**Ratio**:- A ratio is a comparison of two quantities. It is typically expressed as a fraction or in the form “a:b,” where “a” and “b” are two numbers.
- Ratios can be written in different forms. For example, if you have a ratio of 2:3, it can also be expressed as 2/3 or 2 out of 3.

**Proportion**:- A proportion is an equation that states that two ratios are equal. It is typically written as “a:b = c:d” or in fractional form, “a/b = c/d.”
- Proportions are used to solve problems involving unknown values. For example, if you know three values in a proportion, you can use it to find the fourth.

Here are a few examples of how ratios and proportions can be applied:

**1. Scaling Ratios**: Suppose you have a map where 1 inch represents 10 miles. The ratio here is 1:10. You can use this ratio to find the actual distance when given the length on the map.

**2. Mixing Proportions**: In a recipe, you may need to mix ingredients in a specific ratio. For example, a cake recipe might call for 2 parts flour to 1 part sugar. This is a proportion that ensures the right balance of ingredients.

**3. Proportional Relationships**: In finance, you might have a proportion like “Interest earned is directly proportional to the initial investment.” This means that if you double your investment, your interest will also double.

**4. Solving Word Problems**: Many word problems involve ratios and proportions. For instance, finding the time it takes two workers to complete a task when their work rates are in a certain ratio.

To solve problems involving ratios and proportions, you can use cross-multiplication and algebraic methods. When setting up proportions, you ensure that the cross-products of the fractions are equal. This allows you to solve for the unknown values.

In summary, ratios and proportions are fundamental concepts in mathematics that help us compare and relate quantities, whether it’s in everyday tasks, business, or science. They provide a powerful tool for solving a wide range of problems by establishing relationships between different quantities.

Table of Contents

## Ratio and Proportion formula

Ratios and proportions are fundamental concepts in mathematics that involve comparing quantities and setting up equal relationships between ratios. Here are some key formulas and principles related to ratios and proportions:

**1. Ratio Formula:**

- A ratio is typically written as “a:b” or “a to b,” where “a” and “b” are two quantities. The ratio formula is straightforward:

Ratio (a:b) = a / b

**2. Proportion Formula:**

- A proportion is an equation that states that two ratios are equal. It is usually written in the form “a:b = c:d” or “a/b = c/d.” The proportion formula is as follows:

a/b = c/d

You can use this formula to solve for any of the four variables (a, b, c, or d) if you know the other three.

**3. Cross-Multiplication (Method for Solving Proportions):**

- Cross-multiplication is a method for solving proportions. When you have a proportion like a/b = c/d, you can use cross-multiplication as follows:

a * d = b * c

This method helps you find the missing value in a proportion.

**4. Unitary Method:**

- The unitary method is a concept related to proportions. It involves finding the value of one unit to determine the value of the whole quantity. For example, if the price of 3 apples is $6, you can find the price of 1 apple using the unitary method: 6 / 3 = $2 per apple.

**5. Direct Proportion:**

- In a direct proportion, two variables change in the same direction. If one variable increases, the other also increases, and if one decreases, the other decreases. The formula for direct proportion is:

a = k * b

Where “a” and “b” are the two variables, and “k” is the constant of proportionality.

**6. Inverse Proportion:**

- In an inverse proportion, two variables change in opposite directions. If one variable increases, the other decreases, and vice versa. The formula for inverse proportion is:

a * b = k

Where “a” and “b” are the two variables, and “k” is the constant of proportionality.

**7. Means and Extremes in Proportions:**

- In a proportion a/b = c/d, “a” and “d” are called the “extremes,” while “b” and “c” are called the “means.” The product of the means is equal to the product of the extremes:

a * d = b * c

Understanding these formulas and concepts is essential for working with ratios and proportions in various mathematical and real-world applications. They provide a basis for solving problems involving comparisons and relationships between quantities.

## Ratio and Proportion math

Ratio and proportion are fundamental concepts in mathematics that deal with comparing and establishing relationships between quantities. They are often used in various mathematical and real-world problems. Let’s explore these concepts in more detail:

**1. Ratio:**

- A ratio is a way to compare two quantities by dividing one quantity by the other. It is typically expressed as a fraction, a colon, or the word “to.” For example, if you have 3 red marbles and 5 blue marbles, the ratio of red to blue marbles is 3:5 or 3/5.
- Ratios can be simplified by finding their greatest common divisor (GCD) and dividing both parts by it to make the ratio easier to work with.

