RATIO AND PROPORTION

RATIO AND PROPORTION

We compared the two quantities in terms of ‘how many times’. This comparison is known as the Ratio. We denote ratio using symbol ‘ : ’

Ratio:

Comparing two quantities of same kind and same unit by division is called Ratio.

Q1 / Q2 = R

Ratio is represented by : Colon.

Q1 : Q2 = R

Both quantities are known as terms. The first term Q1 is called Antecedent and second term Q2 is known as Consequent.

Two quantities can be compared only if they are in the same unit.

Note

The numbers that form the ratio are called terms of the ratio. The first term Q1 is called Antecedent and second term Q2 is known as Consequent.

★ Ratio can be obtained only for quantities with same units.

★ In ratio first to first and second to second law works.

A ratio shows how many times one number contains another.

A ratio shows the relative sizes of two or more values.

Raahul has 12 red balls and 16 blue balls. Find the ratio of red balls to blue balls.

Red balls = 12

Blue balls = 16

Total balls = 12 + 16 = 28

Ratio can be asked

1. Red balls to blue balls

2. Blue balls to red balls

3. Total balls to blue balls

4. Total balls to red balls

5. Red balls to Total balls

6. Blue balls to Total balls

Sol.

Here In ratio first to first and second to second law works.

The ratio of

1. Red balls to blue balls = 12 : 16.

2. Blue balls to red balls = 16 : 12

3. Total balls to blue balls = 28 : 16

4. Total balls to red balls = 28 : 12

5. Red balls to Total balls = 12 : 28

6. Blue balls to Total balls = 16 : 28

Practice Time 1

1. In a class, there are 40 boys and 30 girls. What is the ratio of the number of boys to the number of girls?

(a) What is the ratio of the number of boys to the number of girls?

(b) Number of girls to the total number of students.

2. Out of 1800 students in a school, 750 opted basketball, 800 opted cricket and

remaining opted table tennis. If a student can opt only one game, find the ratio of

(a) Number of students who opted basketball to the number of students who

opted table tennis.

(b) Number of students who opted cricket to the number of students opting

3. Length and breadth of a rectangular field are 50 m and 15 m respectively.

(a) Find the ratio of the length to the breadth of the field.

(b) Find the ratio of the perimeter to the breadth of the field.

(d) Find the ratio of the area to the breadth of the field.

(e) Find the ratio of the length to the area of the field.

4. There are 20 girls and 15 boys in a class.

(a) What is the ratio of the number of boys to the number of girls?

(b) Number of girls to the total number of students.

(d) What is the ratio of number of girls to the number of boys?

5. Out of 30 students in a class, 6 like football,

12 like cricket and remaining like tennis.

Find the ratio of

(a) Number of students liking football to

number of students liking tennis.

(b) Number of students liking tennis to

total number of students.

(d) Number of students liking cricket to

total number of students.

(e) Number of students liking cricket and football to total number of students.

6. An engineer earns ₹60,000 per month. His monthly expenditure is ₹45,000. Find the ratio of:

(a) his expenditure to his savings.

(b) his income to his expenditure.

7. In a year, Seema earns Rs 1,50,000 and saves Rs 50,000. Find the ratio of

(a) Money that Seema earns to the money she saves.

(b) Money that she saves to the money she spends.

8. In a college, out of 4320 students, 2300 are girls. Find the ratio of

(a) Number of girls to the total number of students.

(b) Number of boys to the number of girls.

9. Present age of father is 42 years and that of his son is 14 years. Find the ratio of

(a) Present age of father to the present age of son.

(b) Age of the father to the age of son, when son was 12 years old.

(d) Age of father to the age of son when father was 30 years old.

10. There are 45 persons working in an office. If the number of females is 25 and the remaining are males, find the ratio of :

(a) The number of females to number of males.

(b) The number of males to number of females.

11. The area of two squares is 16 sq. cm and 25 sq. cm, respectively. Find the ratio of their sides and perimeters.

12. The area of two squares is 36 sq. cm and 64 sq. cm, respectively. Find the ratio of their sides and perimeters.

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Simplest Form or Lowest Term or Standard Form of Ratio

A ratio can be expressed in its Simplest Form or Lowest Term or Standard Form of Ratio

A ratio is expresed into its simplest form like a fraction by dividing the HCF of both terms.

A ratio is written as Q1 : Q2 . Actually Ratios can also be written as fractions Q1 / Q2.

Ratios can also be written as fractions and can be reduced to their simplest form.

For example,

(1)

Convert the ratio 20:50 into simplest form

HCF of 20 and 25 is 5

20 ÷ 5

= ––––––

25 ÷ 5

= ––

Simplest Form of Ratio 20:50 is 4:5

(2)

Ratio of 25 paise to 1 rupee

Sol.

25 paise to 1 rupee = 25 paise / 1 rupee

(Convert the quantities to the same and small unit)

= 25 paise / 100 paise

HCF of 25 and 100 is 25

= (25 ÷ 25)/(100 ÷ 25)

= 1/4

Thus, the ratio = 1/4 ,

(3)

Find the ratio of 5 kg to 350 g

Sol.

