Ratio and Proportion

Here’s a summary of the Ratio and Proportion chapter in 6th-grade math:

Introduction:

• Ratio is a way to compare two quantities of the same kind.
• It expresses the relationship between these quantities.

Representing Ratios:

• Ratios can be written in a few ways:
  • a:b (a to b) – This reads as “a is to b”.
  • a/b (fraction form)
  • a:b = c:d (proportion) – This indicates that the ratio of a to b is equal to the ratio of c to d.

Simplifying Ratios:

• You can simplify ratios by dividing both the numerator and denominator by the greatest common factor (GCD) of the two numbers.

Types of Ratios:

• Equivalent Ratios: Ratios that represent the same comparison and can be obtained by multiplying or dividing the terms of one ratio by the same non-zero number.
• Like Ratios: Ratios comparing quantities of the same kind (e.g., lengths, weights).
• Unlike Ratios: Ratios comparing quantities of different kinds (e.g., length and time).

Rates:

• A rate is a special type of ratio that expresses a change in one quantity relative to another quantity (e.g., price per unit, speed).

Proportion:

• A proportion is a statement that two ratios are equal.
• It can be written as a:b = c:d.

Solving Proportions:

• Cross multiplication is a technique used to solve proportions. It involves multiplying the terms diagonally such that the product of the means (middle terms) equals the product of the extremes (outer terms).

Applications of Ratio and Proportion:

• Ratios and proportions are used in various real-world problems involving scaling, mixing solutions, dividing quantities in a specific ratio, and similar situations.

Key Points:

• Understanding ratios and proportions helps solve problems involving comparisons and proportional relationships.
• Simplifying ratios helps in easier comparison and calculations.
• Cross multiplication is a valuable tool for solving proportion problems.

Learning Objectives:

By studying this chapter, students should be able to:

• Express ratios in different ways.
• Simplify ratios.
• Identify equivalent and like ratios.
• Understand and apply rates.
• Write and solve proportions using cross multiplication.
• Apply ratios and proportions to solve real-world problems.

Exercise 12.1

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

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

(b) What is the ratio of the number of girls to the number of students in the class?

Ans : 

(a) Ratio of girls to boys:

The ratio of girls to boys can be expressed in a few ways:

• 20:15 (girls to boys)
• 20/15 (fraction form) – This simplifies to 4/3

(b) Ratio of girls to total students:

To find the ratio of girls to the total number of students, we need to consider the total number of students:

• Total students = Girls + Boys = 20 girls + 15 boys = 35 students

Therefore, the ratio of girls to total students can be expressed as:

• 20:35 (girls to total students)
• 20/35 (fraction form) – This simplifies to 4/7

2. 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 the number of students liking tennis.

(b) Number of students liking cricket to total number of students.

Ans : 

Number of students in the class = 30
Number of students liking football = 6
Number of students liking cricket = 12
Number of students liking tennis = 30 – (6 + 12) = 30 – 18 = 12

(a) Ratio of the number of the students liking football to the number of students liking tennis

Thus, the required ratio is 1 : 2.

(b) Ratio of the number of students liking cricket to the total number of students

Thus, the required ratio is 2 : 5.

3. See the figure and find the ratio of

(а) Number of triangles to the number of circles inside the rectangle.

(b) Number of squares to all the figures inside the rectangle.

(c) Number of circles to all the figures inside the rectangle.

Ans : 

(a) Number of triangles 3
Number of circles = 2
∴ Ratio of number of triangles to the number of circles

Thus, the required ratio is 3 : 2.

(b) Number of squares = 2
Number of all figures = 7
∴ Ratio of number of squares to the number of all the figures

Thus, the required ratio is 2 : 7.

(c) Ratio of number of circles to the number of all the figures

Thus, the required ratio is 2 : 7.

4. Distances travelled by Hamid and Akhtar in an hour are 9 km and 12 km. Find the ratio of speed of Hamid to the speed of Akhtar.

Ans : 

Speed of Hamid = 9 km / 1 hour = 9 km/h

Speed of Akhtar = 12 km / 1 hour = 12 km/h

Ratio of speeds = Speed of Hamid / Speed of Akhtar

Ratio of speeds = 9 km/h / 12 km/h

Ratio of speeds = 3 km/h / 4 km/h = 3:4

5. Fill in the following blanks:

[Are these equivalent ratios?]

