RATIO PROPORTION AND UNITARY METHOD CLASS 6

Problem 1 :

Two numbers are in the ratio pf 3 : 5. If 9 is subtracted from each, the ratio becomes 12 : 23. Find the greater number ?

Solution :

Let the two numbers be 3x and 5x

When 9 is subtracted from each, 3x - 9 and 5x - 9

(3x - 9) : (5x - 9) = 12 : 23

(3x - 9) / (5x - 9) = 12 / 23

23(3x - 9) = 12(5x - 9)

69x - 207 = 60x - 108

69x - 60x = -108 + 207

9x = 99

x = 99/9

x = 11

3x = 33 and 5x = 55

So, the greater number is 55.

Problem 2 :

10 boys can dig a pitch in 12 hours, How long will 8 boys take to do it ?

Solution :

Number of persons digging : Number of hours working

Let x be the required number of hours.

10 : 12 :: 8 : x

10/12 = 8/x

10x = 8(12)

x = 96/10

x = 9.6 hours.

Problem 3 :

A family consumes 30 kg of sugar in 15 days. How much sugar will be consumed in 275 days ?

Solution :

30 kg sugar = 15 days

Let x be the quantity of sugar.

30 : 15 = x : 275

30/15 = x/275

30(275) = 15x

x = 30(275)/15

x = 550

In 275 days 550 kg sugar will consume.

Problem 4 :

Two numbers are respectively 20% and 50% more than a 3rd number. Find the ratio of two numbers ?

Solution :

Let x be the 3rd number.

First number = 120% of x

Second number = 150% of x

120% of x : 150% of x

= 120x/100 : 150x/100

= 120x : 150x

= 4 : 5

So, the required ratio is 4 : 5.

Problem 6 :

A finishes his work in 15 days while B takes 10 days. How many days will the same work be done if they work together?

Solution :

A will complete his work in 15 days.

Part of work completed by A in one day = 1/15

B will complete his work in 10 days.

Part of work completed by B in one day = 1/10

If they work together,

= 1/15 + 1/10

= (2 + 3)/30

= 5/30

= 1/6

= 6 days.

If they work together, they will complete in 6 days.

Problem 7 :

Cost of 4 pens is $120 and 6 pencils cost is $60. If i want to purchase 6 pens and 2 pencils, how much should i pay ?

Solution :

Cost of 4 pens = $120

Cost of 1 pen = 120/4 ==> 30

Cos of 6 pens = 6(30) ==> 180 ---(1)

Cost of 6 pencils = 60

Cost 1 pencil = 60/6 ==> 10

Cos of 2 pencils = 2(10) ==> 20 ---(2)

Total cost = 180 + 20

= 200

Problem 8 :

When 7 is added to the numerator and denominator of the fraction, then the new ratio becomes 13 : 19, what is the original ratio ?

a)  3 : 6   b) 5 : 6         c) 4 : 7

Solution :

Let x/y be the required fraction.

After adding 7 to the numerator and denominator, the ratio between numerator and denominator.

(x + 7) : (y + 7) = 13 : 19

(x + 7)/(y + 7) = 13/19

19(x + 7) = 13(y + 7)

19x + 133 = 13y + 91

19x - 13y = 91 - 133

19x - 13y = -42

Applying option a,

19(3) - 13(6) = -42

57 - 78 = -42

-21 = -42

Which is multiple of 21. So, option a is correct.

Problem 9 :

If x : y = 7 : 4, then (4x + 7y) : (7x + 4y) ?

Solution :

x : y = 7 : 4

x/y = 7/4

4x = 7y

x = (7y/4)

(4x + 7y) : (7x + 4y) = 

Problem 10 :

Calculate the 3rd proportional to 16 and 32.

Solution :

Let x be the third proportional.

16 : 32 :: 32 : x

16/32 = 32/x

16x = 32 (32)

x = 32(32)/16

x = 64