RS Aggarwal Class 6 Maths Solutions Chapter 10- Ratio, Proportion And Unitary Method Test Paper-10

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Rs Aggarwal Class 6 Math Solution Chapter 10- Ratio, Proportion And Unitary Method

Test Paper-10

A.1. Find the ratio of:

(a) 90 cm to 1.05 m

(b) 35 minutes to an hour

(c) 150 ml to 2 L

(d) 2 dozens to a score

Solution: 

(a) 90 cm : 1.05 m 

or, 90 cm : 105 cm 

     = `frac\{cancel 90^6}{cancel 105^7}`

     = `frac\{6}{7}` =  6:7

(b) 35 minutes to an hour 

or, 35 minutes : 60 minutes 

     = `frac\{cancel 35^7}{cancel 60^12}`

     = `frac\{7}{12}` =  7 : 12

(c) 150 mL to 2 L 

or, 150 L : 2000 L 

     = `frac\{cancel 150^3}{cancel 2000^40}`

     = `frac\{3}{40}` =  3 : 40

(d) 2 dozens to a score 

or, 24 : 20 

     = `frac\{cancel 24^6}{cancel 20^5}`

     = `frac\{6}{5}` =  6 : 5

2. The ratio of zinc and copper in an alloy is 7 : 9. If the weight of copper in the alloy is 2.6 kg, find the weight of zinc in it.

Solution: 

Ratio of zinc and copper in an alloy is = 7 : 9

Let the weight of zinc and copper be (7x) and (9x), respectively.

Now, the weight of a copper = 12.6 kg   (given)

∴ 9x = 12.6

⇒  x = `frac\{12.6}{9}`  = 1.4

∴ Weight of zinc = 7x = 7​ × 1.4 = 9.8 kg

3. Divide Rs. 1400 among A, B and C in the ratio 2 : 3 : 5.

Solution: 

Sum of ratio = 2 + 3 + 5 = 10

Total money = Rs 1400

Then, share of A = `frac\{2}{10}` × Rs 1400 

                        = Rs `frac\{cancel 2800^280}{cancel 10}`  

                        = Rs 280

Share of B =  `frac\{3}{10}` × Rs 1400 

                        = Rs `frac\{cancel 4200^420}{cancel 10}`  

                        = Rs 420

Share of C =  `frac\{5}{10}` × Rs 1400 

                        = Rs `frac\{cancel 7000^700}{cancel 10}`  

                        = Rs 700

4. Prove that (5 : 6) > (3 : 4).

Solution: 

5 : 6 = `frac\{5}{6}` and 3 : 4 = `frac\{3}{4}`

By making their denominators same: 

(Taking the L.C.M. of 6 and 4, which is 24.)

Consider, 5 : 6

         `frac\{5 ​× 4}{6 ​× 4}` =  `frac\{20}{24}` 

And,  `frac\{3 ​× 6}{4 ​× 6}` =  `frac\{18}{24}` 

As 20 > 18

Clearly, (5 : 6) > (3 : 4)

5. 40 men can finish a piece of work in 26 days. How many men will be needed to finish it in 16 days?

Solution: 

Men needed to finish a piece of work in 26 days = 40

Men needed to finish it in 1 day = 26 × 40 = 1040 

Men needed to finish it in 16 days =  `frac\{cancel 1040^65}{cancel 16}` = 65

6. In an army camp, there were provisions for 425 men for 30 days. How long did the provisions last for 375 men?

Solution: 

Provisions last for 425 men = 30 days

Provisions last for 1 men = 30 × 425 = 12750 days. 

Provisions last for 375 men = `frac\{cancel 12750^34}{cancel 375}`  = 34 days

Hence, provisions will last for 34 days for 375 men.

7. Find the value of x when 36 : x : : x : 16.

Solution: 

Product of means = Product of extremes 

  `x × x` = 36 × 16

⇒ `x^2` = 576

⇒ `x^2` = `24^2`

⇒ x = 24

8. Show that 48, 60, 75 are in continued proportion. 

Solution: 

Product of means = 60 × 60 = 3600

Product of extremes = 48 × 75 = 3600

So product of means = Product of extremes

Hence, 48, 60, 75 are in continued proportion.

B. Mark (✔) against the correct answer in each of the following:

9. Two numbers are in the ratio 3 : 5 and their sum is 96. The larger number is

(a) 36

(b) 42

(c) 60

(d) 70

Solution: The correct option is (c) 60

Let the two number be 3x and 5x

A\q,

    3x + 5x = 96

 ⇒ 8x = 96

 ⇒ x =  `frac\{cancel 96^12}{cancel 8}`  = 12

The numbers are:

     3x = 3 ​× 12 = 36

     5x = 5 ​× 12 = 60

The largest number = 60

10. A car travels 288 km in 4 hours and a train travels 540 km in 6 hours. The ratio of their speeds is

(a) 5 : 4

(b) 4 : 5

(c) 5 : 6

(d) 3 : 5

Solution: The correct option is (b) 4 : 5

Speed of the car =  `frac\{distance}{time}`  =  `frac\{cancel 288^72}{cancel 4}` = 72 km/hr

 Speed of the train =  `frac\{distance}{time}`  =  `frac\{cancel 540^90}{cancel 6}` = 90 km/hr

 Ratio of their speeds = 72 : 90 = `frac\{cancel 72^4}{cancel 90^5}` = 4 : 5

11. The first three terms of a proportion are 12,  21 and 8 respectively. The 4th term is

(a) 18

(b) 16

(c) 14

(d) 20

Solution: The correct option is (c) 14

Let the 4th term be x, such that we have:

     12 : 21 : : 8 : x

 Now, we know:

