Exercise 6.4

In an election, 60% of voters cast their votes. Form an equation and draw the graph for this data. Find the following from the graph:

(i) The total number of voters, if 1200 voters cast their votes.

(ii) The number of votes cast, if the total number of voters are 800.

[Hint: If the number of voters who cast their votes be 'x' and the total number of voters be 'y' then x = 60% of y.]

Solution:

Let:

x = number of voters who cast their votes

y = total number of voters

According to the given information, x = 60% of .

Therefore, the equation is:

x = ( 6010060 ) × y

x = (3525) × y

x = y

3y = 5x

y = ( 5335 )x

Graph:

To draw the graph, we can find some points that satisfy the equation:

If x = 0, then y = (5/3) × 0 = .

So, ( , ) is a point.

If x = 300, then y = (5/3) × 300 = .

So, ( , ) is a point.

If x = 600, then y = (5/3) × 600 = .

So, ( , ) is a point.

if x = 1200, then y = (5/3) × 1200 = .

so ( , ) is a point.

(i) If 1200 voters cast their votes (x = 1200), we need to find the total number of voters (y).

Using the equation y = (5/3)x:

y = (5/3) ×

y =

Therefore, if 1200 voters cast their votes, the total number of voters is 2000.

(ii) If the total number of voters is 800 (y = 800), we need to find the number of votes cast (x).

Using the equation x = (3/5)y:

x = (3/5) ×

x =

Therefore, if the total number of voters is 800, the number of votes cast is 480.

2. When Rupa was born, her father was 25 years old. Form an equation and draw a graph for this data. From the graph find:

(i) The age of the father when Rupa is 25 years old. (ii) Rupa’s age when her father is 40 years old.

Solution:

Let:

x = Rupa's age y = Father's age

Since the father was 25 years old when Rupa was born, he will always be years older than Rupa.

Therefore, the equation is:

y = x +

Graph:

To draw the graph, we can find some points that satisfy the equation:

If x = 0 (when Rupa was born), y = 0 + 25 = .

So, ( , ) is a point.

If x = 5, y = 5 + 25 = . So, ( , ) is a point.

If x = 10, y = 10 + 25 = . So, ( , ) is a point.

Using these points, we can draw a straight line graph.

Answers:

(i) To find the father's age when Rupa is 25 years old, we look at the point on the graph where x = 25. It intersect at y = .

Therefore, the father will be 50 years old when Rupa is 25.

(ii) To find Rupa's age when her father is 40 years old, we look at the point on the graph where y = 40. It intersect at x = .

Therefore, Rupa will be 15 years old when her father is 40.

3. An auto charges 15 for the first kilometer and 8 each for each subsequent kilometer. For a distance of ‘x’ km, an amount of ‘y’ is paid. Write the linear equation representing this information and draw the graph. With the help of the graph, find the distance traveled if the fare paid is 55? How much would have to be paid for 7 kilometers?

Solution:

Let:

x = distance traveled in kilometers

y = total fare paid

The fare can be calculated as follows:

First kilometer: 158

Subsequent kilometers: 815 per kilometer

Therefore, the equation representing the fare is:

y = 15 + 8(x - 1) (for x > 1)

Simplifying the equation:

y = 15 + -

y = +

Graph:

To draw the graph, we can find some points that satisfy the equation:

If x = 1, then y = 8(1) + 7 = .

So, (, ) is a point.

If x = 2, then y = 8(2) + 7 = .

So, (, ) is a point.

If x = 3, then y = 8(3) + 7 = .

So, (, ) is a point.

To find the distance traveled if the fare paid is 55, we look at the point on the graph where y = 55. Following the line to where it intersects the x-axis gives us the distance traveled. It should intersect at x = 6.

Therefore, the distance traveled for a fare of 55 is 6 kilometers.

To find how much would have to be paid for 7 kilometers, we look at the point on the graph where x = 7. Following the line to where it intersects the y-axis gives us the fare paid. It should intersect at y = 63.

Therefore, the fare for 7 kilometers would be 63.

4. A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. John paid 27 for a book kept for seven days. If the fixed charge be x and the subsequent per day charge be y, then write the linear equation representing the above information and draw the graph of the same. From the graph, find fixed charges for the first three days if the additional charge for each day thereafter is 4. Find additional charges for each day thereafter if the fixed charge for the first three days is 7.

