Chapter 3 Proportional Reasoning-2 (Class 8 - Latest Maths NCERT (Ganita Prakash II) Solutions)

Given:

Total time of the session = $150$ minutes.

Ratio of activities (Warm-up : Batting : Bowling : Fielding) = $3 : 4 : 3 : 5$.

To Find:

Time spent on each specific activity.

Solution:

First, we find the total number of parts in the given ratio.

Now, we find the value of one part by dividing the total time by the total parts.

By multiplying the value of one part by each ratio term, we get the time for each activity:

1. Warm-up/Cool-down: $3 \times 10 = 30 \text{ minutes}$.

2. Batting: $4 \times 10 = 40 \text{ minutes}$.

3. Bowling: $3 \times 10 = 30 \text{ minutes}$.

4. Fielding: $5 \times 10 = 50 \text{ minutes}$.

Indian Perspective:

In India, cricket is more than just a sport; it is a passion. Professional training academies, such as those run by the BCCI, emphasize systematic practice schedules. This specific ratio ensures that a player is physically ready (Warm-up), develops core skills (Batting/Bowling), and maintains defensive agility (Fielding), which is vital for success in matches at the Gully, District, or National levels.

Given:

Ratio of books (Odiya : Hindi : English) = $3 : 2 : 1$.

Total Odiya books = $288$.

To Find:

The number of Hindi books and English books.

Solution:

Let the common multiplier for the ratio be $x$.

We are given that $3x = 288$. To find $x$:

Now, we can find the number of Hindi and English books:

Summary Table:

Indian Perspective:

India is a land of great linguistic diversity. In states like Odisha, school libraries promote the regional language (Odiya) while also providing resources for the national language (Hindi) and the global language (English). This three-language formula helps students become multilingual, which is a significant advantage in the Indian workforce.

Given:

Total number of coins = $100$.

Ratio of coin counts ($\textsf{₹} 10 : \textsf{₹} 5 : \textsf{₹} 2 : \textsf{₹} 1$) = $4 : 3 : 2 : 1$.

To Find:

Total value of the money in $\textsf{₹}$.

Solution:

Step 1: Find the number of coins of each denomination.

Total parts = $4 + 3 + 2 + 1 = 10$.

Number of coins per part = $\frac{100}{10} = 10$.

1. No. of $\textsf{₹} 10$ coins = $4 \times 10 = 40$.

2. No. of $\textsf{₹} 5$ coins = $3 \times 10 = 30$.

3. No. of $\textsf{₹} 2$ coins = $2 \times 10 = 20$.

4. No. of $\textsf{₹} 1$ coins = $1 \times 10 = 10$.

Step 2: Find the total value of the money.

Value from $\textsf{₹} 10$ coins = $40 \times 10 = \textsf{₹} 400$.

Value from $\textsf{₹} 5$ coins = $30 \times 5 = \textsf{₹} 150$.

Value from $\textsf{₹} 2$ coins = $20 \times 2 = \textsf{₹} 40$.

Value from $\textsf{₹} 1$ coins = $10 \times 1 = \textsf{₹} 10$.

Step 3: Summing up the total amount.

Total Money = $\textsf{₹} 400 + \textsf{₹} 150 + \textsf{₹} 40 + \textsf{₹} 10$.

$\text{Total Money} = \textsf{₹} 600$

Indian Perspective:

In Indian local markets (Bazaars), coins are widely used for daily transactions. The $\textsf{₹} 10$ coin is the highest denomination of coin currently in circulation in India. Managing small change (Chutta) is a common skill developed by Indian students through such practical arithmetic problems.

Given:

The ratio of the sidelengths of a triangle is $3 : 4 : 5$.

To Find:

Whether all triangles drawn with this ratio are congruent and the reason behind it.

Construction Required:

1. To draw one such triangle, let the sides be $3$ cm, $4$ cm, and $5$ cm.

2. Draw a line segment $BC = 4$ cm.

3. Using a compass, draw an arc of radius $3$ cm from point $B$.

4. Draw another arc of radius $5$ cm from point $C$ to intersect the previous arc at point $A$.

5. Join $AB$ and $AC$ to complete the triangle.

Solution:

No, all triangles drawn with the ratio $3 : 4 : 5$ will not be congruent to each other.

1. Similarity vs. Congruency: Triangles with the same ratio of sides are similar because their corresponding angles are equal. However, for triangles to be congruent, they must have the exact same size and shape (identical sidelengths).

2. Scaling: We can have multiple triangles with this ratio by using a constant multiplier $k$:

If $k = 1$, sides are $(3, 4, 5)$.

If $k = 2$, sides are $(6, 8, 10)$.

If $k = 10$, sides are $(30, 40, 50)$.

