Each of our Ganita Prakash Class 8 Worksheet and NCERT Class 8 Maths Chapter 3 Proportional Reasoning 2 Worksheet with Answers focuses on conceptual clarity.
Proportional Reasoning 2 Worksheet Class 8
Class 8 Maths Proportional Reasoning 2 Worksheet
Proportional Reasoning 2 Class 8 Ganita Prakash Worksheet
Proportionality – a quick recap
Ravi crnd Anjali are both making lemonade for a summer party. Ravi mixes 5 glasses of water with 3 cups of lemon juice in one jug, while Anjali mixes 10 glasses of water with 6 cups of lemon juice in another jug.
To see if their lemonades will taste the same, we can represent Ravi’s mixture as 5 : 3 and Anjali’s mixture as 10 : 6. Now, to check if the proportions of water to lemon juice are the same, Ravi and Anjali use the cross-multiplication method.
They know that if the two ratios are proportional, the cross-products should be equal. Let’s see if their lemonades will have the same balance of flavours!
In general, we can say that two ratios a : b and c : d are proportional if
a × d = b × c, or \(\frac{a}{c}=\frac{b}{d}\)
Question 1.
Are Ravi’s and Anjali’s mixtures proportional? Use the cross-multiplication method to check.
Answer:
Yes
Question 2.
Imagine there is another friend, Meera, who mixes 8 glasses of water with 4 cups of lemon juice,
(a) What is the ratio of water to lemon juice in Meera’s mixture?
(b) Check if Meera’s mixture is proportional to Ravi’s and Anjali’s mixtures using the same cross-multiplication method.
Answer:
(a) 2 ; 1
(b) No
Ratios in Maps
Ava and Arun are planning a road trip across the country and decide to use a map to chart their route. As they study the map, Arun notices a number at the bottom right corner of the map, next to the map’s scale. The number is 1 : 60,00,000.
Ava looks closely at the map and smiles. “No secret code, Arun. This is called the Representative Fraction, or RF for short. It tells us the ratio between the distance on the map and the geographical distance on the ground.” Arun looks confused. “So, how does that help us?”
“Let me explain,” Ava continues. “If the ratio is 1 : 60,00,000, it means that 1 cm on the map represents 60,00,000 cm in real life. So, for every 1 cm we measure on the map, we multiply it by 60,00,000 to get the geographical distance. It’s like shrinking the real world to fit onto a piece of paper.”
Question 3.
Imagine you are looking at a map of India with a RF scale 1 : 60,00,000.
(a) What is the distance between Delhi and Patna?
(b) What is the distance between Delhi and Hyderabad?
Answer:
Ratios with More than 2 Terms
When two ratios with two terms are proportional, i.e., a : b :: p : q then \(\frac{a}{p}=\frac{b}{q}\)
Similarly, when two ratios with more than two terms, Jor example, three terms are proportional, i.e.,
a : b : c :: p : q : r then \(\frac{a}{p}=\frac{b}{q}=\frac{c}{r}\)
In general, when two ratios with multiple terms are proportional i.e., a : b : C : d :: p : q : r : s then
\(\frac{a}{p}=\frac{b}{q}=\frac{c}{r}=\frac{d}{s}\)
Question 4.
Red, blue and white paint are mixed in the ratio 1 : 2 : 7 to make lavender paint.
(a) To produce 20 litres of lavender paint, how much red and blue paint should be required, if we have already 14 litres of white paint in stock.
Answer:
Red = 2 litres, Blue = 4 litres
(b) If 1 litre of blue paint is added to the mixture obtained in part (a), find the new ratio of red, blue and white paint?
Answer:
2 : 5: 14
Dividing a Whole in a Given Ratio
When we divide a quantity y in the ratio a : b : c : …, the terms in the ratio are:
\(y \times \frac{a}{(a+b+c+\ldots)}, y \times \frac{b}{(a+b+c+\ldots)}, y \times \frac{c}{(a+b+c+\ldots)}\), and so on.
Question 5.
(a) If a gift packs contain items in the ratio 3:2:5, how many of each item are needed to prepare 300 gift packs?
Answer:
90, 60, 150
(b) For celebration, the organisers want to add a fourth item to the gift packs such that the ratio of the four items is 1 : 3 : 2 : 4. If the total number of items in 300 gift packs is 2700, how many of each item will there be?
Answer:
270, 810, 540, 1080
Question 6.
Can we construct a triangle with sidelengths in the ratio 2 : 3 : 5? Why or why not?