**2. Proportion:**

- A proportion is an equation that states that two ratios are equal. It’s a way of saying that two sets of quantities are in balance or proportional to each other. For example, if you have a/b = c/d, it’s a proportion.
- To solve problems involving proportions, you can use cross-multiplication. If you have a proportion a/b = c/d, you can cross-multiply to get ad = bc.

**3. Solving Proportional Problems:**

- Proportions can be used to solve a wide range of problems. For example, if you need to find the missing value in a proportion, you can use cross-multiplication to solve for it.
- In word problems, ratios and proportions are often used to find missing values, compare quantities, or scale measurements. Be sure to set up your proportions correctly, ensuring that the units are consistent.

**4. Direct Proportion:**

- In a direct proportion, two quantities change in the same direction. If one quantity increases, the other also increases, and if one decreases, the other decreases. The relationship can be expressed as a/b = c/d, where “a” and “b” are the first set of values, and “c” and “d” are the second set of values.

**5. Inverse Proportion:**

- In an inverse proportion, two quantities change in opposite directions. If one quantity increases, the other decreases, and vice versa. The relationship can be expressed as a * b = k, where “a” and “b” are the two quantities, and “k” is a constant.

**6. Solving Real-World Problems:**

- Ratios and proportions are widely used in real-world scenarios. For example, they can be applied in cooking (adjusting ingredient quantities), map scaling, financial calculations (interest rates), and physics (speed, time, and distance problems).

**7. Unitary Method:**

- The unitary method is a practical application of ratios and proportions. It helps you find the value of one unit and, from there, calculate the value of the entire quantity.

### Direct Proportion

Direct proportion, also known as direct variation, is a mathematical relationship between two quantities in which they change in the same direction. This means that as one quantity increases, the other also increases, and as one decreases, the other decreases. In a direct proportion, the ratio of the two quantities remains constant.

The general form of a direct proportion can be expressed as:

$y=kx$

Where:

- $y$ is the dependent variable.
- $x$ is the independent variable.
- $k$ is the constant of proportionality, which remains the same for all data points.

In a direct proportion, as $x$ increases, $y$ increases, and as $x$ decreases, $y$ decreases. The constant of proportionality $k$ represents the rate of change and is typically a positive value.

Here are some key characteristics of direct proportion:

**Constant Ratio**: The ratio of the two quantities remains the same for all values of $x$ and $y$.**Graphical Representation**: When plotted on a graph, the points lie on a straight line that passes through the origin (0,0).**Real-World Examples**: Direct proportion is commonly observed in various real-world scenarios. For example, as you increase the number of hours you work, your income increases proportionally.**Inverse of Direct Proportion**: Inverse proportion is the opposite concept, where one quantity increases as the other decreases. The relationship is expressed as $xy=k$, where $k$ is a constant.

Here’s an example of a direct proportion:

**Example:** The speed of a car is directly proportional to the distance it travels in a given time. If a car travels 100 miles in 2 hours, what is its speed?

**Solution:** In this case, the distance (100 miles) and the time (2 hours) are in a direct proportion to the speed (which is what we want to find).

We can set up the equation as:

$100=2k$

Now, solve for $k$:

$k=100/2=50$

So, the constant of proportionality $k$ is 50. Therefore, the car’s speed is 50 miles per hour (mph).

This means that for every 1 hour (1 unit of time), the car travels 50 miles (1 unit of distance) in a direct proportion.

Direct proportion is a fundamental concept in mathematics and is used in various fields such as science, economics, and engineering to model relationships between different variables.

### Inverse Proportion

Inverse proportion is a mathematical relationship between two variables in which an increase in one variable leads to a decrease in the other, and vice versa. In other words, as one variable increases, the other variable decreases in a consistent manner. In inverse proportion, the product of the two variables remains constant. This relationship can be expressed using the following formula:

$xy=k$

Where:

- $x$ and $y$ are two variables that are inversely proportional.
- $k$ is a constant value (not equal to zero).

