrol = 400 km.

**Question no – (79) **

**Solution : **

**(a) how much does he earn in one year?**

Bachhu Manjhi earns Rs 24000 in 8 months

Bachhu Manjhi earns in 1 month = Rs 24000 / 8

Bachhu Manjhi earns in 1 month = Rs. 3000

In year there are 12 months.

Bachhu Manjhi earns in 12 month = we multiply Rs.3000 with 12

Bachhu Manjhi earns in 12 month = 3000 x 12

Bachhu Manjhi earns in 12 month = Rs.36, 000

**(b) in how many months does he earn Rs 42000?**

Bachhu Manjhi earns in 1 month = Rs. 3000

Months he earn Rs 42000 = we divide Rs 42000 by 3000

Months he earn Rs 42000 = 42,000 / 3000

Months he earn Rs 42000 = 14 months.

**Question no – (80) **

**Solution : **

The yield of wheat from 8 hectares of land is 360 quintals.

The yield of wheat in 1 hectares = we divide 360 quintals by 8.

The yield of wheat in 1 hectares = 360 / 8

we divide both by 8

We get, the yield of wheat in 1 hectares = 45 quintals.

The number of hectares of land required for a yield of 540 quintals =

We divide 540 quintals by 45 quintals.

The number of hectares of land required for a yield of 540 quintals =

= 540 / 45

We divide both by 9

We get,

= 60 / 5

= 12

The number of hectares of land required for a yield of 540 quintals is 12 Hectares.

**Question no – (81) **

**Solution : **

The earth rotates 360° about its axis in about 24 hours.

IN 1 hour earth rotates = 360°/ 24

IN 1 hour earth rotates = 150

Earth rotate in 2 hours = we multiply 2 with 150

Earth rotate in 2 hours = 2 x 150

Earth rotate in 2 hours = 300

**Question no – (82) **

**Solution : **

The cost of 10 iron tablets is Rs 17.

Doctor advised her to take one iron tablet two times a day.

In a day requires 2 tablets.

In 15 day requires = 15 x 2 = 30 tablets.

The cost of 10 iron tablets is Rs 17.

The cost of 30 iron tablets = we multiply Rs 17 with 3.

The cost of 30 iron tablets = 17 x 3 = Rs.51

**Question no – (83) **

**Solution : **

The quarterly school fee in Kendriya Vidyalaya for Class VI is Rs 540

There are 12 months in year.

The quarterly means we have to divide 12 months by 4

= 12 / 4

= 3 months.

3 months fee is Rs. 540

Fee of 1 month = Rs.540 / 3

Fee of 1 month = Rs.180

Fee for seven months = we multiply 7 with Rs.180

Fee for seven months = 180 x 7

Fee for seven months = Rs.1260

**Question no – (84) **

**Solution : **

In an election, the votes cast for two of the candidates were in the ratio 5: 7.

The successful candidate received 20734 votes.

The ratio of successful candidate is 7.

Value of 1 ratio = we divide 20734 votes by 7

Value of 1 ratio = 20,734 / 7

Value of 1 ratio = 2962 votes.

The ratio of unsuccessful candidate is 5.

Votes his opponent receive = we multiply 2962 votes. By 5.

Votes his opponent receive = 2962 x 5

Votes his opponent receive = 14810 votes.

**Question no – (85) **

**Solution : **

A metal pipe 3 metre long was found to weigh 7.6 kg.

Weight of 1 m pipe = 7.6 kg. / 3

Weight of 1 m pipe = 2. 533 kg

The weight of the same kind of 7.8m long pipe = we multiply 2.53 with 7.8 m

The weight of the same kind of 7.8m long pipe

= 2.53 x 7. 8

The weight of the same kind of 7.8m long pipe

= 19. 76 kg

**Question no – (86) **

**Solution : **

A recipe for raspberry jelly calls for 5 cups of raspberry juice and 2 x 1/2 cups of sugar

Ratio of raspberry juice and sugar

= 5 / 2.5

5 / 2.5 = 6 /?

By cross multiplication method,

5 x? = 2.5 x 6

? = 15 /5

? = 3 cups

Therefore, the amount of sugar needed for 6 cups of the juice is 3 cups.

**Question no – (87) **

**Solution : **

A farmer planted 1890 tomato plants in a field in rows each having 63 plants.

To find no. of rows we divide total no. of tomato plant with 63 plants.

Number of rows = 1890 tomato plants / 63 plants.

We divide both by 63,

We get, Number of rows = 30

A certain type of worm destroyed 18 plants in each row.

Plants the worm destroy in the whole field = we multiply 18 plants with 30 rows.

Plants the worm destroy in the whole field,

= 30 x 18

= 540 plants

**Question no – (89) **

**Solution : **

A carpenter had a board which measured 3m × 2m.

Area of rectangular board = Length x Breadth

Area of rectangular board = 3 m x 2 m

Area of rectangular board = 6 m²

Area of rectangular piece = Length x Breadth

Area of rectangular piece

= 2.5 m x 0.9 m (we convert cm into m)

Area of rectangular piece = 2.25 m²

Area of remaining piece = Total area – Area of rectangular piece

Area of remaining piece = 6 – 2.25

Area of remaining piece = 3.75 m²

The ratio of the area of cut out piece and the remaining piece

= 2.25 / 3.75

We divide both by 0.75,

We get, the ratio of the area of cut out piece and the remaining piece = 3/5

**Next Chapter Solution : **