Chapter-8 Ratio And Proportion

Solution:-

(A) 2 : 5

Here, in the above question, we have to find out the ratio.

We know that, to find the ratio, we need to divide the first number by second number in such a way that both the Numerator and denominator remains whole numbers and the symbol use to denote ratio is “:”.

Here, number of books are given, they are 8 and 20.

So, to find ratio, divide 8 by 20, the ratio of 8 books to 20 books 

= 8/20

Now, we need to reduce the value of both the numerator and denominator as much as possible, but remember that they should remains whole number.

So, divide both the value by 4.

= 2/5

Hence, the ratio of 8 books to 20 books = 2 : 5

Solution:-

(A) 1 : 2 to edges = 4/12

Here, in the above question, we have to find out the ratio.

We know that, to find the ratio, we need to divide the first number by second number in such a way that both the Numerator and denominator remains whole numbers and the symbol use to denote ratio is “:”.

Here, we have to find the ratio of the number of sides of a square to the number of edges in cube.

Divide both the numerator and denominator by 4.

= 1/3

Now, we need to reduce the value of both the numerator and denominator as much as possible, but remember that they should remains whole number.

Hence, the ratio of sides of a square to edges of cube = 1 : 3

Solution:-

(D) 1 : 8

Here, in the above question, we have to find out the ratio.

We know that, to find the ratio, we need to divide the first number by second number in such a way that both the Numerator and denominator remains whole numbers and the symbol use to denote ratio is “:”.

Width of a picture = 60 cm

Length of a picture = 1.8 m

Here, we know that 1 m = 100 cm

So, we can write 1.8 m = 180 cm

Now, Perimeter of rectangle = 2 (length + breadth)

= 2 (180 + 60)

= 2 (240)

= 480

Hence, The ratio of its width to its perimeter in lowest form = 60/480

Now, we need to reduce the value of both the numerator and denominator as much as possible, but remember that they should remains whole number.

So, divide both the numerator and denominator by 20.

= 3/24

Again, divide both the numerator and denominator by 3.

= 1/8

= 1 : 8

Solution:-

(B) 1 : 7

Here, in the above question, we have to find out the ratio.

We know that, to find the ratio, we need to divide the first number by second number in such a way that both the Numerator and denominator remains whole numbers and the symbol use to denote ratio is “:”.

Neelam’s annual income = ₹ 288000

Her annual savings amount = ₹ 36000

So, Neelam’s expenditure = 288000 – 36000

= ₹ 252000

Now, the ratio of her savings to her expenditure = 36000/252000

= 36/252

Now, we need to reduce the value of both the numerator and denominator as much as possible, but remember that they should remains whole number.

So, divide both the numerator and denominator by 12.

= 3/21

Again, divide both the numerator and denominator by 3.

= 1/7

Hence, the ratio of her savings to her expenditure = 1 : 7

Solution:-

(C) 3 : 80

Here, in the above question, we have to find out the ratio.

We know that, to find the ratio, we need to divide the first number by second number in such a way that both the Numerator and denominator remains whole numbers and the symbol use to denote ratio is “:”.

Here, in the question,  total number of pages in the Mathematics textbook for Class VI is given as  320 pages

And the chapter ‘symmetry’ runs from page 261 to page 272

So, the Nnmber of pages in chapter ‘symmetry' is 12

Now, the ratio of the number of pages of symmetry chapter to the total number of pages of the book is,

= 12/320

Now, we need to reduce the value of both the numerator and denominator as much as possible, but remember that they should remains whole number.

Now, dividing both the numerator and denominator by 2.

= 6/160

Again, divide both the numerator and denominator by 2.

= 3/80

Hence, the ratio of the number of pages of symmetry chapter to the total number of pages of the book is 3: 80.

Solution:-

(D) 22

Here, in the above question, we have to find out the possible numbers of marbles.

We know that, to find the ratio, we need to divide the first number by second number in such a way that both the Numerator and denominator remains whole numbers and the symbol use to denote ratio is “:”.

Here, in the question, the ratio of red marbles to blue marbles is given as 7:4.

So, let’s suppose that the common factor of 7 and 4 is x.

Now, the total number of marbles in the box should be = 7x + 4x = 11x

Hence, we can say that the numbers of marbles in the box should be 11.

Therefore, the number of marbles in the box is 11 × 2 which is 22 marbles.

Solution:-

(C) 27

Here, in the above question, we have to find out the number of books in the shelf..

We know that, to find the ratio, we need to divide the first number by second number in such a way that both the Numerator and denominator remains whole numbers and the symbol use to denote ratio is “:”.