(Convert the quantities to the same and small unit)

Ratio of 5 kg to 350 g = 5 kg / 350 g

So, 5 × 1000g / 350 g

= (5000g)/(350g)

HCF of 5000 and 350 is 50

= (5000g )/(350g)

= (5000 ÷ 50) / (350 ÷ 50)

= 100/7

The ratio is 100:7

Practice Time 2

1. Find the ratio of the following :

1. Convert each of the following into the simplest form.

(a) 77 to154

(b) 66 to 88

(d) 23 to 92

(e) 81 to 108

(f) 98 to 63

(g) 7 to154

(h) 68 to 88

(i) 14 to 56

(j) 33 to 108

2. Express the following as ratios in simplest form.

(a) 12 kg to 84 kg

(b) 5 kg to 500 g

(d) 80 kg to 10 kg

(e) 30 minutes to 45 minutes

(f) 55 minutes to 35 minutes

(g) 4 minutes to 48 seconds

(h) 5 minutes to 30 seconds

(i) 45 minutes to 1 hour

(j) 70 minutes to 2 hour

(k) 90 paise to 1 Rs

(l) 75 paise to 1Rs

(m) 30 paise to 2 Rs

(n) 1Rs to 30 paise

(o) 2 Rs to 50 paise

(p) 2 years to 4 months

(q) 1 years to 8 months

(r) 33 km to 120 km

(s) 35 km to 105 km

(t) 3 km to 500 m

(u) 1 km to 250 m

(v) 75 cm to 2 m

(w) 2 m to 75 cm

(x) 1 m to 25 cm

(y) 30 minutes to 1.5 hours

(z) 40 cm to 1.5 m

(aa) 55 paise to Re 1

(ab) 500 ml to 2 litres

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Always remember it

★ A ratio is a comparison between two quantities of the same kind. It is read as a to b' or 'a is to b. It is written as a : b or in fraction form a/b.

★ The first term of the ratio is called Antecedent, and the second term of the ratio is called Consequent.

★ The order of quantities is very important in a ration, so x : y is not the same as y : x.

★ A ratio is usually expressed in its simplest form.

★ To find ratio, quantities of different units are converted into the same units.

★ A ratio has no units, since the two units are of the same kind.

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Equivalent Ratios

When the ratios represent the same ratio when expressed in their simplest forms are are called equivalent ratios.

We can find equivalent ratios by multiplying or dividing the antecedent and consequent of the given ratio by the same non-zero numbers.

Practice Time 3

Find five equivalent ratios for each of the following:

(a) 1 : 2

(b) 4 : 6

(d) 10 : 20

(e) 15 : 25

(f) 12 : 20

(g) 4 : 5

(h) 5 : 6

(i) 30 : 20

(j) 100 : 200

(k) 9:15

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Comparing Ratios

We follow the given steps to compare the ratio.

We can compare the ratio as greater than, less than or equal to.

★ Write the given ratios as fractions in the simplest form.

★ Find the LCM of denominators.

★ Convert into like fractions by making denominators equal to the LCM.

★ Compare the numerators.

★ The ratio having the smaller numerator is smaller and big numerator is bigger.

1. Compare 5:6 and 4:9,

Sol.

Writing the ratios as fractions:

= 5/6 and 4/9

LCM of 6 and 9 is 18.

=> 5/6 ........ 4/9

5/6

= (5 × 3)/(6 × 3)

= 15/18

4/9

= (4 × 2)/(9 × 2)

= 8/18

As. 15 > 8,

15/18 > 8/18

This

5/6 > 4/9

Alternatively, cross multiply the ratios:

5/6 or 4/9

5 * 9 or 4 * 6

= 45 > 24

5/6 > 4/9

Since, 45 > 24,

we get 5.6 > 4.9

Practice Time 4

1. Compare the following ratios.

Which following are the pair of equivalent ratios?

Compare with >, < and =.

(a) 14:49 and 6:21

(b) 6:11 and 9:44

(d) 3:10 and 2:15

(e) 56:21 and 16:6

(f) 3:4 and 51:68

(g) 1 : 5 and 3 : 15

(h) 2 : 9 and 18 : 81

(i) 15 : 45 and 5 : 25

(j) 4 : 12 and 9 : 27

(k) Rs 10 to Rs 15 and 4 to 6

Concept Highlights

Consider two ratios a:b and c:d

Cross multiply the ratios

ad > bc

then a / b > c / d

ad < bc

then a:b < cd

If two ratios are equal, we say that they are in proportion and use the symbol ‘ : : ’ or ‘ = ’ to equate the two ratios.

If two ratios are not equal, then we say that they are not in proportion.

In a statement of proportion, the four quantities involved when taken in order are known as respective terms.

First and fourth terms are known as extreme terms. Second and third terms are known as middle terms.

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Proportion: A proportion is an equality of two ratios.

Four quantities are in proportion if Product of extreme terms product of middle terms.

★ If a: b:: c: d

then a × d = b × с.

Four quantities are said to be in proportion, if the ratio of the first and the second quantities is equal to the ratio of the third and the fourth quantities.

1. Are 30, 40, 45 and 60 in proportion?

Sol.

