Question 1

Q1: Following are the car parking charges near a railway station upto 4 hours ₹ 60 8 hours ₹ 100 12 hours ₹ 140 24 hours ₹ 180 Check if the parking charge...

Accepted Answer

To Check: If the parking charges are in direct proportion to the parking time. Condition for Direct Proportion: Two quantities, say $x$ and $y$, are in direct proportion if the ratio $\frac{x}{y}$ remains constant. Here, let the parking time be $x$ (in hours) and the parking charges be $y$ (in ₹). W...

Question 2

Q10: A loaded truck travels 14 km in 25 minutes. If the speed remains the same, how far can it travel in 5 hours?

Accepted Answer

Given: The speed of the truck is constant. Therefore, the distance travelled is in direct proportion to the time taken. Distance travelled = 14 km Time taken = 25 minutes To Find: Distance the truck can travel in 5 hours. Let: Let the distance travelled in 5 hours be $x$ km. Units Conversion: We nee...

Question 3

Q2: A mixture of paint is prepared by mixing 1 part of red pigments with 8 parts of base. In the following table, find the parts of base that need to be a...

Accepted Answer

Given: A paint mixture is prepared by mixing 1 part of red pigments with 8 parts of base. This means the ratio of parts of red pigment to parts of base is constant and equal to $\frac{1}{8}$. This is a case of direct proportion. Let: Let the parts of red pigment be $x$ and the parts of base be $y$....

Question 4

Q3: In Question 2 above, if 1 part of a red pigment requires 75 mL of base, how much red pigment should we mix with 1800 mL of base?

Accepted Answer

Given: 1 part of a red pigment requires 75 mL of base. Total volume of base available = 1800 mL. The amount of red pigment and the amount of base are in direct proportion. Let: Let the required parts of red pigment be $x$. Setting up the proportion: We can set up a proportion based on the ratio of r...

Question 5

Q4: A machine in a soft drink factory fills 840 bottles in six hours. How many bottles will it fill in five hours?

Accepted Answer

Given: A machine fills 840 bottles in 6 hours. The number of bottles filled is directly proportional to the time taken. Let: Let the number of bottles filled in 5 hours be $x$. Setting up the proportion: We can set up a proportion based on the ratio of bottles to time. $$\frac{	ext{Number of bottle...

Question 6

Q5: A photograph of a bacteria enlarged 50,000 times attains a length of 5 cm. What is the actual length of the bacteria? If the photograph is enlarged 20...

Accepted Answer

Part 1: Find the actual length of the bacteria. Given: Enlarged length = 5 cm Enlargement factor = 50,000 times Formula: Actual length = $\frac{	ext{Enlarged length}}{	ext{Enlargement factor}}$ Solution: Actual length = $\frac{5}{50000} 	ext{ cm}$ Actual length = $\frac{1}{10000} 	ext{ cm}$ Actu...

Question 7

Q6: In a model of a ship, the mast is 9 cm high, while the mast of the actual ship is 12 m high. If the length of the ship is 28 m, how long is the model...

Accepted Answer

Given: The dimensions of the model ship and the actual ship are in direct proportion. Model mast height = 9 cm Actual mast height = 12 m Actual ship length = 28 m To Find: The length of the model ship. Let: Let the length of the model ship be $x$ cm. Units Conversion: To maintain consistency, we sho...

Question 8

Q7: Suppose 2 kg of sugar contains $9 	imes 10^6$ crystals. How many sugar crystals are there in (i) 5 kg of sugar? (ii) 1.2 kg of sugar?

Accepted Answer

Given: Weight of sugar and the number of crystals are in direct proportion. 2 kg of sugar contains $9 	imes 10^6$ crystals. Let: Let the weight of sugar be $w$ and the number of crystals be $n$. The ratio $\frac{n}{w}$ is constant. Constant ratio = $\frac{9 	imes 10^6}{2} = 4.5 	imes 10^6$ crysta...

Question 9

Q8: Rashmi has a road map with a scale of 1 cm representing 18 km. She drives on a road for 72 km. What would be her distance covered in the map?

Accepted Answer

Given: The map scale is 1 cm : 18 km. This means the distance on the map and the actual distance are in direct proportion. Actual distance driven = 72 km. Let: Let the distance covered in the map be $x$ cm. Setting up the proportion: $$\frac{	ext{Distance on map (cm)}}{	ext{Actual distance (km)}}...

Question 10

Q9: A 5 m 60 cm high vertical pole casts a shadow 3 m 20 cm long. Find at the same time (i) the length of the shadow cast by another pole 10 m 50 cm high...