Here, in the question, ratio of green cover books to red cover books is given as 2:3 and there are total 18 green cover books in the shelf.

So, let’s suppose that the common factor of 2 and 3 as x.

Then, 2x = 18

x = 18/2

Now, we need to reduce the value of both the numerator and denominator as much as possible, but remember that they should remains whole number.

Divide both the numerator and denominator by 2.

x = 9

Hence, the number of books with brown covers in the shelf is = 3x = 3 × 9

= 27

Solution:-

(D) 40 : 25

Here, in the above question, we have to find out the ratio.

We know that, to find the ratio, we need to divide the first number by second number in such a way that both the Numerator and denominator remains whole numbers and the symbol use to denote ratio is “:”.

Now, consider the given ratios, 2 : 3, 5 : 8, 75 : 121 and 40 : 25.

The simplified form of 2: 3 = 2/3 = 0.67

The simplified form of 5 : 8 = 5/8 = 0.625

Simplified form of 75: 121 = 75/121 = 0.61

The simplified form of 40: 25 = 40/25 = 1.6

Hence, the greatest ratio among the given ratios is 40 : 25

Solution:-

(A) b/(b + g)

Here, in the above question, we have to find out the ratio.

We know that, to find the ratio, we need to divide the first number by second number in such a way that both the Numerator and denominator remains whole numbers and the symbol use to denote ratio is “:”.

Here, in the above question the numbers of boys and the girls are given as a and b.

Number of boys in the class = b

Number of girls in the class = g

So, the total number of students in the class = b + g

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

= b/(b + g)

Solution:-

(C) 5 : 8

Here, in the above question, we have to find out the ratio.

We know that, to find the ratio, we need to divide the first number by second number in such a way that both the Numerator and denominator remains whole numbers and the symbol use to denote ratio is “:”.

Here, in the above question, speed and time both are given for both the trains.

So, by using velocity equation, we can find out the distance travelled by both the trains.

The bus travels 160 km in 4 hours

The train travels 320 km in 5 hours

So, distance travelled by bus in an hour = 160/4 = 40 km/h

Distance travelled by train in an hour = 320/5 = 64 km/h

Then the ratio of the distances travelled by them in one hour is = 40/64

Now, we need to reduce the value of both the numerator and denominator as much as possible, but remember that they should remains whole number.

So, divide both the numerator and denominator by 8.

= 5/8

Hence, the ratio of the distances travelled by bus and train in one hour is 5: 8.

Solution:-

Here, in the above question, we have to find out the missing number from the ratio.

We know that, to find the ratio, we need to divide the first number by second number in such a way that both the Numerator and denominator remains whole numbers and the symbol use to denote ratio is “:”.

So, to find the missing number, let's suppose that the missing number is x.

Now, (3/5) = (x/20)

More solving the above equation, we get the value of x,

(3 × 20)/5 = x

x = 60/5

Now, we need to reduce the value of both the numerator and denominator as much as possible, but remember that they should remains whole number.

Divide both the numerator and denominator by 5.

x = 12

Hence, 3/5 = [12]/20

Solution:-

Here, in the above question, we have to find out the missing number from the ratio.

We know that, to find the ratio, we need to divide the first number by second number in such a way that both the Numerator and denominator remains whole numbers and the symbol use to denote ratio is “:”.

So, to find the missing number, let’s suppose that the missing number is x.

Now, x/18 = 2/9

More solving the above equation, we get the value of x,

x = (2 × 18)/9

x = 36/9

Now, we need to reduce the value of both the numerator and denominator as much as possible, but remember that they should remains whole number.

Divide both the numerator and denominator by 9.

x = 4

Hence, [4]/18 = 2/9

Solution:-

Here, in the above question, we have to find out the missing number from the ratio.

We know that, to find the ratio, we need to divide the first number by second number in such a way that both the Numerator and denominator remains whole numbers and the symbol use to denote ratio is “:”.

So, to find the missing number, let’s suppose that the missing number is y.

Now, 8/y = 3.2/4

More solving the above equation, we get the value of y,

Y = (8 × 4)/3.2

Y = 32/3.2

Y = 320/32

Now, we need to reduce the value of both the numerator and denominator as much as possible, but remember that they should remains whole number.

Divide both the numerator and denominator by 32.

y = 10

Hence, 8/[10] = 3.2/4

Solution:-

Here, in the above question, we have to find out the missing number from the ratio.