Solution:

Let:

x = fixed charge for the first three days

y = additional charge per day thereafter

The total charge for keeping a book for seven days can be calculated as follows:

Fixed charge for the first three days: x Additional charge for the remaining four days: 4 ×

Therefore, the equation representing the total charge is:

= x +

Graph:

To draw the graph, we can find some points that satisfy the equation:

If x = 0, then 27 = + 4y, so y = .

So, ( , ) is a point.

If x = 3, then 27 = + 4y, so y = .

So, ( , ) is a point.

If x = 7, then 27 = + 4y, so y = .

So, ( , ) is a point.

Using these points, we can draw a straight line graph.

To find the fixed charges for the first three days if the additional charge for each day thereafter is ` 4, we look at the point on the graph where y = 4. Following the line to where it intersects the x-axis gives us the fixed charge. It should intersect at x = 11.

Therefore, the fixed charges for the first three days would be 11 if the additional charge per day is 4.

To find the additional charges for each day thereafter if the fixed charge for the first three days is ` 7, we look at the point on the graph where x = 7. Following the line to where it intersects the y-axis gives us the additional charge per day. It should intersect at y = 5.

Therefore, the additional charge per day would be 5 if the fixed charge for the first three days is 7.

5. The parking charges of a car in Hyderabad Railway station for the first two hours is 50 and 10 for each subsequent hour. Write down an equation and draw the graph. Find the following charges from the graph:

(i) For three hours

(ii) For six hours

(iii) How many hours did Rekha park her car if she paid ` 80 as parking charges?

Solution:

Let:

x = number of hours parked

y = total parking charges

The parking charges can be calculated as follows:

First two hours:

Subsequent hours: per hour

Therefore, the equation representing the parking charges is:

y = 5010 + 1050(x - 2) (for x > 2)

Simplifying the equation:

y = 50 + -

y = 10x +

Graph:

To draw the graph, we can find some points that satisfy the equation:

If x = 2, then y = 10(2) + 30 = .

So, (, ) is a point.

If x = , then y = 10(3) + 30 = 60.

So, (, ) is a point.

If x = 4, then y = 10(4) + 30 = .

So, (, ) is a point.

Using these points, we can draw a straight line graph.

Answers:

(i) To find the charges for three hours, we look at the point on the graph where x = 3. Following the line to where it intersects the y-axis gives us the parking charges. It should intersect at y = 60.

Therefore, the parking charges for three hours would be ` 60.

(ii) To find the charges for six hours, we look at the point on the graph where x = 6. Following the line to where it intersects the y-axis gives us the parking charges. It should intersect at y = 90.

Therefore, the parking charges for six hours would be ` 90.

(iii) To find how many hours Rekha parked her car if she paid ` 80 as parking charges, we look at the point on the graph where y = 80. Following the line to where it intersects the x-axis gives us the number of hours parked. It should intersect at x = 5.

Therefore, Rekha parked her car for 5 hours if she paid ` 80 as parking charges.

6. Sameera was driving a car with a uniform speed of 60 kmph. Draw a distance-time graph. From the graph find the distance traveled by Sameera in

(i) 1 1/2 hours

(ii) 2 hours

(iii) 3 1/2 hours

Solution:

Since Sameera is driving at a uniform speed, the distance-time graph will be a straightCurve line. The slope of the line represents the speed.

To draw the graph, we can use the following points:

If time (t) = 0, then distance (d) = .

So, ( , ) is a point.

If t = 1 hour, then d = km.

So, ( , ) is a point.

If t = 2 hours, then d = km.

So, (, ) is a point.

Using these points, we can draw a straight line graph.

Answers:

(i) To find the distance traveled in 1 1/2 hours, we look at the point on the graph where t = hours.

Following the line to where it intersects the y-axis gives us the distance. It should intersect at d = km.

Therefore, Sameera travels 90 km in 1 1/2 hours.

(ii) To find the distance traveled in 2 hours, we look at the point on the graph where t = hours. It should intersect at d = km.

Therefore, Sameera travels 120 km in 2 hours.

(iii) To find the distance traveled in 3 1/2 hours, we look at the point on the graph where t = hours. It should intersect at d = km.

Therefore, Sameera travels 210 km in 3 1/2 hours.

7. The ratio of molecular weight of Hydrogen and Oxygen in water is 1:8. Set up an equation between Hydrogen and Oxygen and draw its graph. From the graph find the quantity of Hydrogen if Oxygen is 12 grams. And quantity of oxygen if hydrogen is 3/2 gms?

