Using 3 bags of corn kernels, one can make 21 small packages of popcorn. Which of the cases represent the same ratio

1 bag of corn 7 packages of popcorn

6 bags of corn 3 packages of popcorn

1 bag of corn 4 packages of popcorn

2 bags of corn 10 packages of popcorn

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?

Swimmer D Swimmer A Swimmer C Swimmer B

Swimmer A Swimmer C Swimmer D Swimmer B

Swimmer B Swimmer D Swimmer A Swimmer C

Swimmer C + A Swimmer B Swimmer D

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

Detailed explanation

Practice Inverse Proportion

Test your knowledge with 28 quizzes

In the clothing factory there are two t-shirt machines

Machine A produces 30 t-shirts in 3 minutes,
Machine B produces 16 t-shirts in 2 minutes.

Which machine will produce more t-shirts in 10 minutes?

Examples with solutions for Inverse Proportion

Step-by-step solutions included

Exercise #1

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, $\text{apples:oranges} = \frac{15}{10} = 3:2$ .
Thus, the ratio of apples to oranges is $3:2$ .

Answer:

$3:2$

Exercise #2

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 $\text{flour:sugar} = \frac{400}{200} = 2:1$ .
Therefore, the ratio of flour to sugar is $2:1$ .

Answer:

$3:2$

Exercise #3

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 $10$ .
Step 2: The typical number of toes on a human is $10$ .
Step 3: The ratio of fingers to toes is $10:10$ .
Step 4: Simplifying this ratio $10:10$ gives us $1:1$ .

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

Answer:

$1:1$

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

Answer:

$4$ liters/minute

Exercise #5

According to a recipe, one cup of flour is needed for 3 cookies. How many cups of flour are needed for six cookies?

Step-by-Step Solution

To solve this problem, let's determine how many cups of flour are needed to make six cookies using proportions.

Initially, we know that 1 cup of flour produces 3 cookies. Our task is to determine how many cups ( $x$ ) will be necessary for 6 cookies.

We can set up a proportion based on the information given:

\frac{1}{3} = \frac{x}{6}

To solve for $x$ (the unknown number of cups), we cross-multiply:

(1 \times 6) = (3 \times x)

This simplifies to:

6 = 3x

Next, divide both sides of the equation by 3 to isolate $x$ :

x = \frac{6}{3} = 2

Therefore, 2 cups of flour are needed for six cookies.

The solution to the problem is 2 cups.

Answer:

2 cups

Frequently Asked Questions

Everything you need to know about Inverse Proportion

What is inverse proportion and how do I recognize it?

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?

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?

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?

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?

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?

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?

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?

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.

Continue Your Math Journey

Master Inverse Proportion first, then advance to these related topics that build on your fraction skills

Topics Learned in Later Sections

Scale Factors, Ratio and Proportional Reasoning Ratio Equivalent Ratios Division in a given ratio Direct Proportion

Practice by Question Type

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Inverse Proportion Practice Problems and Solutions

Understanding Inverse Proportion

Inverse Proportion

Let's see an example to illustrate this concept.

Practice Inverse Proportion

Examples with solutions for Inverse Proportion

Frequently Asked Questions

What is inverse proportion and how do I recognize it?

How do I solve inverse proportion problems step by step?

What's the difference between direct and inverse proportion?

What are common real-world examples of inverse proportion?

How do I check if a table shows inverse proportionality?

What does the graph of inverse proportion look like?

How do I find the constant in inverse proportion problems?

What are the most common mistakes in inverse proportion problems?

More Inverse Proportion Questions

Ratio

Continue Your Math Journey

Topics Learned in Later Sections

Practice by Question Type