Ratio of 30 : 40 = 45 : 60

3 : 4 = 3 : 4.

Since, 30 : 40 = 45 : 60.

Therefore, 30, 40, 45, 60 are in proportion.

Practice Time 5

1. Determine if the following are in proportion.

(a) 15, 45, 40, 120

(b) 33, 121, 9,96

(d) 32, 48, 70, 210

(e) 4, 6, 8, 12

(f) 33, 44, 75, 100

(g) 16, 4, 32, 8

(h) 3,4,2,6

(i) 18,10,9,15

(j) 16, 4, 64, 16

(k) 3,4,12,16

(l) 18,10,9,5

2. Circle the following statements of proportion that are true.

(i) 4 : 7 :: 12 : 21

(ii) 8 : 3 :: 24 : 6

(iii) 7 : 12 :: 12 : 7

(iv) 21 : 6 :: 35 : 10

(v) 12 : 18 :: 28 : 12

(vi) 24 : 8 :: 9 : 3

3. Write True ( T ) or False ( F ) against each of the following statements :

(a) 16 : 24 :: 20 : 30

(b) 21: 6 :: 35 : 10

(d) 8 : 9 :: 24 : 27

(e) 5.2 : 3.9 :: 3 : 4

(f) 0.9 : 0.36 :: 10 : 4

4. Are the following statements true?

(a) 40 persons : 200 persons = Rs 15 : Rs 75

(b) 7.5 litres : 15 litres = 5 kg : 10 kg

(d) 32 m : 64 m = 6 sec : 12 sec

(e) 45 km : 60 km = 12 hours : 15 hours

5. Determine if the following ratios form a proportion. Also, write the middle terms

and extreme terms where the ratios form a proportion.

(a) 25 cm : 1 m and Rs 40 : Rs 160

(b)39 litres : 65 litres and 6 bottles : 10 bottles

(d) 200 ml : 2.5 litre and Rs 4 : Rs 50

6. Find the value of x in the following proportions.

(a) 7:15:x:30

(b) x:3:: 12:18

(d) 8:16:25:x

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Unitary Method

The method in which first we find the value of one unit and then the value of required number of units is known as Unitary Method.

If the cost of 6 cans of juice is Rs 210, then what will be the cost of 4 cans of juice?

Sol.

Cost of 6 cans of juice = Rs 210

Therefore,

cost of 1 can of juice = 210/6

= Rs 35

Therefore,

cost of 4 cans of juice = Rs 35 × 4

= Rs 140.

Thus, cost of 4 cans of juice is Rs 140.

A motorbike travels 220 km in 5 litres of petrol. How much distance will it cover in 1.5 litres of petrol?

Sol.

In 5 litres of petrol, motorbike can travel = 220 km.

Therefore,

in 1 litre of petrol, motor bike travels = 220/5 km

= Rs 44

Therefore,

in 1.5 litres, motorbike travels = 44 ×1.5 km

= 66 km.

Thus, the motorbike can travel 66 km in 1.5 litres of petrol.

If the cost of a dozen soaps is Rs 153.60, what will be the cost of 15 such soaps?

Sol.

We know that 1 dozen = 12

Since,

cost of 12 soaps = Rs 153.60

cost of 1 soaps = Rs 153.60/12

= Rs 12.8

cost of 1 soaps = Rs 12.8 × 15

= Rs 192

A 36. सामान्य अंकगणित (एकक या ऐकिक नियम)

लेखक

ॐ जितेन्द्र सिंह तोमर

M.A., B.Ed., DNYS, MASSCOM

23/4/11/11/2021

A 36. सामान्य अंकगणित (एकक या ऐकिक नियम)

एकक या ऐकिक नियम

अनेक वस्तुओं की कीमत ज्ञात होने पर एक वस्तु की अनेक वस्तुओं की कीमत ज्ञात करने की धारणा को एकक या 1 नयम की धारणा कहते हैं।

A 36. General Arithmetic (Unit or Unit Rule)

Unit or Unitary Rule

The concept of finding the price of one commodity when the price of many commodities is known is called the concept of unit or Unitary rule.

Finding the price of one commodity when the price of many commodities is known.

अनेक वस्तुओं की कीमत ज्ञात होने पर एक वस्तु की कीमत ज्ञात करना।

(A)

यदि 1 वस्तु का मूल्य ₹ x है तो N वस्तुओं का मूल्य ज्ञात करने के लिए।

If the price of 1 item is ₹ x then to find the price of N items.

(B)

यदि N वस्तुओं का मूल्य ₹ x है तो 1 वस्तु का मूल्य ज्ञात करने के लिए।

To find the price of 1 item if the price of n items is ₹ x.

(C)

यदि N वस्तुओं का मूल्य ₹ x है तो M वस्तुओं का मूल्य ज्ञात करने के लिए।

To find the price of M items if the price of N items is ₹ x.

(A) Type

एक वस्तु की कीमत ज्ञात होने पर अनेक वस्तुओं की कीमत ज्ञात करना।

Finding the price of many items when the price of one item is known.

[यदि 1 वस्तु का मूल्य ₹ x है तो N वस्तुओं का मूल्य ज्ञात करने के लिए।]

If the price of 1 item is ₹ x then to find the price of N items.