[Hint: If the quantities of hydrogen and oxygen are ‘x’ and ‘y’ respectively, then x : y = 1:8 ⇒ 8x = y]

Solution:

Let:

x = quantity of Hydrogen

y = quantity of Oxygen

The given ratio 1:8 means that for every 1 gram of Hydrogen, there are grams of Oxygen.

Therefore, the equation is:

y =

Graph:

To draw the graph, we can use the following points:

If x = 0, then y = .

So, (, ) is a point.

If x = 1, then y = .

So, (, ) is a point.

If x = 2, then y = .

So, (, ) is a point.

Using these points, we can draw a straight line graph.

Answers:

To find the quantity of Hydrogen if Oxygen is 12 grams, we look at the point on the graph where y = 12. Following the line to where it intersects the x-axis gives us the quantity of Hydrogen. It should intersect at x = grams.

Therefore, there are 1.5 grams of Hydrogen if Oxygen is 12 grams.

To find the quantity of Oxygen if Hydrogen is 3/2 grams, we look at the point on the graph where x = 3/2 = . It should intersect at y = grams.

Therefore, there are 12 grams of Oxygen if Hydrogen is 3/2 grams.

8. In a mixture of 28 liters, the ratio of milk and water is 5:2. Set up the equation between the mixture and milk. Draw its graph. By observing the graph, find the quantity of milk in the mixture.

[Hint: Ratio between mixture and milk = 5 + 2 : 5 = 7 : 5]

Solution:

Let:

x = quantity of milk in the mixture

y = total quantity of the mixture (28 liters)

The given ratio 5:2 means that for every 5 liters of milk, there are liters of the mixture.

Therefore, the equation is:

y = (7/5)x

Graph:

To draw the graph, we can use the following points:

If x = 0, then y = .

So, ( , ) is a point.

If x = 5, then y = .

So, ( , 7) is a point.

If x = 10, then y = .

So, ( , ) is a point.

Using these points, we can draw a straight line graph.

Answer:

To find the quantity of milk in the mixture, we look at the point on the graph where y = 28 liters (the total quantity of the mixture).

Following the line to where it intersects the x-axis gives us the quantity of milk. It should intersect at x = liters.

Therefore, there are 20 liters of milk in the 28-liter mixture.

9. In countries like the USA and Canada, temperature is measured in Fahrenheit whereas in countries like India, it is measured in Celsius. Here is a linear equation that converts Fahrenheit to Celsius: F = (9/5)C + 32

Solution:

Let:

x = temperature in Celsius (C)

y = temperature in Fahrenheit (F)

The equation is:

y = (9/5)x + 32

(i) Draw the graph of the above linear equation having Celsius on the x-axis and Fahrenheit on the Y-axis.

Graph:

To draw the graph, we can use the following points:

If x = 0, then y = .

So, ( , ) is a point.

If x = 5, then y = .

So, ( , ) is a point.

If x = 10, then y = .

So, ( , ) is a point.

Using these points, we can draw a straight line graph.

(ii) If the temperature is 30°C, what is the temperature in Fahrenheit?

Temperature in Fahrenheit when Celsius is 30°C:

To find the temperature in Fahrenheit when Celsius is 30°C, we look at the point on the graph where x = 30.

Following the line to where it intersects the y-axis gives us the temperature in Fahrenheit. It should intersect at y = 86°F.

Therefore, when the temperature is 30°C, it is 86°F.

(iii) If the temperature is 95°F, what is the temperature in Celsius?

(iii) Temperature in Celsius when Fahrenheit is 95°F:

To find the temperature in Celsius when Fahrenheit is 95°F, we look at the point on the graph where y = 95. Following the line to where it intersects the x-axis gives us the temperature in Celsius. It should intersect at x = 35°C.

Therefore, when the temperature is 95°F, it is 35°C.

(iv) Is there a temperature that has numerically the same value in both Fahrenheit and Celsius? If yes, find it?

(iv) Temperature with the same numerical value in Fahrenheit and Celsius:

To find the temperature that has the same numerical value in both Fahrenheit and Celsius, we look for the point on the graph where the x and y coordinates are equal. This point lies on the line y = x.

By solving the equations y = (9/5)x + 32 and y = x simultaneously, we get:

x = (9/5)x + 32

(5/5)x - (9/5)x = 32

(-4/5)x = 32

x = -40

Therefore, -40°C is numerically the same as -40°F.