Answer:
No, The sum of the lengths of any two sides must be greater than the third side.
Question 7.
Can we construct a triangle with angles in the ratio 2:3:5? Why or why not?
Answer:
Yes
Question 8.
I have 300 coins in the ratio – no. of ₹ 20 coins: no. of ₹ 10 coins: no. of ₹ 5 coins: no. of ₹ 2 coins:no.of ₹ 1 coins :: 5 : 4 : 3 : 2 : 1. How much money do I have in coins?
Answer:
₹ 3200
A Slice of The Pie
A pie chart shows different proportions of a whole. The angles in a pie chart are proportional to the quantities they represent. Ratios used for pie charts should be reduced to their simplest form before calculating angles. To draw a pie chart, we first convert the data into degrees for the circle. Since a circle has 360°, the formula for the angle of each slice or sector is:
Angle of secret = \(\left(\frac{\text { Value of category }}{\text { Total value }}\right) \times 360^{\circ}\)
Question 9.
The following table shows the favourite after school activtttes of class VIII students.
| After school activities | Watching TV | Outdoor games | Video games | Reading |
| Number of students | 24 | 36 | 24 | 12 |
Draw a pie chart, for the information given in the table.
Answer:
Question 10.
500 students of a school were surveyed for their favourite activity. The information is represented in the pie chart.
Observe the chart and answer the following questions.
(a) What does the pie chart represent?
(b) Which is the least favourite activity of students?
(c) What percentage of the students like music?
(d) What percentage of the students like playing outdoor games?
Answer:
(a) The Pie chart represents favourite activity of 500 students.
(b) Reading
(c) 20%
(d) 26%
Question 11.
A survey of 720 people showed their favourite fruits: Mango (240), Apple (180), Orange (120), and Banana (180). Calculate the central angles and draw a pie chart to show this information.
Answer:
Inverse proportions
Two quantities are in direct proportion if they increase or decrease together by the same factor. In direct proportion, the ratio of corresponding values remains constant. For example, if x and y are two quantities that are directly proportional, and (x1 x2, x3, …) and (y1, y2, y3, …) are the corresponding values of x and y, then, \(\frac{x_1}{y_1}=\frac{x_2}{y_2}=\frac{x_3}{y_3}\) = … = k, where k is constant.
When two ratios are direct proportional, i.e., when a : b :: c : d, then d = \(\frac{b c}{a}\).
Two quantities are in inverse proportion, if one quantity increases while the other decreases by the same factor.
In inverse proportion, when one quantity changes by a factor n, the other quantity changes by the inverse \(\frac{1}{n}\). For example, if x and y are two quantities that are inversely proportional, and (x1, x2, x3,…) (y1, y2, y3…..) are the corresponding values of x and y, then x1y1 = x2y2 = x3y3 = … = n, where n is constant.
Question 12.
Identify which of the following pairs of quantities are La direct proportion or in inverse proportion.
(a) The distance between two cities on a map and the actual geographical distance between them.
(b) The number of students in a hostel and the number of days the food stock lasts.
(c) The number of spokes in a wheel and the angle between each pair of consecutive spokes.
(d) The amount of money deposited in a bank and the simple interest earned over a fixed period.
Answer:
(a) Direct proportion
(b) Inverse proportion
(c) Inverse proportion
(d) Direct proportion
Question 13.
If 8 workers can build a wall that is 20 meters long in a day, how many workers would be needed to build a wall 50 meters long in the same amount of time?
Answer:
20 workers
Question 14.
It takes 5 identical pipes about 80 minutes to fill a large swimming pool. If 2 of the pipes get clogged and stop working, how long will it take the remaining 3 pipes to fill the same pool?
Answer:
\(133 \frac{1}{3}\) minutes
Question 15.
Identify which of these are in inverse proportion?
Answer:
(a) Not in inverse proportion
(b) In inverse proportion
Question 16.
Fill in the empty cells if x and y are in inverse proportion.
Answer:
| x | 5 | 7 | 7 | 1 |
| y | 21 | 15 | 35 | 105 |
Question 17.
Shabnam takes 20 minutes to reach her school, when she travels at a speed of 6 km/h. If she wants to reach school in 24 minutes, what should her speed be?
Answer:
5 km/hr
Question 18.
In a scout camp, there is food provision for 350 cadets for 36 days. If 50 cadets leave the camp, for how many days will the provisions last?
Answer:
42 days
Question 19.
A train moving at a uniform speed of 72 km/h reaches its destination in 20 hours,
(a) How long will it take if it runs at the speed of 90 km/h?