Q. यदि 1 वस्तुओं का मूल्य ₹ x है तो N वस्तुओं का मूल्य ज्ञात करने के लिए।

Sol.

1 वस्तु का मूल्य = ₹ x

N वस्तुओं का मूल्य = ₹ x × N

Q. If the price of 1 item is ₹ x then to find the price of N items.

Sol.

Price of 1 item = ₹ x

Price of N items = ₹ x × N

अभ्यास 06

1. यदि 1 किलोग्राम चीनी का मूल्य 40 रु० है तो 17 किलोग्राम चीनी का मूल्य ज्ञात कीजिए।

Sol.

1 किलोग्राम चीनी का मूल्य = ₹ 40
17 किलोग्राम चीनी का मूल्य = ₹ 40 × 17

= ₹ 680
चीनी का मूल्य = ₹ 680

2. यदि 1 किलोग्राम अनाज का मूल्य 24 रु० है तो 15 किलोग्राम अनाज का मूल्य ज्ञात कीजिए।

Sol.

1 किलोग्राम चीनी का मूल्य = ₹ 24
1 किलोग्राम चीनी का मूल्य = ₹ 24 × 15

= ₹ 360

अनाज का मूल्य = ₹ 360

3. एक प्रेश 1 दिन में 120 अखबार छापता है। तो 12 दिन में वह कितने अखबार छापेगा?

4. कोलकाता से नई दिल्ली का 1 व्यक्तियों का रेल भाड़ा ₹ 900 है। 7 व्यक्ति का किराया कितना होगा?

5. एक कॉपी का मूल्य ₹ 18 है तो एक दर्जन कापियों का मूल्य कितना होगा?

6. एक प्रेश 1 दिन में 10 पुस्तक बाईंड करती है। तो 15 दिन में वह कितने पुस्तक बाईंड करेगी?

7. 1 प्रति का दाम ₹ 200 है तो ग्यारह प्रतियों की दाम कितना होगा?

8. 1 पेन्सिल का मूल्य 7 रुपये हो तो 15 पेन्सिल का मूल्य ज्ञात कीजिये।

9. 1 पेनों का मूल्य ₹ 30 है तो 21 पेनों का मूल्य कितना होगा?

10. 1 किग्रा तेल का मूल्य ₹ 180 है, 16 किग्रा तेल का मूल्य कितना होगा?

11. 1 पैकिट में 20 बिस्कुट हैं। 19 पैकिटों में कितने बिस्कुट होंगे?

12. एक फैक्टरी में 1 दिन में 750 पेंच तैयार होते हैं। 11 दिन में कितने पेंच तैयार होंगे?

13. एक मशीन 1 मिनट में 7 खिलौने बनाती है। 120 मिनट में कितने खिलौने बनायेगी?

14. एक नौकर 1 दिन में ₹ 420 की मजदूरी करता है। एक सप्ताह में उसे कितने ₹ मिलेंगे?

15. 1 शॉल का मूल्य ₹ 150 है तो 19 शाॅलों का मूल्य कितना होगा?

Exercise 06

1. If the cost of 1 kg of sugar is Rs.40, then find the cost of 17 kg of sugar.

Sol.

Price of 1 kg sugar = ₹ 40

Price of 17 kg sugar = ₹ 40 × 17

= ₹ 680

Price of sugar = ₹ 680

2. If the cost of 1 kg of grain is Rs.24, then find the cost of 15 kg of grain.

Sol.

Price of 1 kg sugar. = ₹ 24

Price of 1 kg sugar. = ₹ 24 × 15

= ₹ 360

Price of grain = ₹ 360

3. A press prints 120 newspapers in a day. So how many newspapers will he print in 12 days?

4. The train fare for 1 person from Kolkata to New Delhi is ₹ 900. How much will be the fare for 7 persons?

5. If the cost of one copy is ₹ 18, then what will be the cost of a dozen copies?

6. One press binds 10 books in a day. So how many books will she bind in 15 days?

7. If the cost of 1 copy is ₹ 200, then what will be the cost of eleven copies?

8. If the cost of 1 pencil is Rs 7, then find the cost of 15 pencils.

9. If the cost of 1 pen is ₹ 30, then what will be the cost of 21 pens?

10. The cost of 1 kg of oil is ₹ 180, what will be the cost of 16 kg of oil?

11. There are 20 biscuits in 1 packet. How many biscuits will be there in 19 packets?

12. In a factory 750 screws are manufactured in a day. How many screws will be ready in 11 days?

13. A machine makes 7 toys in 1 minute. How many toys will she make in 120 minutes?

14. A servant earns ₹ 420 in a day. How much Rs will he get in a week?

15. If the cost of 1 shawl is ₹ 150, then what will be the cost of 19 shawls?

(B) Type

[यदि N वस्तुओं का मूल्य ₹ x है तो 1 वस्तु का मूल्य ज्ञात करने के लिए।]

[To find the price of 1 item if the price of n items is ₹ x.]

Q. यदि N वस्तुओं का मूल्य ₹ x है तो 1 वस्तु का मूल्य ज्ञात करने के लिए।

Sol.