Ans : 

Now the fractions, we have

6. Find the ratio of the following:

(a) 81 to 108

(b) 98 to 63

(c) 33 km to 121 km

(d) 30 minutes to 45 minutes

Ans : 

(a) 81 to 108:

The greatest common factor (GCD) of 81 and 108 is 27. Dividing both numbers by 27 gives:

• 81 / 27 = 3
• 108 / 27 = 4

Therefore, the ratio of 81 to 108 is 3:4.

(b) 98 to 63:

The GCD of 98 and 63 is 7. Dividing both numbers by 7 gives:

• 98 / 7 = 14
• 63 / 7 = 9

Therefore, the ratio of 98 to 63 is 14:9.

(c) 33 km to 121 km:

Since kilometers (km) are the same unit on both sides, we can directly find the ratio:

• Ratio = 33 km / 121 km = 3:11

(d) 30 minutes to 45 minutes:

Minutes (min) are the same unit on both sides, so the ratio is:

• Ratio = 30 minutes / 45 minutes = 2:3

7. Find the ratio of the following:

(a) 30 minutes to 1.5 hours

(b) 40 cm to 1.5 m

(c) 55 paise to ₹ 1

(d) 500 mL to 2 litres

Ans : 

(a) 30 minutes to 1.5 hours:

• There are 60 minutes in 1 hour.
• To compare minutes to hours, we need to convert either minutes to hours or hours to minutes.

We can convert 1.5 hours to minutes:

• 1.5 hours * 60 minutes/hour = 90 minutes

Therefore, the ratio of 30 minutes to 1.5 hours is 30 minutes : 90 minutes (or you can simplify it to 1:3 if units are not required).

(b) 40 cm to 1.5 m:

• We can directly compare these if both are in the same unit (meters).
• Convert centimeters (cm) to meters (m):
• 40 cm / 100 cm/m = 0.4 m

Therefore, the ratio of 40 cm to 1.5 m is 0.4 m : 1.5 m (or you can simplify it to 2:7 if units are not required).

(c) 55 paise to ₹ 1:

• Paise (p) is a subunit of the Indian Rupee (₹).
• 1 rupee is equal to 100 paise.

Therefore, the ratio of 55 paise to ₹ 1 is:

• 55 paise / (1 rupee * 100 paise/rupee) = 55/100

We can simplify this to 11/20. So, the ratio is 55 paise : ₹ 1 (or 11:20 if units are not required).

(d) 500 mL to 2 litres:

• Liters (L) and milliliters (mL) are units of volume.
• 1 liter is equal to 1000 milliliters.

Therefore, the ratio of 500 mL to 2 liters is:

• 500 mL / (2 liters * 1000 mL/liter) = 500/2000

We can simplify this to 1/4. So, the ratio is 500 mL : 2 L (or 1:4 if units are not required).

8. In a year, Seema earns ₹ 1,50,000 and saves ₹ 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.

Ans : 

a) Money earned by Seema = ₹ 1,50,000
Money saved by Seema= ₹ 50,000
∴ Money spent by Seema= ₹ 1,50,000 – ₹ 50,000 = ₹ 1,00,000
∴ Ratio of money earned by Seema to the money saved by her

(b) Ratio of money saved by Seema to the money

9. There are 102 teachers in a school of 3300 students. Find the ratio of the number of teachers to the number of students.

Ans : 

Ratio of teachers to students = 102 teachers : 3300 students

Ratio of teachers to students = 102 teachers / 3300 students

102 teachers / 3300 students = (102 / 6) teachers / (3300 / 6) students 

= 17 teachers : 550 students

10. In a college, out of 4320 students, 2300 are girls, find the ratio of

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

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

(c) Number of boys to the total number of students.

Ans : 

(a) Ratio of girls to total students:

There are 2300 girls and 4320 total students. The ratio can be expressed in two ways:

• 2300 girls : 4320 students (actual number of girls to total students)
• 115 girls : 216 students (simplified ratio)

We can simplify the ratio by finding the greatest common factor (GCD) of 2300 and 4320, which is 20. Divide both sides by 20:

• 2300 girls / 20 : 4320 students / 20 = 115 girls : 216 students

(b) Ratio of boys to girls:

There are 2300 girls (already given) and 4320 total students (given). The number of boys can be found by subtracting the number of girls from the total number of students:

• Number of boys = Total students – Number of girls
• Number of boys = 4320 students – 2300 girls = 2020 boys

Therefore, the ratio of boys to girls is:

• 2020 boys : 2300 girls

(c) Ratio of boys to total students:

We already found the number of boys (2020) and the total number of students (4320). The ratio can be expressed as:

• 2020 boys : 4320 students

11. 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

(а) 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 basketball.