(b) At what speed must it travel to reach the destination in 24 hours?
Answer:
(a) 16 hours
(b) 60 km/hr
Worksheet On Proportional Reasoning 2 Class 8
A. Choose the correct option.
1. If the ratio of cement, sand, and gravel is 1 : 1.5 : 3, how many bags of sand are required for every 6 bags of cement?
(a) 3 bags
(b) 4.5 bags
(c) 9 bags
(d) 6 bags
Answer:
(c) 9 bags
2. A map has a scale of 1 : 60,00,000. What does this mean in terms of distance?
(a) 1 cm on the map represents 60 cm on the ground.
(b) 1 cm on the map represents 60,00,000 cm on the ground.
(c) 1 km on the map represents 60 km on the ground.
(d) 1 cm on the map represents 6,00,000 cm on the ground.
Answer:
(b) 1 cm on the map represents 60,00,000 cm on the ground.
3. If the ratio of red, blue, and white paint needed for a shade of lavender is 2 : 3 : 5, how much red paint is required to produce 100 litres of lavender paint?
(a) 10 litres
(b) 15 litres
(c) 12 litres
(d) 20 litres
Answer:
(d) 20 litres
4. In an inverse proportion, if the number of workers is doubled, what happens to the time taken to complete the work?
(a) It remains the same
(b) It halves
(c) It doubles
(d) It becomes four times
Answer:
(b) It halves
5. A triangle has angles tn the ratio 4 : 3 : 3. What is the measure of each angle?
(a) 72°, 54°, 36°
(b) 36°, 72°, 36°
(c) 40°, 40°, 100°
(d) 72°, 54°, 54°
Answer:
(d) 72°, 54°, 54°
6. When dividing a quantity in the ratio 2 : 3 : 5, how do you find the amount of the first part if the total quantity is 60?
(a) Multiply the total by
(b) Multiply the total by
(c) Multiply the total by
(d) Add the parts and divide the total by 10
Answer:
(a) Multiply the total by
Directions. (For Q.7 – 8): In the following questions, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option as:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
(c) Assertion (A) is true but Reason (R) is false.
(d) Assertion (A) is false but Reason (R) is true.
7. Assertion (A): Ia a direct proportion, if one quantity increases by a factor, the other quantity also increases by the same factor.
Reason (R): In direct proportion, the ratio of the two quantities remains constant.
Answer:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
8. Assertion (A): In inverse proportion, if one quantity doubles, the other quantity is halved.
Reason (R): In inverse proportion, the product of the two quantities remains constant.
Answer:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
B. Fill in the blanks.
1. In a proportional relationship, when one quantity increases by a factor, the other quantity _____ by the same factor.
Answer:
increases
2. The total angle in a circle is _____ degrees, which is used when constructing pie charts.
Answer:
360
3. A map’s Representative Fraction (RF) of 1:10,00,000 means that 1 cm on the map represents _____ cm on the ground.
Answer:
10,000,000
4. When dividing a whole in the ratio 3 : 2, we first find the total sum of the ratio as _____.
Answer:
5
5. In inverse proportionality, if the number of workers doubles, the time taken to complete the work _____.
Answer:
halved
C. State whether the following statements are true (T) or False (F).
1. The sum of the angles in a pie chart always equals 360°.
Answer:
True
2. In inverse proportion, if the speed increases, the time taken to travel the same distance also increases.
Answer:
False
3. When dividing a quantity in a ratio, you can multiply each part by the total sum of the ratio.
Answer:
False
4. A pie chart can only represent data with two categories.
Answer:
False
5. When mixing paint in a ratio, the quantities of the components should always be in whole numbers.
Answer:
False
D. Solve the following.
Question 1.
If 5 workers can complete a task in 8 hours, how long will it take for 10 workers to complete the same task, assuming they work at the same rate?
Answer:
4 hours
Question 2.
A car travels 240 km in 4 hours. If the speed is doubled, how much time will it take to travel the same distance?
Answer:
2 hours
Question 3.
In a recipe, 3 cups of flour are needed for 4 servings. How much flour is required for 12 servings?
Answer:
9 cups of flour
Question 4.
A baker has enough ingredients to make 24 cakes in 6 hours. How many hours will it take to make the same number of cakes if the number of bakers is doubled?
Answer:
3 hours
Question 5.
The cost of 5 kg of apples is ₹ 250. What will be the cost of 10 kg of apples?
Answer:
₹ 500