We know that, to find the ratio, we need to divide the first number by second number in such a way that both the Numerator and denominator remains whole numbers and the symbol use to denote ratio is “:”.

So, to find the missing number in the first ratio, let’s suppose that the missing number is P.

Now, P/45 = 16/40

More solving the above equation, we get the value of P,

P = (16 × 45)/40

P = 720/40

P = 72/4

Now, we need to reduce the value of both the numerator and denominator as much as possible, but remember that they should remains whole number.

Divide both the numerator and denominator by 4.

P = 18

Hence, [18]/45 = 16/40

Now, consider the last two ratios, 16/40 = 24/[ ]

Let’s suppose that the missing number is Q.

Now, 16/40 = 24/Q

More solving the above equation, we get the value of Q,

Q = (24 × 40)/16

Q = 960/16

Now, we need to reduce the value of both the numerator and denominator as much as possible, but remember that they should remains whole number.

Divide both the numerator and denominator by 16.

P = 60

Hence, 16/40 = 24/[60]

Solution:-

Here, in the above question, we have to find out the missing number from the ratio.

We know that, to find the ratio, we need to divide the first number by second number in such a way that both the Numerator and denominator remains whole numbers and the symbol use to denote ratio is “:”.

So, to find the missing number in the first ratio, let’s suppose that the missing number is P.

Now, 16/36 = P/63

More solving the above equation, we get the value of P,

P = (16 × 63)/36

P = 1008/36

Now, we need to reduce the value of both the numerator and denominator as much as possible, but remember that they should remains whole number.

Divide both the numerator and denominator by 36.

P = 28

Hence, 16/36 = [28]/63

Now, consider the middle two ratios, 28/63 = 36/[ ]

Let’s suppose that the missing number is Q.

Now,  28/63 = 36/Q

More solving the above equation, we get the value of P,

Q = (36 × 63)/28

Q = 2268/28

Now, we need to reduce the value of both the numerator and denominator as much as possible, but remember that they should remains whole number.

Divide both the numerator and denominator by 28.

P = 81

Hence, 28/63 = 36/[81]

Consider the last two ratios 36/81 = [ ]/117

Let’s suppose that the missing number is R.

Now, 36/81 = R/117

More solving the above equation, we get the value of P,

R = (36 × 117)/81

R = 4212/81

Now, we need to reduce the value of both the numerator and denominator as much as possible, but remember that they should remains whole number.

Divide both the numerator and denominator by 81.

P = 52

Hence, 36/81 = [52]/117

So, 16/36 = [28]/63 = 36/[81] = [52]/117

Solution:-

True.

We know that, in the ratio, the proportion of the values doesn't change. 

Here, they have multiplied both the numerator and denominator by 5.

So, by multiplication we get, 

(3×5)/(8×5) = 15/40

Now, by dividing 15/40 by 5 we again get the same value as  

3/8

Hence, 3/8 = 15/40

Solution:-

True.

We know that, in the ratio, the proportion of the values doesn’t change. 

Here, they have multiplied both the numerator and denominator by 5.

So, by multiplication we get,

= (4×5)/(7×5) = 20/35

Now, by dividing 20/35 by 5 we again get the same value as,

= 4/7

Hence, 4/7 = 20/35

Solution:-

False.

We know that, in the ratio, the proportion of the values doesn’t change. 

Here, the ratio in the left hand side if given as 0.2/5 and on the right hand side it is given as 2/0.5.

So, by multiplying 10 to the denominator we get 2 which is on the right hand side, but according to the proportionality the numerator should also multiplied by 10.

So, the Numerator should become 50, but on the right hand side it is given as 0.5, which is not correct.

Hence, 0.2/5 ≠ 2/0.5

Solution:-

False.

We know that, in the ratio, the proportion of the values doesn’t change. 

Here, the ratio in the left hand side if given as 3/33 and on the right hand side it is given as 33/333

So, by multiplying 11 to the denominator we get 33 which is on the right hand side, but according to the proportionality the numerator should also multiplied by 11.

So, the Numerator should become 363, but on the right hand side it is given as 333, which is not correct.

Hence, 3/33 ≠ 33/333

Solution:-

False.

We know that, in the ratio, the proportion of the values doesn’t change. 

Here, the ratio in the left hand side if given as 12m/40m and on the right hand side it is given as 35m/65m.

Now, let's solve this by dividing method.

So, divide both the side and compare the result, if both the value are save than the ratio is correct but if there is difference than the ratio is incorrect.

15/40=0.375

35/65=0.538

Here, 0.375≠0.538

Hence, 15m/40m≠35m/65m.