While all these triangles are right-angled, their areas and perimeters differ, so they cannot be congruent.

Indian Perspective:

The ratio $3 : 4 : 5$ is the most famous Baudhāyana Triple (Pythagorean Triple). In the ancient Sulba Sutras, Indian mathematicians used this specific ratio for the construction of rectangular sacrificial altars (Vedis). They understood that while the size of the altar could change (scaling), the "right-angled" property remained constant as long as the ratio was maintained.

Given:

The ratio of the sidelengths is $1 : 3 : 5$.

To Find:

Whether it is possible to construct such a triangle.

Solution:

No, it is impossible to construct a triangle with sidelengths in the ratio $1 : 3 : 5$.

1. Triangle Inequality Theorem: According to this fundamental geometric rule, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

2. Testing the Ratio: Let the side lengths be $x$, $3x$, and $5x$, where $x$ is a positive multiplier.

Sum of the two smaller sides = $x + 3x = 4x$.

Length of the third (longest) side = $5x$.

Now, we compare the sum with the third side:

Since the sum of the two shorter sides is less than the third side, the two sides will never meet to form a vertex; they will lie flat along the longest side without closing the figure.

Indian Perspective:

This principle is a core part of the NCERT and State Board mathematics curriculum in India. Students are taught early on that geometry is governed by strict logical constraints. For example, in civil engineering and architecture projects across India, ensuring these inequalities is basic to structural stability before any physical construction begins.

Given:

Total number of people = $360$

Number of people who like Summer = $90$

Number of people who like Rainy season = $120$

To Find:

1. Number of people who like Winter.

2. Central angles for each season to construct a pie chart.

Solution:

Step 1: Calculate the number of people who like Winter.

Step 2: Calculate Central Angles for the Pie Chart.

The central angle for a component is calculated as: $\text{Central Angle} = \frac{\text{Component Value}}{\text{Total Value}} \times 360^\circ$

Indian Perspective:

In India, these three seasons are traditionally known as Grishma (Summer), Varsha (Rainy), and Shishir/Hemant (Winter). The Rainy season is particularly vital for the Indian economy as it brings the monsoon, which is essential for our farmers and agriculture.

Given:

Percentage of viewers for different channels:

Entertainment = $50\%$

Sports = $25\%$

News = $15\%$

Information = $10\%$

To Find:

Central angles for each channel type to draw the pie chart.

Solution:

To find the central angle from a percentage, we use: $\text{Angle} = \frac{\text{Percentage}}{100} \times 360^\circ$

Construction Detail:

1. Draw a circle of any convenient radius.

2. Draw a horizontal radius.

3. Use a protractor to mark $180^\circ$ for Entertainment, then $90^\circ$ for Sports, and so on.

Solution (Sample Survey):

Let us consider a typical Indian classroom of $60$ students and conduct a sample survey to fill the table.

Observation:

In this sample data, Maths is the most popular subject with a central angle of $90^\circ$. Subjects like Science and Social Science are also highly preferred, which reflects the academic focus often seen in Indian schools affiliated with boards like CBSE or ICSE.

To Find: Identify which of the given data tables represent an inverse proportion.

Solution:

In the Indian mathematical context, two quantities $x$ and $y$ are said to be in inverse proportion if their product remains constant. That is:

Let us check the products for each case:

Case (i):

$40 \times 20 = 800$

$80 \times 10 = 800$

$25 \times 32 = 800$

$16 \times 50 = 800$

Since the product $x \times y$ is constant ($800$) for all pairs, table (i) is in inverse proportion.

Case (ii):

$40 \times 20 = 800$

$80 \times 10 = 800$

$25 \times 12.5 = 312.5$

Since the product is not constant ($800 \neq 312.5$), table (ii) is not in inverse proportion.

Case (iii):

$30 \times 15 = 450$

$90 \times 5 = 450$

$150 \times 3 = 450$

$10 \times 45 = 450$

Since the product $x \times y$ is constant ($450$) for all pairs, table (iii) is in inverse proportion.

Conclusion:

The tables in (i) and (iii) represent inverse proportions.

Given: The quantities $x$ and $y$ are in inverse proportion.

Solution:

For an inverse proportion, the product of the corresponding values must be a constant $k$.

From the first column, we have $x = 16$ and $y = 9$.

We will use this constant $k = 144$ to find the missing values in the other cells.

Step 1: Finding $y$ when $x = 12$

Step 2: Finding $x$ when $y = 48$

Step 3: Finding $y$ when $x = 36$

Final Table:

The filled values are highlighted in bold in the table above.

Solution:

Two quantities are in inverse proportion if an increase in one leads to a proportional decrease in the other, and vice versa.

(i) Inverse Proportion: Increasing the number of taps will decrease the time required to fill the tank.

