Practice Questions

Define direct proportion with an example.

Examine the following table and determine if $x$ and $y$ are in direct proportion. Justify your answer.

Justify whether the relationship between the number of family members and the monthly grocery bill is a direct proportion. Propose a reason why it might not be.

Examine the following table and determine if $a$ and $b$ are in inverse proportion. Justify your answer.

If 15 workers can build a wall in 8 days, calculate how many workers would be required to build the same wall in 6 days.

If 12 notebooks cost ₹180, calculate the cost of 16 such notebooks.

A recipe for a cake that serves 6 people requires 300 g of flour. Calculate the amount of flour needed if the cake is to serve 15 people.

Describe the relationship between the speed of a car and the time taken to cover a fixed distance. Is this a direct or inverse proportion?

Explain why the amount of petrol consumed by a car and the distance it travels (at a constant speed) are in direct proportion.

State the mathematical condition for two quantities, $x$ and $y$, to be in inverse proportion.

A school is organizing a trip. The cost per student is inversely proportional to the number of students attending. If 30 students go, the cost is ₹800 per student. Formulate an equation for this relationship and use it to propose the number of students required to bring the cost down to ₹600 per student.

A car travels a distance of 150 km. A student creates the following table relating speed and time. Evaluate the data in the table. Is the relationship an inverse proportion? Justify your answer by identifying and correcting any error.

A vertical pole of height 6 m casts a shadow 4 m long on the ground. At the same time, a tower casts a shadow 28 m long. Calculate the height of the tower.

Identify if the following situation represents direct or inverse proportion: The time taken to fill a water tank and the number of taps used to fill it.

Name the type of proportion where if one quantity is doubled, the other quantity is also doubled.

If $x_1, y_1$ and $x_2, y_2$ are two corresponding values of an inverse proportion, state the formula that relates them.

A table shows the values of two variables, $p$ and $q$. Describe the steps you would take to check if $p$ and $q$ are in direct proportion.

List three real-world examples of inverse proportion.

Recall the introductory problem about Mohan preparing tea. He uses 300 mL of water, 2 spoons of sugar, 1 spoon of tea leaves and 50 mL of milk for two persons. Explain how you would determine the quantity of each item needed for five persons using the concept of direct proportion.

Describe a real-life situation that represents direct proportion. Create a table with at least four pairs of values for this situation and explain why the ratio of the corresponding values remains constant.

6 pipes can fill a tank in 1 hour 40 minutes. Calculate how long it will take to fill the same tank if only 4 pipes of the same type are used.

Design a real-world scenario where two quantities, $P$ and $Q$, are related such that $P$ is directly proportional to $Q$ and also directly proportional to another quantity $R$. Formulate a single equation that describes the relationship between $P, Q,$ and $R$.

A recipe for a fruit punch requires fruit juice and soda water in the ratio 3:5. A caterer has 12 litres of fruit juice. They also have a large container of soda water. They start making the punch but discover their 12-litre juice container was only 75% full. Evaluate the situation and calculate the correct amount of soda water they must now add to maintain the recipe's proportion.

Design an experiment to verify the relationship between the number of identical pipes used to fill a swimming pool and the time taken. Your design should include:

1. A hypothesis.
2. The variables to be controlled and measured.
3. The procedure to collect data.
4. A method to analyze the data to justify your conclusion.

A batch of bottles was packed in 30 boxes with 16 bottles in each box. If the same batch is packed using 24 bottles in each box, calculate how many boxes would be filled.

A car travelling at a speed of 60 km/h covers a distance in 5 hours. To cover the same distance in 4 hours, what speed should the car maintain? Analyze the relationship between speed and time.

A hostel has enough food for 125 students for 16 days. How many students must leave the hostel so that the same food will last for 20 days?

A train travels 180 km in 3 hours with a uniform speed. Calculate: (i) The distance it will travel in 5 hours. (ii) The time it will take to cover 450 km.

A factory needs 42 machines to produce a certain number of articles in 56 days. The factory owner wants to complete the production in 49 days. (i) Analyze whether more or fewer machines will be required. (ii) Calculate the number of extra machines that must be employed.

The scale on a map is 1 : 25,000,000. (i) Calculate the actual distance in km between two cities that are 6 cm apart on the map. (ii) Calculate the distance on the map in cm if the actual distance between two other cities is 1500 km.

A student states, "If quantity A doubles and quantity B is halved, they must be in inverse proportion." Critique this statement. Is it always true? Justify your answer.

Evaluate if the area of a square is in direct proportion to the length of its side. Justify your conclusion mathematically.

A taxi fare consists of a fixed charge of ₹50 plus a variable charge of ₹15 per km. Formulate an equation for the total fare ($y$) for a journey of $x$ km. Justify why the total fare is not in direct proportion to the distance travelled.

Propose a simple modification to the problem of 'number of students' and 'time to arrange chairs' from the source text that would make it NOT an inverse proportion. Justify your proposal.

A contractor undertook a project to complete a road in 40 days and employed 30 men. After 25 days, he found that only half of the road was built. To complete the project on time, calculate the number of additional men he must employ.

Two variables, $x$ and $y$, are related by the equation $y = \frac{k}{x^2}$, where $k$ is a constant. Critique the statement: '$x$ and $y$ are in inverse proportion'. Justify your reasoning.

A team of 10 workers can complete a project in 24 days. They work for 6 days. After 6 days, 4 more workers join the team. Formulate a plan to calculate the total time it will take to complete the project and justify each step.

A garrison has provisions for 300 soldiers for 90 days. After 20 days, 50 soldiers leave the garrison. However, at the same time, the daily ration per soldier is increased by 25%. Evaluate how long the remaining food will last for the remaining soldiers. Justify your multi-step calculation.

A contractor estimates that a project can be completed by 5 skilled workers in 16 days or by 8 apprentice workers in 25 days. He decides to hire 2 skilled workers and 4 apprentices. Formulate a model to predict the time it will take for this mixed team to complete the project. Justify each step of your model.

The simple interest on a certain sum for 2 years is ₹400. Assuming the rate of interest is constant, calculate the simple interest on the same sum for 5 years.

Describe a real-life situation that represents inverse proportion. Create a table with at least four pairs of values for this situation and explain why the product of the corresponding values remains constant.

Three taps A, B, and C can fill a tank in 6, 8, and 12 hours respectively. If all three taps are opened together, formulate a method to find the time it will take to fill the tank. Justify why you cannot simply average the times.

Explain why the height of a person and their age are not in direct proportion, even though both increase over time.

Summarize the key differences between direct proportion and inverse proportion. Include their mathematical forms and how one variable changes with respect to the other.

Explain, using two different tables, why 'Simple Interest' is in direct proportion with 'Time Period' (if Principal and Rate are fixed), but 'Compound Interest' is not.