Solution:-

True

We know that, in the ratio, the proportion of the values doesn’t change. 

Here, the ratio in the left hand side if given as 27cm²/57cm² and on the right hand side it is given as 18cm²/38cm².

Now, let’s solve this by dividing method.

So, divide both the side and compare the result, if both the value are save than the ratio is correct but if there is difference than the ratio is incorrect.

27/57=0.473

18/38=0.473

Here, 0.473=0.473

Hence, 27cm²/57cm²=18cm²/38cm².

Solution:-

False.

We know that, in the ratio, the proportion of the values doesn’t change. 

Here, the ratio in the left hand side if given as 5kg : 7.5kg and on the right hand side it is given as Rs 7.50 : Rs 5

Now, let’s solve this by dividing method.

So, divide both the side and compare the result, if both the value are save than the ratio is correct but if there is difference than the ratio is incorrect.

5/7.5=0.67

7.5/5=1.5

Here, 0.67≠1.5

Hence, 5kg : 7.5kg ≠ Rs 7.50 : Rs 5

Solution:-

True

We know that, in the ratio, the proportion of the values doesn’t change. 

Here, the ratio in the left hand side if given as 20g : 100g  and on the right hand side it is given as 1metre : 500cm

Now, let’s solve this by dividing method.

So, divide both the side and compare the result, if both the value are save than the ratio is correct but if there is difference than the ratio is incorrect.

20/100= 0.2

100/500= 0.2

Here, 0.2=0.2

Hence, 20g : 100g = 1metre : 500cm

Solution:-

True

We know that, in the ratio, the proportion of the values doesn’t change. 

Here, the ratio in the left hand side if given as 12 hours : 30 hours  and on the right hand side it is given as 8km : 20km

Now, let’s solve this by dividing method.

So, divide both the side and compare the result, if both the value are save than the ratio is correct but if there is difference than the ratio is incorrect.

12/30=0.4

8/20=0.4

Here, 0.4=0.4

Hence, 12 hours : 30 hours = 8km : 20km.

Solution:-

True.

We know that, in the ratio, the proportion of the values doesn’t change. 

Here, the ratio in the left hand side if given as 10/100 and on the right hand side it is given as 1/10.

Now, let’s solve this by dividing method.

So, divide both the side and compare the result, if both the value are save than the ratio is correct but if there is difference than the ratio is incorrect.

10/100=1/10

Hence, the ratio of 10kg to 100kg is 1:10.

Solution:-

False

We know that, in the ratio, the proportion of the values doesn’t change. 

Here, the ratio in the left hand side if given as 150cm/1m which is 150cm/100cm and on the right hand side it is given as 1/1.5.

Now, let’s solve this by dividing method.

So, divide both the side and compare the result, if both the value are save than the ratio is correct but if there is difference than the ratio is incorrect.

150/100=1.5

1/1.5=0.67

Here, 1.5≠0.67

Hence, the ratio of 150cm to 1 metre is not 1:1.5.

Solution:-

True.

We know that, in the ratio, the proportion of the values doesn’t change. 

Here, the ratio in the left hand side if given as 25kg : 20g  and on the right hand side it is given as 50kg : 40g.

Now, let’s solve this by dividing method.

So, divide both the side and compare the result, if both the value are save than the ratio is correct but if there is difference than the ratio is incorrect.

25000/20=1250

50000/40=1250

Here, 1250=1250

Hence, 25kg : 20g = 50kg : 40g.

Solution:-

False

We know that, in the ratio, the proportion of the values doesn’t change. 

Here, the ratio in the left hand side if given as 1 hour to1 day which can be written as 1/24 and on the right hand side it is given as 1/1.

Now, let’s solve this by dividing method.

So, divide both the side and compare the result, if both the value are save than the ratio is correct but if there is difference than the ratio is incorrect.

1/24=0.42

1/1=1

Here, 0.42≠1

Hence, the ratio of 1 hour to one day is not 1:1.

Solution:-

False

Here, the ratio is given as 4:16

Now, we know that by multiplied 4 to 4 we get 16.

So, the numerator and denominator both can by divided by 4

= 4/16

= 1/4

Hence, the lowest form of 4: 16 is 1/4

Solution:-

True.

We know that, in the ratio, the proportion of the values doesn’t change. 

Here, the ratio in the left hand side if given as 5:4 and on the right hand side it is given as 4:5.

Now, let’s solve this by dividing method.

So, divide both the side and compare the result, if both the value are save than the ratio is correct but if there is difference than the ratio is incorrect.