     Product of extremes = Product of means

                           12x = 21 × 8 

                               x =  `frac\{cancel 168^14}{cancel 12}` = 14

12. The ratio 92 : 115 in simplest form is

(a) 23 : 25

(b) 18 : 23

(c) 3 : 5

(d) 4 : 5

Solution: The correct option is (d) 4 : 5

92 : 115

`frac\{92}{115}`  =  `frac\{92 ​÷ 23}{115 ​÷ 23}`   = `frac\{4}{5}`

13. If 57 : x : : 51 : 85. then the value of x is

(a) 95

(b) 76

(c) 114

(d) none of these

Solution: The correct option is (a) 95

Given :  

57 : x : : 51 : 85

We know:

Product of means = Product of extremes

                   51x = 57 × 85

                      x =  `frac\{cancel 4845^95}{cancel 51}`  = 95

14. If 4 : 5 : : x : 45, then the value of x is

(a) 54

(b) 60

(c) 36

(d) 30

Solution: The correct option is (c) 36

Given:

4 : 5 : : x : 45

We know:

Product of mean = Product of extremes

                    5x = 4 ​× 45

                      x = `frac\{cancel 180^36}{cancel 5}` = 36

15. If a, b, c are in proportion, then

(a) `a^2` = bc

(b) `b^2` = ac

(c) `c^2` = ab

(d) none of these

Solution: The correct option is (b) `b^2` = ac

Given:

a, b, c are in proportion, such that we have:

  a : b : : b : c

Now, we know:

  Product of means = Product of extremes

                    b ​× b = a ​× c

                  `b^2` = ac

16. 10 boys can dig a pitch in 12 hours. How long will 8 boys take to do it?

(a) 9 hrs 36 min

(b) 15 hrs

(c) 6 hrs 40 min

(d) 13 hrs 10 min

Solution: The correct option is (b) 15 hrs.

Time taken by 10 boys to dig a pitch = 12 hours

Time taken by 1 boy to dig a pitch = 12 × 10 = 120 hours   

Time taken by 8 boys to dig a pitch =  `frac\{cancel 120^15}{cancel 8}`  = 15 hours

17. In covering 148 km, a car consumes 8 litres of petrol. How many kilometres will it go in 10 litres of petrol?

(a) 172 km

(b) 185 km

(c) 205 km

(d) 266.4 km

Solution: The correct option is (b) 185 km

Distance covered in 8 litres of petrol = 148 km

Distance covered in 1 litre of petrol =  `frac\{148}{8}`  km

Distance covered in 10 litres of petrol = 10 × `frac\{148}{8}` 

= `frac\{cancel 1480^185}{cancel 8}` 

= 185 km

C. 18. Fill in the blanks.

(i) `frac\{14}{ 21}` = `frac\{⬜}{3}` =  `frac\{6}{⬜}`

(ii) 90 cm: 1.5 m = .......

(iii) If 36 : 81 : : x : 63, then x = ........

(iv) If 25, 35,  x are in proportion, then x = …...

(v) If 9, x, x, 49 are in proportion, then x = ....... .

Solution: 

(i) `frac\{14}{ 21}` = `frac\{⬜}{3}`

⬜ = `frac\{14 × 3}{21}` = `frac\{cancel 42^2}{cancel 21}` = 2

and, `frac\{2}{3}` =  `frac\{6}{⬜}`

⬜ = `frac\{6 × 3}{2}` = `frac\{cancel 18^9}{cancel 2}` = 9

(ii) 90 cm : 1.5 m 

or,  90 cm : 150 cm 

      = `frac\{90}{150}`  

      = `frac\{90 ÷ 30}{150 ÷ 30}` 

      = `frac\{3}{5}`

(iii) If 36 : 81 : : x : 63

      Product of means = Product of extremes

                          81x = 36 × 63

                             x =  `frac\{cancel 2268^28}{cancel 81}`

                             x = 28

(iv) Given:

      25, 35, x are in proportion.

       25 : 35 : : 35 : x

     Now, we know:

     Product of extremes = Product of means

                        25 × x = 35 ​× 35

                            25x = 1225

                              x =  `frac\{cancel 1225^49}{cancel 25}`  = 49

(v) Given:

     9, x, x, 49 are in proportion.

         9 : x : : x : 49

     Now, we know:

     Product of means = Product of extremes

                      `x ​× x` = 9 ​× 49

                     `x^2` = 441

                     `x^2` = `21^2`

                            x = 21

D.19. Write 'T' for true and 'F' for false for each of the statements given below:

(i) 30, 40, 45, 60 are in proportion.

(ii) 6 : 8 and 9 : 12 are equivalent ratios of 3 : 4.

(iii) a dozen : a score = 5 : 3.

(iv) 60 p : Rs. 3 = 1 : 5.

Solution: 

(i) 30, 40, 45, 60 

      `frac\{cancel 30^3}{cancel 40^4}`  = `frac\{3}{4}`

    `frac\{cancel 45^3}{cancel 60^4}`  = `frac\{3}{4}`    

  

They are in proportion.

  Hence, true.

(ii)  `frac\{cancel 6^3}{cancel 8^4}`  = `frac\{3}{4}`

    `frac\{cancel 9^3}{cancel 12^4}`  = `frac\{3}{4}`    

  

They are in proportion.

  Hence, true.

(iii) 1 dozen : 1 score = 12 : 20

      `frac\{12}{20}`  =  `frac\{cancel 12^3}{cancel 20^5}` = `frac\{3}[5}`   

 

Hence, false.

(iv) 60p : Rs 3 = 60p : 300p 

`frac\{60}{300}`  =  `frac\{cancel 60^1}{cancel 300^5}` = `frac\{1}[5}`

Hence, true.