N वस्तुओं का मूल्य = ₹ x

1 वस्तु का मूल्य = ₹ x / N

Q. If the price of N items is ₹ x then to find the price of 1 item.

Sol.

Price of N items = ₹ x

Price of 1 item = ₹ x / N

अभ्यास 07

1. यदि 8 किलोग्राम चीनी का मूल्य 240 रु० है तो 1 किलोग्राम चीनी का मूल्य ज्ञात कीजिए।

Sol.

8 किलोग्राम चीनी का मूल्य = ₹ 240
1 किलोग्राम चीनी का मूल्य = ₹ 240 /8

= ₹ 30
चीनी का मूल्य = ₹ 240 प्रति किलो

2. यदि 12 किलोग्राम अनाज का मूल्य 240 रु० है तो 1 किलोग्राम अनाज का मूल्य ज्ञात कीजिए।

Q. 2. यदि 12 किलोग्राम अनाज का मूल्य 240 रु० है तो 1 किलोग्राम अनाज का मूल्य ज्ञात कीजिए।

12 किलोग्राम चीनी का मूल्य = ₹ 240
1 किलोग्राम चीनी का मूल्य = ₹ 240 /12

= ₹ 20

अनाज का मूल्य = ₹ 240 प्रति किलो

3. एक प्रेश 3 दिन में 1200 अखबार छापता है। तो 1 दिन में वह कितने अखबार छापेगा?

4. कोलकाता से नई दिल्ली का 4 व्यक्तियों का रेल भाड़ा ₹ 900 है। 1 व्यक्ति का किराया कितना होगा?

5. ₹ 108 प्रति दर्जन की दर से 1 कापी का मूल्य कितना होगा?

6. एक प्रेस 12 दिन में 1200 पुस्तक छापती है। तो 1 दिन में वह कितनी पुस्तक छापेगी ?

7. ₹ 8 प्रतियों का दाम ₹ 200 है तो एक प्रति की दाम कितना होगा?

8. 5 पेन्सिल का मूल्य 35 रुपये हो तो एक पेन्सिल का मूल्य ज्ञात कीजिये।

9. 10 पेनों का मूल्य ₹ 30 है तो 1 पेनों का मूल्य कितना होगा?

10. 15 किग्रा तेल का मूल्य ₹ 315 है, 1 किग्रा तेल का मूल्य कितना होगा?

11. 15 पैकिटों में 600 बिस्कुट हैं। 1 पैकिटों में कितने बिस्कुट होंगे?

12. एक फैक्टरी में 3 दिन में 750 पेंच तैयार होते हैं। 1 दिन में कितने पेंच तैयार होंगे?

13. एक मशीन 7 मिनट में 70 खिलौने बनाती है। 1 मिनट में कितने खिलौने बनायेगी?

14. एक नौकर 1 सप्ताह में ₹ 4200 की मजदूरी करता है। एक दिन में उसे कितने ₹ मिलेंगे?

15. 30 शॉल का मूल्य ₹ 15000 है तो 1 शाल का मूल्य कितना होगा?

Exercise 07

1. If the cost of 8 kg of sugar is Rs 240, then find the cost of 1 kg of sugar.

Sol.

Price of 8 kg sugar = ₹ 240

Price of 1 kg sugar = ₹ 240 /8

= ₹ 30

Price of 1 kg sugar = ₹ 30 per kg

2. If the cost of 12 kg of grain is Rs.240, then find the cost of 1 kg of grain.

Sol.

Price of 12 kg sugar = ₹ 240

Price of 1 kg sugar = ₹ 240 /12

= ₹ 20

Price of 1 kg grain = ₹ 20

3. A press prints 1200 newspapers in 3 days. So how many newspapers will he print in 1 day?

4. The train fare of 4 persons from Kolkata to New Delhi is ₹ 900. How much will be the fare for 1 person?

5. What will be the cost of 1 copy at the rate of ₹ 108 per dozen?

6. A press prints 1200 books in 12 days. So how many books will she print in 1 day?

7. If the cost of ₹ 8 copies is ₹ 200, then what will be the cost of one copy?

8. If the cost of 5 pencils is Rs 35, then find the cost of one pencil.

9. If the cost of 10 pens is ₹ 30, then what will be the cost of 1 pen?

10. The cost of 15 kg of oil is ₹ 315, what will be the cost of 1 kg of oil?

11. There are 600 biscuits in 15 packets. How many biscuits will be in 1 packet?

12. In a factory, 750 screws are manufactured in 3 days. How many screws will be ready in 1 day?

13. A machine makes 70 toys in 7 minutes. How many toys will she make in 1 minute?

14. A servant earns ₹ 4200 in a week. How much Rs will he get in a day?

15. If the cost of 30 shawls is ₹ 15000, then what will be the cost of 1 shawl?

(C) Type

[यदि N वस्तुओं का मूल्य ₹ x है तो M वस्तु का मूल्य ज्ञात करने के लिए।]

Q. यदि N वस्तुओं का मूल्य ₹ x है तो M वस्तु का मूल्य ज्ञात करने के लिए।

Sol.

