Inverse Proportion Practice Problems and Solutions

Master inverse proportionality with step-by-step practice problems. Learn to identify inverse relationships, solve equations, and apply concepts to real-world scenarios.

📚What You'll Master in This Practice Session
  • Identify inverse proportional relationships from tables and real-world examples
  • Apply the inverse proportion function Y = a/X to solve problems
  • Calculate how variables change when one increases and another decreases proportionally
  • Verify inverse proportionality by checking multiplication and division patterns
  • Solve practical problems involving speed, time, distance, and resource consumption
  • Graph inverse proportion relationships and interpret their hyperbolic curves

Understanding Inverse Proportion

Complete explanation with examples

Inverse Proportion

Inverse proportionality describes a situation in which, when one term is multiplied by a certain number of times, the second term is decreased by the same number of times. This also occurs in reverse, if one term decreases by a certain number of times, the second term increases by the same number of times.

Let's see an example to illustrate this concept.

Given the following table:

We see two values, X X and Y Y . It can be very clearly seen that, when the value of X X increases by 2 2 , the value of Y Y also decreases 2 2 times. Therefore, it can be said that there is inverse proportionality here.

Inverse Proportion

Detailed explanation

Practice Inverse Proportion

Test your knowledge with 27 quizzes

During a swimming contest, four swimmers completed different distances in varying times:

Swimmer A - 50m in 25 seconds.

Swimmer B - 75m in 50 seconds.

Swimmer C - 20m in 10 seconds.

Swimmer D - 100m in 80 seconds.

Which swimmer had the fastest pace?

Examples with solutions for Inverse Proportion

Step-by-step solutions included
Exercise #1

What is the ratio between the number of fingers and the number of toes?

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Identify the number of fingers, which is typically 10.
  • Step 2: Identify the number of toes, which is also typically 10.
  • Step 3: Write the ratio of fingers to toes.
  • Step 4: Simplify the ratio.

Now, let's work through each step:
Step 1: The typical number of fingers on a human is 10 10 .
Step 2: The typical number of toes on a human is 10 10 .
Step 3: The ratio of fingers to toes is 10:10 10:10 .
Step 4: Simplifying this ratio 10:10 10:10 gives us 1:1 1:1 .

Therefore, the solution to the problem is 1:1 1:1 , which corresponds to answer choice 4.

Answer:

1:1 1:1

Exercise #2

In a basket, there are 15 apples and 10 oranges. What is the ratio of apples to oranges?

Step-by-Step Solution

To find the ratio of apples to oranges, divide the number of apples by the number of oranges.
Therefore, apples:oranges=1510=3:2 \text{apples:oranges} = \frac{15}{10} = 3:2 .
Thus, the ratio of apples to oranges is 3:2 3:2 .

Answer:

3:2 3:2

Exercise #3

A recipe calls for 400g of flour and 200g of sugar. What is the ratio of flour to sugar in the recipe?

Step-by-Step Solution

To find the ratio of flour to sugar, divide the amount of flour by the amount of sugar.
Thus, we have flour:sugar=400200=2:1 \text{flour:sugar} = \frac{400}{200} = 2:1 .
Therefore, the ratio of flour to sugar is 2:1 2:1 .

Answer:

3:2 3:2

Exercise #4

A tank fills with water at a rate of 20 liters every 5 minutes.
What is the flow rate of the water in liters per minute?

Step-by-Step Solution

The total volume of water that fills the tank is 20 20 liters over 5 5 minutes. The flow rate is given by the volume divided by time:
Flow Rate=Total VolumeTime=205=4 \text{Flow Rate} = \frac{\text{Total Volume}}{\text{Time}} = \frac{20}{5} = 4
Thus, the water flows at a rate of 4 4 liters per minute.

Answer:

4 4 liters/minute

Exercise #5

On one tree, 8 oranges grow in 4 days.
What is the growth rate of the oranges?

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Identify the total number of oranges that grow, which is 8.
  • Step 2: Note the total number of days in which the 8 oranges grow, which is 4 days.
  • Step 3: Apply the formula for the growth rate: Growth rate=Total number of orangesTotal number of days\text{Growth rate} = \frac{\text{Total number of oranges}}{\text{Total number of days}}.
  • Step 4: Calculate the growth rate by dividing 8 by 4.

Now, let's work through each step:
Step 1: The problem gives us a total of 8 oranges.
Step 2: These oranges grow over a period of 4 days.
Step 3: Using the formula, we find the growth rate: 84=2\frac{8}{4} = 2 oranges per day.

Therefore, the solution is that the growth rate is 2 oranges per day.

Answer:

2 oranges per day

Frequently Asked Questions

What is inverse proportion and how do I recognize it?

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Inverse proportion occurs when one variable increases while another decreases by the same factor. For example, if X doubles, Y is halved. You can recognize it by checking if the product of the two variables remains constant (X × Y = k).

How do I solve inverse proportion problems step by step?

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Follow these steps: 1) Identify the inverse relationship, 2) Find the constant k using k = X₁ × Y₁, 3) Use the formula Y = k/X to find unknown values, 4) Verify your answer by checking that the product remains constant.

What's the difference between direct and inverse proportion?

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In direct proportion, both variables increase or decrease together (Y = kX). In inverse proportion, when one variable increases, the other decreases (Y = k/X). Direct proportion graphs as a straight line, while inverse proportion creates a hyperbola.

What are common real-world examples of inverse proportion?

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Common examples include: • Speed and travel time (faster speed = less time) • Number of workers and completion time • Brightness and distance from light source • Gas consumption and remaining fuel • Price per item and quantity bought with fixed budget

How do I check if a table shows inverse proportionality?

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Multiply each pair of X and Y values. If all products equal the same constant, it's inverse proportion. For example, if (5,20), (10,10), (20,5) gives products 100, 100, 100, then it shows inverse proportionality with k=100.

What does the graph of inverse proportion look like?

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The graph of Y = a/X is a hyperbola with two curved branches. It never touches the X or Y axes, approaches them asymptotically, and lies in the first and third quadrants when 'a' is positive.

How do I find the constant in inverse proportion problems?

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Use any complete pair of values from your data: k = X × Y. This constant remains the same for all pairs in an inverse relationship. Once you find k, you can use Y = k/X to solve for any missing values.

What are the most common mistakes in inverse proportion problems?

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Common mistakes include: confusing inverse with direct proportion, forgetting to multiply X and Y to find the constant, not checking that products are equal, and misreading whether one variable increases or decreases when solving word problems.

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