(ii) Inverse Proportion: Hiring more painters will reduce the number of days needed to complete the work.

(iii) Direct Proportion: The more petrol there is in the tank, the more distance the car can cover.

(iv) Inverse Proportion: If the cyclist increases their speed, the time taken to cover the fixed route will decrease.

(v) Direct Proportion: Buying more cloth results in paying a higher total price.

(vi) Direct Proportion: A book with more pages will take more time to read at a constant speed.

Indian Perspective:

In Indian construction sites or daily household chores, we often see inverse proportion in action. For example, when more laborers are engaged for a Government project like building a local road, the work gets completed much faster. Similarly, in an Indian kitchen, if you use two burners instead of one to boil water, the time taken is significantly reduced.

Given:

Cost of $24$ pencils = $\textsf{₹} 120$

To Find:

Cost of $20$ pencils.

Solution:

This is a case of Direct Proportion because if the number of pencils decreases, the cost will also decrease.

Step 1: Find the cost of one pencil (Unitary Method).

Step 2: Find the cost of 20 pencils.

Alternate Solution:

Using the ratio method: $\frac{x_1}{y_1} = \frac{x_2}{y_2}$

$\frac{24}{120} = \frac{20}{y_2}$

$y_2 = \frac{20 \times 120}{24} = 20 \times 5 = \textsf{₹} 100$.

Indian Perspective:

When buying stationery for school students in local Indian stationery shops, shopkeepers often give a slight discount for bulk purchases, but mathematically, the base price follows a direct proportion logic.

Given:

Initial number of families ($x_1$) = $20$

Initial number of days ($y_1$) = $6$

New number of families ($x_2$) = $20 + 10 = 30$

To Find:

Number of days the water will last for $30$ families ($y_2$).

Solution:

This is a case of Inverse Proportion because as the number of families increases, the water will be consumed faster, and thus it will last for fewer days.

In inverse proportion, $x_1 y_1 = x_2 y_2$.

Now, solve for $y_2$:

Assumptions:

1. Each family consumes the same amount of water per day.

2. The daily consumption rate per family remains constant throughout the period.

3. There is no additional water being added to the tank during these days.

Indian Perspective:

In many Indian housing societies (Apartments), water management is a critical task for the Resident Welfare Association (RWA). Understanding inverse proportion helps in calculating the capacity needed for overhead tanks or planning water tanker requirements during peak summers when consumption increases.

Given:

A list of average sleep hours: $15, 2.5, 20, 8, 3.5, 13, 10.5, 18$.

A set of pie charts representing a 24-hour day where the shaded (purple) part represents the sleeping time.

Solution:

To find the hours, we estimate the fraction of the circle shaded and multiply by 24 hours. In the Indian perspective, understanding these cycles helps in appreciating biodiversity and the biological clocks (circadian rhythms) of different species.

Observation: We can see that the Giraffe sleeps the least while the Bat sleeps the most among the given list. This proportional reasoning is similar to how Indian school students are taught to interpret data in science and geography.

Given:

Total angle of a pie chart = $360^\circ$

Angle for Bus = $120^\circ$

Angle for Walk = $90^\circ$ (indicated by the right-angle symbol)

Angle for Cycle = $60^\circ$

Angle for Two-wheeler = $60^\circ$

To Find:

(i) Most common mode.

(ii) Fraction for Car.

(iii) Total number of children if 18 travel by car, and number of taxi users.

(iv) Two modes with equal number of children.

Solution:

First, let us calculate the central angle for the 'Car' sector.

(i) Most common mode of transport

The mode with the largest central angle is the most common.

Central angle for Bus = $120^\circ$.

Therefore, Bus is the most common mode of transport.

(ii) Fraction of children travelling by car

The fraction is $\frac{1}{12}$.

(iii) Survey total and Taxi users

Let the total number of children be $x$.

Since the pie chart does not show a sector for "Taxi", the number of children using taxis is $0$.

(iv) Modes with equal numbers of children

Modes with identical central angles will have the same number of children.

Angle for Cycle = $60^\circ$

Angle for Two-wheeler = $60^\circ$

Therefore, Cycle and Two-wheeler have equal numbers of children.

Indian Perspective:

In Indian cities like Delhi, Mumbai, or Bengaluru, school buses (yellow buses) are the most common mode of organized transport. Walking and cycling are also prevalent in many suburban and rural areas, reflecting the diverse socio-economic fabric of Indian schools.

Given:

Initial number of workers ($x_1$) = $3$

Time taken to paint the fence ($y_1$) = $4$ days

Final number of workers ($x_2$) = $3 + 1 = 4$

To Find:

Number of days required by $4$ workers ($y_2$).

Solution:

This is a case of Inverse Proportion. As the number of workers increases, the time required to complete the same task decreases.

