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

1 bag of corn 7 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 D Swimmer A Swimmer C Swimmer B

Swimmer B Swimmer D Swimmer A Swimmer C

Swimmer C + A Swimmer B Swimmer D

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

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Let's look at an example from everyday life

Imagine traveling in some vehicle while the roads are quite empty, without any traffic jams.

As you travel more kilometers and more time passes, the amount of gasoline decreases.

We can say that, as the distance increases the amount of gasoline decreases.

Let's see a graphical representation of inverse proportionality:

The function: $Y=\frac{a}{X}$

represents inverse proportionality.

As the $X$ increases the $Y$ decreases.

How can we check if there is inverse proportionality?

To find out if there is inverse proportionality, we will examine if, when one term is multiplied by a certain amount of times, the second term is decreased by the same amount of times.

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Test your knowledge

Question 1

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

Question 2

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

Question 3

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

Let's look at an example

Given the following table:

Let's check if every time $X$ increases by a specific amount, $Y$ also decreases in the same proportion.

If this occurs, it means there is inverse proportionality. If not, then there isn't.

Let's ask:

By how much does $X$ increase from $5$ to $10$ ?

The answer is it multiplies by $2$ .

And by how much does $Y$ decrease from $20$ to $10$ ?

The answer is it divides by $2$ .

Let's continue,

By how much does $X$ increase from $5$ to $20$ ? The answer is it multiplies by $4$ .

And by how much does $Y$ decrease from $20$ to $5$ ?

The answer is it divides by $4$ .

We will continue examining and discover that indeed every time $X$ multiplies by a certain number, $Y$ also decreases divided by the same number.

We will see it in the following way:

Examples and exercises with solutions on inverse proportionality

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

Do you know what the answer is?

Question 1

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

Question 2

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

Question 3

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?

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Inverse Proportion