5/4=1.2

4/5=0.8

Here, 1.2≠0.8

Hence, the ratio 5 : 4 is different from the ratio 4 : 5.

Solution:-

False.

We know that, when two value are given in the p/q form.

The value of the ratio will be more than 1 if numerator if more than denominator and less than 1 is numerator is less than denominator.

So, it can be more or less than 1.

Solution:-

True.

We know that, when two value are given in the p/q form.

The value of the ratio will be more than 1 if numerator if more than denominator and less than 1 is numerator is less than denominator.

But if both numerator and denominator are equal than it is equal to 1.

Example: 2: 2 = 2/2 = 1

Solution:-

False

We know that, when the ratio of first and second terms are equal to the ratio of third and fourth terms than they are in proportion.

Solution:-

False.

We know that, to find the ratio both the denominator and numerator should be in the same unit if they are not than they need to convert them first into the save unit form.

Solution:-

A ratio is a form of comparison by division.

We know that, the ratio are the expressions which have numerator and denominator showing comparison in the form of division.

Solution:-

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

Here, we need to solve the above given ratio.

So, let's suppose that the blank space as P.

 Note,

20m : 70m = ₹ 8 : ₹ P

20/70 = 8/P

P = (70 × 8)/20

P = 560/20

P = 56/2

P = 28

Hence, 20m : 70m = Rs 8 : Rs 28.

Solution:-

There is a number in the box [ ] such that [ ], 24, 9, 12 are in proportion. The number in the box is 18.

Here, we have to find the number in the bracket.

So, let's suppose that the number in the bracket as ‘P’

Now, P, 24, 9, 12

P: 24 = 9: 12

P/24 = 9/12

Here, we can simplify right side further by dividing both the numerator and denominator by 3.

So, P/24 = 3/4

P = (3 × 24)/4

P = 72/4

P = 18

Hence, the missing number in the bracket is 18.

Solution:-

If two ratios are equal, then they are in proportion.

We know that, when the ratio of first and second terms are equal to the ratio of third and fourth terms than they are in proportion.

Use Fig. 8.2 (In which each square is of unit length) for questions 39 and 40:

Solution:-

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

Here, from the figure we can find our the perimeter of the shaded petition.

= 1 + 2 + 1 + 2 = 6 units

We know that the perimeter of the whole figure is

= 3 + 4 + 3 + 4 = 14 units

Now, to find the ratio of the perimeter of the boundary of the shaded portion to the perimeter of the whole figure 

= 6/14

= 3/7

= 3: 7

Solution:-

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

Here, from the figure we can find out the area of the shaded figure 

= 2 × 1

= 2 sq. units

We know that the area of whole figure 

= 3 × 4 = 12 sq. units

Not, to find the ratio of the area of the shaded portion to that of the whole figure is 

= 2: 12

= 2/12

= 1/6

= 1: 6

Solution:-

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

Here, from the figure, we can say that,

The sleeping time of python = 18 hours and 

Awaking time = 24 – 18 = 6 hours

So, the ratio of sleeping time to awaking time is =18/6

= 3/1

= 3: 1

Solution:-

A ratio expressed in the lowest form has no common factor other than one in its terms.

We know that, the lowest form is the form which is further not divisible by any number in both numerator and denominator.

Hence, the common factor for such terms are 1.

Solution:-

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

We know that, to find the ratio both the denominator and numerator should be in the same unit if they are not than they need to convert them first into the save unit form.

Solution:-

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

Here, we have to solve the problem to get the answer.

So, let's first create the ratio

Now, in the question it is given that

5 paise : 25 paise = 20 paise: [ ]

So, let's suppose that the black space has Q.

5 paise : 25 paise = 20 paise: Q

5/25 = 20/Q

Q = (20 × 25)/5

Q = 500/5

Q = 100

Hence, the ratio of 5 paise to 25 paise is the same as the ratio of 20 paise to 100 paise

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 the lowest form is 149: 160.

Here, in the question days if given as

Saturn takes 9 hours 56 minutes for one spin on their axes

Now, 1 hour = 60 minutes

So, time taken by Saturn 

=(9 × 60) + 56 

= 540 + 56 

= 596 minutes

Find taken by Jupiter takes 10 hours 40 minutes for one spin on their axes

= (10 × 60) + 40

= 600 + 40

= 640 minutes

So, to find the ratio of the time taken by Saturn and Jupiter in lowest form is 

= 596/640

Here, we can further divide numerator and denominator by 2,

= 298/320

Again, divide both the numerator and denominator by 2,

= 149/160

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