(c) Number of students who opted basketball to the total number of students.

Ans : 

(a) Ratio of Basketball to Table Tennis:

1. Find the number of students who opted for table tennis:

We know 1800 students are total, 750 opted for basketball, and 800 opted for cricket. Since students can choose only one game, the remaining opted for table tennis.

• Number of table tennis players = Total students – Basketball players – Cricket players
• Number of table tennis players = 1800 students – 750 students – 800 students = 250 students

2. Calculate the ratio:

• Ratio of Basketball to Table Tennis = Number of Basketball players / Number of Table Tennis players
• Ratio of Basketball to Table Tennis = 750 students / 250 students = 3:1

(b) Ratio of Cricket to Basketball:

The ratio of Cricket to Basketball can be expressed as:

• Ratio of Cricket to Basketball = Number of Cricket players / Number of Basketball players
• Ratio of Cricket to Basketball = 800 students / 750 students = 16:15

(c) Ratio of Basketball to Total Students:

The ratio of Basketball to Total Students can be expressed as:

• Ratio of Basketball to Total Students = Number of Basketball players / Total Students
• Ratio of Basketball to Total Students = 750 students / 800 students = 5:12 

12 . Cost of a dozen pens is ₹180 and cost of 8 ball pens is ₹56. Find the ratio of the cost of a pen to the cost of a ball pen.

Ans : 

1. Cost per Pen:

We are given the cost of a dozen pens (12 pens) and not the individual pen cost. To find the cost of a single pen, we can divide the total cost by the number of pens:

• Cost per Pen = Cost of a dozen pens / Number of pens in a dozen
• Cost per Pen = ₹180 / 12 pens = ₹15 per pen

2. Cost per Ball Pen:

We are directly given the cost of 8 ball pens (₹56). To find the cost of a single ball pen, we can again divide:

• Cost per Ball Pen = Cost of 8 ball pens / Number of ball pens
• Cost per Ball Pen = ₹56 / 8 ball pens = ₹7 per ball pen

3. Ratio of Cost (Pen to Ball Pen):

Now that you know the cost of each pen and ball pen, you can find the ratio:

• Ratio of Cost (Pen:Ball Pen) = Cost per Pen / Cost per Ball Pen
• Ratio of Cost (Pen:Ball Pen) = ₹15 / ₹7 = 15:7

Therefore, the ratio of the cost of a pen to the cost of a ball pen is 15:7. This means a pen costs ₹15 while a ball pen costs ₹7, and the pen is more expensive than the ball pen by a ratio of 15:7.

13. Consider the statement : Ratio of breadth and length of a hall is 2 : 5. Complete the following table that shows some possible breadths and lengths of the hall.

Ans : 

14. Divide 20 pens between Sheela and Sangeeta in the ratio of 3 : 2.

Ans : 

1. Total ratio parts: The ratio is 3:2, which means there’s a total of 3 + 2 = 5 parts.
2. Pens per part: Divide the total number of pens (20) by the total ratio parts (5): 20 pens / 5 parts = 4 pens/part.
3. Pens for Sheela: Sheela’s share is 3 parts of the ratio. So, she gets 3 parts * 4 pens/part = 12 pens.
4. Pens for Sangeeta: Sangeeta’s share is 2 parts of the ratio. So, she gets 2 parts * 4 pens/part = 8 pens.

Therefore, Sheela gets 12 pens and Sangeeta gets 8 pens.

15. Mother wants to divide ₹ 36 between her daughters Shreya and Bhoomika in the ratio of their ages. If age of Shreya is 15 years and age of Bhoomika is 12 years, find how much Shreya and Bhoomika will get?

Ans : 

Money got by Shreya : Money got by Bhoomika = 15 : 12

∴ Sum = 15 + 12 = 27

16. 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.

(c) Age of father after 10 years to the age of son after 10 years.