In inverse proportion, the product of the variables is constant ($x_1 y_1 = x_2 y_2$).

Assumptions:

1. Each worker works at the same constant speed or efficiency.

2. The workers do not hinder each other's work when more people join the team.

3. The size of the fence and the environmental conditions remain the same.

Indian Perspective:

In India, daily wage labor (Mazdoori) is a common way to execute domestic or public works. Contractors often use this logic to speed up projects like house painting before festivals like Diwali by hiring more hands to ensure the work is done in fewer days.

Given:

Number of tanks ($x_1$) = $2$

Time taken ($y_1$) = $6$ hours

Target number of tanks ($x_2$) = $5$

To Find:

Time required to fill $5$ tanks ($y_2$).

Solution:

This is a case of Direct Proportion. If we increase the number of tanks to be filled, the time taken by the pump will also increase proportionally.

In direct proportion, the ratio remains constant ($\frac{y_1}{x_1} = \frac{y_2}{x_2}$).

Alternate Solution (Unitary Method):

Time taken for $1$ tank = $\frac{6}{2} = 3 \text{ hours}$.

Time taken for $5$ tanks = $5 \times 3 = 15 \text{ hours}$.

Indian Perspective:

This is a very practical problem for Indian farmers who use diesel or electric pumps to irrigate their fields. They often calculate the time needed to fill overhead water storage tanks or village ponds based on the discharge capacity of their pump sets.

Given:

Initial rows ($R_1$) = $25$

Chairs per row ($C_1$) = $12$

New chairs per row ($C_2$) = $20$

Solution:

Since the total number of chairs is fixed, the number of rows and the number of chairs per row are in Inverse Proportion.

Total number of chairs = $25 \times 12 = 300$.

Let the new number of rows be $R_2$.

Indian Perspective:

Organizing seating for a large gathering like a school assembly, a wedding function (Shaadi), or a community event in a Panchayat Bhawan involves such rearrangements to fit the audience into the available space.

Given:

Number of periods ($n_1$) = $8$

Duration of each period ($t_1$) = $45$ minutes

New number of periods ($n_2$) = $9$

To Find:

New duration of each period ($t_2$).

Solution:

The total school time remains constant. Therefore, the number of periods and the duration of each period are in Inverse Proportion. More periods mean shorter time for each.

Indian Perspective:

This is a common scenario in Indian schools (CBSE or State Boards) where the principal might increase the number of periods to accommodate a new subject like Computer Science or Physical Education without extending the overall school dismissal time.

Given:

Time taken by small pump = $3$ hours

Time taken by large pump = $2$ hours

To Find:

Time taken to fill the tank when both pumps work together.

Solution:

In Indian arithmetic, we solve "Time and Work" problems by finding the work done in one unit of time (1 hour in this case).

When both pumps are used together, their work rates are added:

The time taken to fill the whole tank is the reciprocal of the work done in one hour:

To convert this into minutes: $0.2 \times 60 = 12 \text{ minutes}$.

Final Answer: The tank will take $1$ hour and $12$ minutes to fill.

Indian Perspective:

This situation is very common in Indian households where multiple motors or pumps are used to fill overhead tanks from underground sumps. Knowing the combined time helps in managing electricity usage and preventing water overflow.

Given:

Initial number of machines ($x_1$) = $42$

Initial number of days ($y_1$) = $63$

Target number of days ($y_2$) = $54$

To Find:

Number of machines required ($x_2$).

Solution:

This is a case of Inverse Proportion. If the work needs to be completed in fewer days, the factory must employ more machines.

In inverse proportion, the product $x \times y$ remains constant.

Solving for $x_2$:

Simplifying the fraction:

Final Answer: $49$ machines are required.

Indian Perspective:

Small-scale toy industries in India, like those in Channapatna (Karnataka) or Kondapalli (Andhra Pradesh), often face tight deadlines during festive seasons like Dasara or Diwali. Understanding this proportionality helps factory managers plan machinery upgrades to meet market demand on time.

Given:

Initial Speed ($s_1$) = $60$ km/h

Initial Time ($t_1$) = $2$ hours

Final Speed ($s_2$) = $80$ km/h

To Find:

New time taken ($t_2$).

Solution:

Distance is constant. Therefore, Speed and Time are in Inverse Proportion. Higher speed results in less time taken.

Solving for $t_2$:

To convert $0.5$ hours to minutes: $0.5 \times 60 = 30 \text{ minutes}$.

Final Answer: The car will take $1$ hour and $30$ minutes.

Indian Perspective:

With the expansion of National Highways and Expressways in India (like the Delhi-Mumbai Expressway), vehicles can now maintain higher average speeds. This mathematical logic explains how better infrastructure significantly reduces travel time for Indian commuters and logistics.