N वस्तुओं का मूल्य = ₹ x

1 वस्तु का मूल्य = ₹ x / N

M वस्तुओं का मूल्य = ₹ (x / N ) × M

अभ्यास 08

1. यदि 8 किलोग्राम चीनी का मूल्य 240 रु० है तो 11 किलोग्राम चीनी का मूल्य ज्ञात कीजिए।

Sol.

8 किलोग्राम चीनी का मूल्य = ₹ 240

1 किलोग्राम चीनी का मूल्य = ₹ 240 /8

= ₹ 30

11 किलोग्राम चीनी का मूल्य = ₹ 30 × 11

= ₹ 330

चीनी का मूल्य = ₹ 330

2. यदि 12 किलोग्राम अनाज का मूल्य 240 रु० है तो 15 किलोग्राम अनाज का मूल्य ज्ञात कीजिए।

Sol.

12 किलोग्राम अनाज का मूल्य = ₹ 240
1 किलोग्राम अनाज का मूल्य = ₹ 240 /12

= ₹ 20

15 किलोग्राम अनाज का मूल्य = ₹ 20 /15

अनाज का मूल्य = ₹ 300

3. एक प्रेश 3 दिन में 1200 अखबार छापता है। तो 7 दिन में वह कितने अखबार छापेगा?

4. कोलकाता से नई दिल्ली का 4 व्यक्तियों का रेल भाड़ा ₹ 900 है। 15 व्यक्ति का किराया कितना होगा?

5. ₹ 108 प्रति दर्जन की दर से 11 कापियों का मूल्य कितना होगा?

6. एक प्रेश 12 दिन में 1200 पुस्तक छापती है। तो 30 दिन में वह कितनी पुस्तक छापेगी ?

7. ₹ 8 प्रतियों का दाम ₹ 200 है तो 20 प्रतियों की दाम कितना होगा?

8. 5 पेन्सिल का मूल्य 35 रुपये हो तो 12 पेन्सिल का मूल्य ज्ञात कीजिये।

9. 10 पेनों का मूल्य ₹ 30 है तो 18 पेनों का मूल्य कितना होगा?

10. 15 किग्रा तेल का मूल्य ₹ 315 है, 16 किग्रा तेल का मूल्य कितना होगा?

11. 15 पैकिटों में 600 बिस्कुट हैं। 13 पैकिटों में कितने बिस्कुट होंगे?

12. एक फैक्टरी में 3 दिन में 750 पेंच तैयार होते हैं। 17 दिन में कितने पेंच तैयार होंगे?

13. एक मशीन 7 मिनट में 70 खिलौने बनाती है। 17 मिनट में कितने खिलौने बनायेगी?

14. एक नौकर 1 सप्ताह में ₹ 4200 की मजदूरी करता है। 15 दिन में उसे कितने ₹ मिलेंगे?

15. 30 शॉल का मूल्य ₹ 15000 है तो 19 शाॅलों का मूल्य कितना होगा?

Exercise 08

1. If the cost of 8 kg of sugar is Rs 240, then find the cost of 11 kg of sugar.

Sol.

Price of 8 kg sugar = ₹ 240

Price of 1 kg sugar = ₹ 240 /8

= ₹ 30

Price of 11 kg sugar = ₹ 30 × 11

= ₹ 330

Price of sugar = ₹ 330

2. If the cost of 12 kg of grain is Rs.240, then find the cost of 15 kg of grain.

Sol.

Price of 12 kg grain = ₹ 240

Price of 1 kg grain = ₹ 240 /12

= ₹ 20

Price of 15 kg grain = ₹ 20 /15

Price of grain = ₹ 300

3. A press prints 1200 newspapers in 3 days. So how many newspapers will he print in 7 days?

4. The train fare of 4 persons from Kolkata to New Delhi is ₹ 900. How much will be the fare for 15 persons?

5. What will be the cost of 11 notebooks at the rate of ₹ 108 per dozen?

6. A press prints 1200 books in 12 days. Then how many books will she print in 30 days?

7. If the cost of 8 copies is Rs 200, then what will be the cost of 20 copies?

8. If the cost of 5 pencils is Rs 35, then find the cost of 12 pencils.

9. If the cost of 10 pens is ₹ 30, then what will be the cost of 18 pens?

10. The cost of 15 kg of oil is ₹ 315, what will be the cost of 16 kg of oil?

11. There are 600 biscuits in 15 packets. How many biscuits will be there in 13 packets?

12. In a factory, 750 screws are manufactured in 3 days. How many screws will be ready in 17 days?

13. A machine makes 70 toys in 7 minutes. How many toys will she make in 17 minutes?

14. A servant earns ₹ 4200 in a week. How much Rs will he get in 15 days?

15. If the cost of 30 shawls is ₹ 15000, then what will be the cost of 19 shawls?

Exercise 9

1. If 12 book costs ₹144, find the cost of 5 books,

2. If 15 pens cost 230, find the cost of 23 pens.

3. In a hostel, 80 students consume 1440 kg of rice per month. What quantity of rice will be consumed by 100 students in a month?