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

Ans : 

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

• Ratio = Father’s present age : Son’s present age
• Ratio = 42 years : 14 years
• We can simplify this by dividing both sides by 2 (their greatest common factor): 3:1

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

• Son’s age then = 12 years
• Father’s age then = Present age – Age difference (since their age difference remains constant)
• Father’s age then = 42 years – (14 years – 12 years) = 40 years
• Ratio = Father’s age then : Son’s age then
• Ratio = 40 years : 12 years
• We can simplify this by dividing both sides by 4: 10:3

(c) Age of father after 10 years to the age of son after 10 years:

• Son’s age after 10 years = Son’s present age + 10 years = 14 years + 10 years = 24 years
• Father’s age after 10 years = Father’s present age + 10 years = 42 years + 10 years = 52 years
• Ratio = Father’s age after 10 years : Son’s age after 10 years
• Ratio = 52 years : 24 years
• We can simplify this by dividing both sides by 4: 13:6

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

• Father’s age then = 30 years
• Son’s age then = Present age – Father’s age difference (since their age difference remains constant)
• Son’s age then = 42 years – (42 years – 30 years) = 30 years (This is a coincidence, but it doesn’t affect the ratio)
• Ratio = Father’s age then : Son’s age then
• Ratio = 30 years : 30 years
• This simplifies to 1:1 (since their ages were the same at that point).

Exercise 12.2

1. Determine if the following are in proportion,

(a) 15, 45, 40, 120

(b) 33, 121, 9, 96

(c) 24, 28, 36, 48

(d) 32, 48, 70, 210

(e) 4, 6, 8, 12

(f) 33, 44, 75, 100

Ans : 

(a) 15, 45, 40, 120

• Yes, these are in proportion.
• We can simplify 15:45 to 1:3 and 40:120 to 1:3.
• Since both ratios have the same proportion (1:3), they are proportional.

(b) 33, 121, 9, 96

• No, these are not in proportion.
• Simplifying 33:121 is difficult, but 9:96 reduces to 1:12.
• Since the ratios are not the same (1:3 vs 1:12), they are not proportional.

(c) 24, 28, 36, 48

• No, these are not in proportion.
• Simplifying 24:28 is 6:7, and 36:48 is 3:4.
• The ratios are not the same (6:7 vs 3:4), so they are not proportional.

(d) 32, 48, 70, 210

• No, these are not in proportion.
• We can try simplifying each ratio, but none will result in the same proportion.
• For example, 32:48 can be simplified to 8:12, but 70:210 cannot be simplified to the same ratio.
• Therefore, they are not proportional.

(e) 4, 6, 8, 12

• Yes, these are in proportion.
• Both ratios (4:6 and 8:12) simplify to 2:3.
• Since they have the same proportion, they are proportional.

(f) 33, 44, 75, 100

• Yes, these are in proportion.
• Simplifying 33:44 is 3:4, and 75:100 is 3:4.
• Both ratios have the same proportion (3:4), so they are proportional.

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

(a) 16 : 24 :: 20 : 30

(b) 21 : 6 :: 35 : 10

(c) 12 : 18 :: 28 : 12

(d) 8 : 9 :: 24 : 27

(e) 5.2 : 3.9 :: 3 : 4

(f) 0.9 : 0.36 :: 10 : 4

Ans : 

(a) 16 : 24 :: 20 : 30 – True (T)

• Both ratios simplify to 2:3. (16/24 = 2/3 and 20/30 = 2/3)

(b) 21 : 6 :: 35 : 10 – False (F)

• Simplification: 21/6 = 7/2 and 35/10 = 7/2 (They are equal, but the proportion isn’t 2:1)

(c) 12 : 18 :: 28 : 12 – False (F)

• Simplification: 12/18 = 2/3 and 28/12 = 7/3 (Ratios are not the same)

(d) 8 : 9 :: 24 : 27 – False (F)

• Simplification: 8/9 = 8/9 and 24/27 = 8/9 (Ratios are the same, but not a proportion since both terms are the same)

(e) 5.2 : 3.9 :: 3 : 4 – False (F)

• Ratios don’t have a common simplification

(f) 0.9 : 0.36 :: 10 : 4 – False (F)

• Ratios don’t have a common simplification

3. Are the following statements true?

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

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

(c) 99 kg : 45 kg = ₹ 44 : ₹ 20

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

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

Ans : 

(a) 40 persons : 200 persons

∴ Statement (a) is true.

