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

Equivalent Ratios

To easily solve ratio problems and to gain a better understanding of the topic in general, it is convenient to know about equivalent ratios.

Equivalent ratios are, in fact, ratios that seem different, are not expressed in the same way but, by simplifying or expanding them, you arrive at exactly the same relationship.

Think of it this way,

Detailed explanation

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What is the ratio between the orange and gray parts in the drawing?

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If you have two fractions in front of you:
$\frac{2000}{4000}$

and

$\frac{2}{4}$

we can simplify the larger fraction and arrive at

and even more, we can simplify the smaller fraction and arrive at:

In fact, we can say that:

$\frac{4000}{2000}=\frac{2}{4}=\frac{1}{2}$

All the expressions are equivalent ratios.

Do you remember we said that a ratio can be shown in the form of a fraction?

Therefore, the same rule also applies to the ratios we have learned.

We can reduce both terms of the ratio or amplify them and arrive at equivalent ratios.

To solve this type of problem easily we will always try to arrive at the smallest ratio.

We will ask ourselves by what number we can divide both terms of the ratio, in this way we will arrive at the most reduced equivalent ratio possible.

How can you tell if they are equivalent ratios?

We will ask ourselves: Will we arrive at the same ratio by reduction or by amplification?

Let's see some examples:

$2:5$

$6:16$

We have managed to demonstrate that by multiplying both terms by $3$ we arrive at the same ratio. Therefore, they are equivalent ratios!

Are these ratios equivalent?

$1:3$

$2:6$

$6:18$

Yes! The first ratio is equivalent to the second: we multiply both terms by $2$ .

The first ratio is also equivalent to the third, we multiply by $6$ .

The second ratio is equivalent to the third: multiplication of both terms by $3$ .

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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?

Example

José and Dani have notebooks and pencils. José has $4$ notebooks and $8$ pencils.

The ratio between the notebooks and pencils that Dani has is the same as José's. Dani has $6$ notebooks. We are asked to calculate how many pencils Dani has.

We see that the number of pencils José has is double the number of notebooks he has. Since we already know that the ratio between notebooks and pencils that José and Dani have is identical, we can deduce that Dani has $12$ pencils ( $6$ times $2$ , so that the number of pencils is double the number of notebooks).

Examples and exercises with solutions of Equivalent Ratios

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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Equivalent Ratios

Test yourself on ratio!

How can you tell if they are equivalent ratios?

Example

Examples and exercises with solutions of Equivalent Ratios

Exercise #1

Step-by-Step Solution

Answer

Exercise #2

Step-by-Step Solution

Answer

Exercise #3

Step-by-Step Solution

Answer

Exercise #4

Step-by-Step Solution

Answer

Exercise #5

Step-by-Step Solution

Answer

More Questions

Ratio