Accepted Answer

Given: At the same time, the height of an object and the length of its shadow are in direct proportion. Height of first pole = 5 m 60 cm Length of its shadow = 3 m 20 cm Units Conversion: Let's convert all measurements to a single unit, centimeters. Height of first pole = $5 	ext{ m} + 60 	ext{ cm...

Question 11

Q1: Which of the following are in inverse proportion? (i) The number of workers on a job and the time to complete the job. (ii) The time taken for a journ...

Accepted Answer

Condition for Inverse Proportion: Two quantities are in inverse proportion if an increase in one quantity causes a proportional decrease in the other, such that their product remains constant ($x 	imes y = k$). Analysis of each case: (i) The number of workers on a job and the time to complete the j...

Question 12

Q10: Two persons could fit new windows in a house in 3 days. (i) One of the persons fell ill before the work started. How long would the job take now? (ii)...

Accepted Answer

Given: Number of persons = 2 Time taken = 3 days The number of persons and the time taken are in inverse proportion. Step 1: Calculate the total work in 'person-days'. Total work = (Number of persons) $	imes$ (Time taken) Total work = $2 	imes 3 = 6$ person-days. (i) One person fell ill. New numbe...

Question 13

Q11: A school has 8 periods a day each of 45 minutes duration. How long would each period be, if the school has 9 periods a day, assuming the number of sch...

Accepted Answer

Given: Initial number of periods = 8 Duration of each period = 45 minutes The number of periods and the duration of each period are in inverse proportion, as the total school time is constant. Step 1: Calculate the total duration of school hours in minutes. Total school time = (Number of periods) $\...

Question 14

Q2: In a Television game show, the prize money of ₹ 1,00,000 is to be divided equally amongst the winners. Complete the following table and find whether t...

Accepted Answer

Given: Total prize money = ₹ 1,00,000. This amount is to be divided equally among the winners. Let: Let the number of winners be $x$ and the prize for each winner be $y$. Relationship: The product of the number of winners and the prize for each winner must equal the total prize money. $$x 	imes y =...

Question 15

Q3: Rehman is making a wheel using spokes. He wants to fix equal spokes in such a way that the angles between any pair of consecutive spokes are equal. He...

Accepted Answer

Given: The total angle in a circle (at the center of the wheel) is $360^\circ$. This angle is divided equally by the spokes. Let: Let the number of spokes be $x$ and the angle between a pair of consecutive spokes be $y$. Relationship: The product of the number of spokes and the angle between them is...

Question 16

Q4: If a box of sweets is divided among 24 children, they will get 5 sweets each. How many would each get, if the number of the children is reduced by 4?

Accepted Answer

Given: Number of children initially = 24 Number of sweets each child gets = 5 The number of children and the number of sweets each gets are in inverse proportion, because the total number of sweets is constant. Step 1: Calculate the total number of sweets. Total sweets = (Number of children) $	imes...

Question 17

Q5: A farmer has enough food to feed 20 animals in his cattle for 6 days. How long would the food last if there were 10 more animals in his cattle?

Accepted Answer

Given: Initial number of animals = 20 Number of days the food lasts = 6 The number of animals and the number of days the food lasts are in inverse proportion, as the total amount of food is constant. Step 1: Calculate the total food supply in 'animal-days'. Total food = (Number of animals) $	imes$...

Question 18

Q6: A contractor estimates that 3 persons could rewire Jasminder's house in 4 days. If, he uses 4 persons instead of three, how long should they take to c...

Accepted Answer

Given: Initial number of persons = 3 Time taken = 4 days The number of persons and the time taken to complete a job are in inverse proportion, as the total amount of work is constant. Step 1: Calculate the total work in 'person-days'. Total work = (Number of persons) $	imes$ (Number of days) Total...

Question 19

Q7: A batch of bottles were packed in 25 boxes with 12 bottles in each box. If the same batch is packed using 20 bottles in each box, how many boxes would...

Accepted Answer

Given: Number of boxes = 25 Number of bottles in each box = 12 The number of boxes and the number of bottles per box are in inverse proportion for the same batch, as the total number of bottles is constant. Step 1: Calculate the total number of bottles in the batch. Total bottles = (Number of boxes)...

Question 20

Q8: A factory requires 42 machines to produce a given number of articles in 63 days. How many machines would be required to produce the same number of art...

Accepted Answer

Given: Initial number of machines = 42 Time taken = 63 days The number of machines and the number of days to produce a fixed number of articles are in inverse proportion. Step 1: Calculate the total work in 'machine-days'. Total work = (Number of machines) $	imes$ (Number of days) Total work = $42...

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