(b) 7.5 litres : 15 litres

∴ Statement (b) is true.

∴ Statement (c) is true.

∴ Statement (d) is true.

∴ Statement (e) is not true.

4. 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 ₹ 40 : ₹ 160

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

(c) 2 kg : 80 kg and 25 g : 625 g

(d) 200 mL : 2.5 litres and ₹ 4 : ₹ 50

Ans : 

(a) 25 cm : 1 m = 25 cm : 100 cm [∵ 1 m = 100 cm]

∴ The given ratios are in proportion.
Extreme terms are 25 cm and ₹ 160.
Middle terms are 1 m and ₹40.

∴ The given ratios are in proportion.
Extreme terms are 39 litres and 10 bottles.
Middle terms are 65 litres and 6 bottles.

∴ The given ratios are not in proportion.

(d) 200 mL : 2.5 litres = 2.5 litres = 2.5 x 1000 mL = 2500 mL

∴ The given ratios are in proportion.

Extreme terms are 200 mL and ₹ 50

Middle terms are 2.5 litres and ₹ 4.

Exercise 12.3

1. If the cost of 7 m of cloth is ₹ 294, find the cost of 5 m of cloth.

Ans : 

7 meters: Costs ₹294

Unknown length (x meters): We want to find the cost for 5 meters (x = 5)

7 meters : 5 meters :: ₹294 : y

(7 meters) * y = (5 meters) * ₹294

y = (5 meters * ₹294) / 7 meters y ≈ ₹210

Therefore, 5 meters of cloth cost approximately ₹210.

2. Ekta earns ₹ 1500 in 10 days. How much she will earn in 30 days?

Ans : 

Ekta works for 10 days (initial condition).

We want to find her earnings for 30 days.

Ratio of days = 10 days : 30 days

10 days : 30 days :: ₹1500 : E

(10 days) * E = (30 days) * ₹1500

E = (30 days * ₹1500) / 10 days

E = ₹4500

Therefore, Ekta will earn ₹4500 in 30 days.

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

Ans : 

We know the total rain in 3 days is 276 mm. To find the rain per day, divide the total rain by the number of days:

Rain per day = Total rain in 3 days / Number of days 

Rain per day = 276 mm / 3 days = 92 mm/day

Since the rain rate is assumed constant, we can multiply the rain per day by the number of days in a week to find the total rain for 7 days:

Rain in a week = Rain per day * Number of days in a week 

Rain in a week = 92 mm/day * 7 days = 644 mm

4. Cost of 5 kg of wheat is ₹ 30.50.

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

(b) What quantity of wheat can be purchased in ₹ 61?

Ans : 

(a) Cost of 8 kg of wheat:

1. Cost per kg: We know the cost of 5 kg of wheat, but we need the cost per kg. To find this, we can divide the total cost by the number of kilograms:

Cost per kg = Total cost of 5 kg wheat / Number of kg Cost per kg = ₹30.50 / 5 kg = ₹6.10 per kg

2. Cost of 8 kg: Now that you know the cost per kg, you can multiply it by the desired quantity (8 kg) to find the total cost for 8 kg:

Cost of 8 kg wheat = Cost per kg * Quantity Cost of 8 kg wheat = ₹6.10 per kg * 8 kg = ₹48.80

Therefore, 8 kg of wheat will cost ₹48.80.

(b) Quantity of wheat for ₹61:

1. Calculate using cost per kg: We can again use the cost per kg (₹6.10) to find out how much wheat we can buy for ₹61.

Quantity of wheat (kg) = Total money / Cost per kg Quantity of wheat (kg) = ₹61 / ₹6.10 per kg ≈ 10 kg (approximately)

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?

Ans : 

1. Calculate the temperature drop per day:

• Total temperature drop = 15 degrees Celsius
• Number of days = 30 days
• Temperature drop per day = Total temperature drop / Number of days
• Temperature drop per day = 15 degrees Celsius / 30 days = 0.5 degrees Celsius per day

2. Find the temperature drop in 10 days:

• Number of days for which we want to predict the drop = 10 days
• Temperature drop per day = 0.5 degrees Celsius per day (as calculated earlier)
• Temperature drop in 10 days = Temperature drop per day * Number of days
• Temperature drop in 10 days = 0.5 degrees Celsius per day * 10 days = 5 degrees Celsius

Therefore, if the temperature drop remains constant, the temperature will drop by 5 degrees Celsius in the next ten days.