3. If Ram earns ₹750 in 3 days, how much money will he earn in 5 days?

The cost of 5 chocolates is 225. What is the cost of 9 such chocolates?

4. If 12 boxes of apples weigh 50 kg, what will be the weight of 8 boxes?

5. 6 books cost ₹900. What is the cost of 19 books?

The cost of 5 pair of shoes is 4250 what is the cost of 15 pair of shoes.

Cost of a desert pulse is rupees 72 what is the cost of a score (20 apples) of apples.

The cost of a dozen books is 1920 what is the cost of 10 such books

A bag contains 7 clothes of rice how many bags will be required to pack 240 kg of rice.

Ram reads 42 phases of a group in a day how long will she take to finish the book that has 1428 pages?

2. A train covers 882 km in 9 hours. What distance will it cover in 6 hours?

5. A loaded truck travels 18 km in 25 minutes. How far will it travel in 5 hours?

Exercise 10

14: Cost of 105 envelopes is₹ 350. How many envelopes can be ₹100?

11: If the cost of 6 cans of juice is 210, then what will be the cost of 14 cans of juice?

12: A motorbike travels 220 km in 5 litres of petrol. How much distance will it cover in 1.5 litres of petrol?

13: If the cost of a dozen soaps is ₹ 153.60, what will be the cost of 15 such soaps?

1. If the cost of 7 m of cloth is 1470, find the cost of 5 m of cloth.

2. Ekta earns 3000 in 10 days. How much will she earn in 30 days?

3. If it has rained 276 mm in the last 3 days, how many cm of rain will fall in one full week (7 days)? Assume that the rain continues to fall at the same rate.

5. The temperature dropped 15 degree celsius in the last 30 days. If the rate of temperature drop remains the same, how many degrees will the temperature drop in the next ten days?

6. Shaina pays 15000 as rent for 3 months. How much does she has to pay for a whole year, if the rent per month remains same?

7. Cost of 4 dozen bananas is 180. How many bananas can be purchased for ₹90?

8. The weight of 72 books is 9 kg. What is the weight of 40 such books?

9. A truck requires 108 litres of diesel for covering a distance of 594 km. How much diesel will be required by the truck to cover a distance of 1650 km?

11. Anish made 42 runs in 6 overs and Anup made 63 runs in 7 overs. Who made more runs per over?

10. Raju purchases 10 pens for 150 and Manish buys 7 pens for₹ 84. Can you say who got the pens cheaper?

4. Cost of 5 kg of wheat is 91.50.

(a) What will be the cost of 8 kg of wheat?

(b) What quantity of wheat can be purchased in 183?

15: A car travels 90 km in 2½ hours.

(a) How much time is required to cover 30 km with the same speed?

(b) Find the distance covered in 2 hours with the same speed.

If 15 chairs cost ₹2400, then find the cost of 70 chairs. [11200]

4. The cost of 1 dozen bananas is 24. How much do 50 bananas cost?

5. Sujata drives her car at a constant speed. If she travels 25 km in 30 minutes, how long will she take to travel 120 km?

6. On a map, 600 km between two points A and B is represented by 5 cm. What is the actual distance between points C and D if they are 8 cm apart on the map?

7. The ratio of the breadth to the length of a lawn is 2:3. If the breadth of the lawn is 32 m, find its length.

8. Critical Thinking Kishi has a recipe for making cookies that requires 2 cups of flour and 1 cup of sugar. If she wants to make twice as many cookies, how would she adjust the amounts of flour and sugar while maintaining the same ratio?

9. Critical Thinking In a village, the sex ratio is reported as 105 males for every 100 females. If the total population of the village is 5125, how many males and females are there in the village?

14: If a car travels 180 km in 3 hours, how far will it travel in 10 minutes?

If a car covers 276 km in 12 L of petrol, find the distance it will cover in 20 L.

DIRECT AND INVERSE PROPORTIONS

Points to Remember

In direct proportion, two quantities 'x' and 'y' increase or decrease together, the ratio of their corresponding values remains constant.

i.e.k

ork, where k is a positive number.

e.g. Ramesh covers a distance of 4 kilometres in 2 hours and with same speed he can cover 8 kilometres in 4 hours. We can say that when time is increased, distance also increases. So, it is a case of direct proportion.

In inverse proportion, increase in a quantity 'x', causes a proportional decrease in quantity 'y' or increase in a quantity 'y', causes a proportional decrease in quantity

'x', the product of their corresponding values remains constant.

ie. xy=x2 y2=k

ork, where k is a positive number.

e.g. Four men complete a construction work in 10 days and 5 men complete the same work in 8 days. When the number of workers is increased, the number of days to complete the same work decreases. So, it is a case of inverse proportion.

COMPARING QUANTITIES

pints to remember:

When two quantities of same kind with same units are compared by division, ratio of two quantities is obtained.

A ratio has no unit.

A ratio is said to be in simplest form, if its two terms have no common factor other than one.

Comparison of two ratios is done by making them fractions with equal denominators and these ratios are called equivalent or proportional.

Another method of comparison is percentage.

Any simple fraction, decimal fraction or ratio can be converted into percentage and any percentage can be converted into simple fraction, decimal or ratio.

TRICKS

A number which, when added to terms of a:b makes it equal to the cid is ad-be e-d

A number which, when subtracted from the terms of ratio a:b makes it equal to ratio c:d is be e-d

If the sum of two numbers is A and their difference is B, then the ratio of numbers is given by A+ B: A-B

Ifa:b:c: d, then a x d = bx c.

Exercise 7.2

Concept and Skills

1. Fill in the blanks to make the ratios equivalent.

(a) 15; 80 = 60

(b) 3.6% * 0.6 = 12

1. Which of the following are in proportion?

(a) 16, 4, 32, 8

(b) 3,4,2,6

2. Find the value of x in the following proportions.

(a) 7:15:x:30

(b) x:3:: 12:18

(d) 8:16:25:x

3. If the first, second, and third terms of a proportion are 10, 12, and 15 respectively, find its fourth term.

Problem Solving

4. The cost of 1 dozen bananas is 24. How much do 50 bananas cost?

5. Sujata drives her car at a constant speed. If she travels 25 km in 30 minutes, how long will she take to travel 120 km?

6. On a map, 600 km between two points A and B is represented by 5 cm. What is the actual distance between points C and D if they are 8 cm apart on the map?

7. The ratio of the breadth to the length of a lawn is 2:3. If the breadth of the lawn is 32 m, find its length.

9. Critical Thinking In a village, the sex ratio is reported as 105 males for every 100 females. If the total population of the village is 5125, how many males and females are there in the village?

Problem Sulsing

4. Sandhya attempted 15 GK quizzes, out of which she won in 9 quizzes. Find the ratio of

(a) the number of quizzes attempted to the number of quizzes lost

(b) the number of quizzes won to the number of quizzes lost.

Which is the bigger ratio?

5. Anisha is preparing bags of candies for gifting. She wants to maintain the ratio of chocolate to gummies as 3:5. She has put 12 chocolates in the bag. How many gummies should she put to keep the required ratio?

6. Critical Thinking The ratio of apples to oranges sold on Monday was 2:3 and on Tuesday was 3:4. The total number of fruits sold on Monday was 60 and on Tuesday was 84. Which day had a higher ratio of apples to oranges?

Concept and Skills

1. Divide the given quantities into the mentioned ratios.

(a) 110 km in the ratio 2:3

(b) ₹1500 in the ratio 7:8

Problem Solving

2. A 1500 cm rope is to be divided into a ratio of 1:3. Find the length of each part.

3. Divide 1200 between Ram and Shyam in the ratio 7:5.

4. If the angles of a triangle are in the ratio 4:4:2, find the value of each angle.

5. Sumedha splits her collection of 450 beads between her and her friends, Nidhi and Megha in the ratio 2:3:4. Find the number of beads with each of them.

6. A container has 60 L of a mixture of milk and water in the ratio 3:2. Find the quantity of milk

and water.

7. Critical Thinking A company makes a protein powder by mixing soy, pea, and whey proteins in the ratio 2:3:5. They prepare a sample of 50 g for testing. Find the quantity of each ingredient in

the sample.

After the feedback, they modify the mixture so that the new ratio becomes 4:6:9, How much of each ingredient needs to be added to the original mixture to achieve the new ratio?

UNITARY METHOD

Suppose you bought 2 kg mangoes for ₹160. But your mom called and asked you to buy 5 kg mangoes instead. So, how much will you pay for 5 kg mangoes?

One way is to first find the cost of 1 kg mangoes and then use it to find the cost of 5 kg mangoes.

127

Cost of 2 kg mangoes 160

Cost of 1 kg mangoes₹1602-780

Cost of 5 kg mangoes₹80×5400

The above method of finding the value of one unit first and then finding the value of required units is called the Unitary Method.

The unitary method is based on proportionality. If cost is proportional to weight, then:

Cost of n kgnx Cost of 1 kg

Example 12: If a car covers 276 km in 12 L of petrol, find the distance it will cover in 20 L.

Solution: Distance covered in 12 L of petrol = 276 km Distance covered in 1 L of petrol = 276-23 km Distance that will be covered in 20 L=23 x 20 = 460 km Hence, the car covers 460 km in 20 L of petrol.

Example 13: A travel agent charges 30,000 to make travel arrangements for 25 people. How much will the same arrangement cost for 40 people? Solution: Travel cost for 25 people ₹30,000 Travel cost for 1 person = ₹30000 ₹1200 25

Hence, the travel cost for 40 people = (1200×40)=48,000.

Role of Unit Conversion

Sometimes, problems involve different units-like minutes and hours, grams and kilograms, centimetres and metres. In such problems, unit conversion becomes very important.

Solution: We know that 3 hours=3x60 180 minutes.

So, the car travels 180 km in 180 minutes, distance travelled in 1 minute = 180 = 1 km

180 distance travelled in 10 minutes = 1 x 10 = 10 km

Note: Without converting hours into minutes, the calculation would have gone wrong.

The steps for solving problems using the unitary method are as follows:

Find the value of one unit of the item by division.

Find the value of the required number of items by multiplying it with the value of one unit.

Always convert all the values in the same units.

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RATIO AND PROPORTION

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