6. Shaina pays ₹ 7500 as rent for 3 months. How much does she has to pay for a whole year, if the rent per month remains same?

Ans : 

1. Months in a year: There are 12 months in a year.
2. Cost per month: We are given that Shaina pays ₹7500 for 3 months.
3. Total rent for a year: To find the total rent for a year, we can multiply the monthly rent by the number of months in a year:

Total rent for a year = Monthly rent * Number of months 

Total rent for a year = ₹7500 per 3 month * 12 months 

Total rent for a year = ₹30000

Therefore, Shaina has to pay ₹30,000 for a whole year’s rent if the rent remains the same each month.

7. Cost of 4 dozen bananas is ₹ 60. How many bananas can be purchased for ₹ 12.50?

Ans : 

Total cost = ₹60

Number of dozens = 4 dozen

Cost per dozen = Total cost / Number of dozens 

Cost per dozen = ₹60 / 4 dozen = ₹15 per dozen

Since 1 dozen = 12 bananas, 4 dozen translates to 4 * 12 = 48 bananas. 

We are essentially buying 48 bananas for ₹60.

This implies that 12 bananas (1 dozen) would cost ₹60 / 4 = ₹15.

Now that you know the cost per dozen (₹15), you can find the price of a single banana:

Cost per dozen = ₹15

Number of bananas in a dozen = 12

Price per banana = Cost per dozen / Number of bananas in a dozen 

Price per banana = ₹15 / 12 = ₹1.25 per banana

Number of bananas = Total money / Price per banana 

Number of bananas = ₹12.50 / ₹1.25 per banana = 10 bananas

Therefore, you can purchase 10 bananas for ₹12.50.

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

Ans : 

1. Weight per book: We don’t directly know the weight of a single book. Let’s find the weight of 1 book:
   • Total weight = 9 kg
   • Number of books = 72
2. Weight per book = Total weight / Number of books Weight per book = 9 kg / 72 books = 1/8 kg per book
3. Weight of 40 books: Now that you know the weight per book (1/8 kg), you can find the weight of 40 books:
   Number of books = 40 books Weight per book = 1/8 kg per book
   Weight of 40 books = Number of books * Weight per book Weight of 40 books = 40 books * (1/8 kg per book) = 5 kg

Therefore, the weight of 40 such books is 5 kg.

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?

Ans : 

To cover 594 km, the amount of diesel required is 108 liters.

To cover 1 km, the amount of diesel required is 108/594 liters.

To cover 1650 km, the amount of diesel required is 108×1650/liters, which equals 300 liters.

Thus, the required amount of diesel is 300 liters.

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

Ans : 

Cost per pen for Raju:

Cost of 10 pens = ₹150

Number of pens = 10

Cost per pen for Raju = Total cost / Number of pens

Cost per pen for Raju = ₹150 / 10 pens = ₹15 per pen

Cost per pen for Manish:

Cost of 7 pens = ₹84

Number of pens = 7

Cost per pen for Manish = Total cost / Number of pens

Cost per pen for Manish = ₹84 / 7 pens = ₹12 per pen

Comparison:

Cost per pen for Raju = ₹15

Cost per pen for Manish = ₹12

Since ₹12 is less than ₹15, Manish paid less per pen. Therefore, Manish got the pens cheaper.

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

Ans : 

1. Runs per over for Anish:
   • Runs scored = 42 runs
   • Overs bowled = 6 overs

Runs per over for Anish = Runs scored / Overs bowled Runs per over for Anish = 42 runs / 6 overs = 7 runs/over

2. Runs per over for Anup:
   • Runs scored = 63 runs
   • Overs bowled = 7 overs

Runs per over for Anup = Runs scored / Overs bowled Runs per over for Anup = 63 runs / 7 overs = 9 runs/over

3. Comparison:
   • Runs per over for Anish = 7 runs/over
   • Runs per over for Anup = 9 runs/over

Since 9 runs/over is more than 7 runs/over, Anup made more runs per over than Anish.