Key characteristics of inverse proportion:

**Product is Constant**: In an inversely proportional relationship, the product of the two variables remains constant. As one variable increases, the other decreases in such a way that their product (xy) stays the same.**Inverse Variation**: Inverse proportion is also known as inverse variation. It is the opposite of direct proportion, where an increase in one variable corresponds to an increase in the other.**Graphical Representation**: When plotted on a graph, inverse proportion forms a hyperbola. The graph never intersects the axes, and the product of the x and y coordinates is a constant value.**Real-World Examples**: Inverse proportion is often encountered in real-world scenarios. For example, the time it takes to complete a task is inversely proportional to the number of people working on it. As the number of people increases, the time required decreases, and vice versa.**Mathematical Representation**: In problems involving inverse proportion, you can use the formula $xy=k$ to find unknown values.

Here’s an example of inverse proportion:

**Example:** Suppose you are traveling at a constant speed, and you want to calculate the time it takes to cover a certain distance. If your speed is inversely proportional to the time taken and you can cover 60 miles in 3 hours, how long will it take to cover 120 miles at the same speed?

**Solution:** In this example, speed and time are inversely proportional, and their product ($speed×time$) remains constant.

- First, find the constant of proportionality ($k$) using the initial values: $speed×time=k$ $60miles/hour×3hours=k$ $180=k$
- Now that you have the constant value ($k$), you can use it to solve for the new time required to cover 120 miles: $speed×time=180$ $60miles/hour×new time=180$ $60×new time=180$

Now, solve for the new time: $new time=60180 =3hours$

So, it will take 3 hours to cover 120 miles at the same speed.

Inverse proportion is a fundamental concept in mathematics and has practical applications in various fields, including physics, engineering, and economics.

### Ratio and Proportion Solving Techniques

Solving problems involving ratios and proportions involves setting up and solving equations that describe the relationships between quantities. Here are some techniques for solving ratio and proportion problems:

**1. Identifying the Relationship:**

- Begin by identifying the nature of the relationship. Is it a direct proportion, inverse proportion, or a simple ratio comparison? This will guide your approach to solving the problem.

**2. Setting Up Proportions:**

- When dealing with proportions, set up an equation where two ratios are equal. For example, if you have a problem involving two ratios a/b and c/d, you would set up the proportion a/b = c/d.

**3. Cross-Multiplication:**

- To solve proportions, use cross-multiplication. Cross-multiplication means multiplying the numerator of one ratio by the denominator of the other ratio and setting them equal to each other. For the proportion a/b = c/d, you cross-multiply to get ad = bc.

**4. Solving for the Unknown:**

- Once you have a proportion set up, you can solve for the unknown value. For instance, if you have a proportion like 2/5 = 4/x, you can cross-multiply to get 2x = 20 and then solve for x by dividing both sides by 2, resulting in x = 10.

**5. Units Consistency:**

- Ensure that units are consistent in your ratios and proportions. If you’re dealing with measurements, make sure that all quantities are in the same units before setting up the proportion.

**6. Check for Extraneous Solutions:**

- After solving for the unknown value, it’s a good practice to check your solution in the original problem to make sure it makes sense and is not an extraneous solution.

**7. Scaling and Measurements:**

- In scaling problems, if you’re asked to scale up or down, use ratios to determine the new values based on the given scale factor. For example, if a map scale is 1:10, you can use this to find real-world distances by scaling up.

**8. Word Problems:**

- Many ratio and proportion problems come in the form of word problems. Read the problem carefully, identify the known and unknown quantities, and translate the information into mathematical expressions.

**9. Practice and Familiarity:**

- Like many math skills, solving ratio and proportion problems becomes easier with practice. Try various problems to become more familiar with different scenarios and techniques.

**10. Consistency and Precision:**

- When working with proportions, be consistent in the order of your ratios. For example, if you set up a proportion as a/b = c/d, make sure you always multiply diagonally in that order (ad = bc) when using the cross-multiplication method.

Remember that ratio and proportion problems are commonly used in a wide range of fields, so developing proficiency in solving them can be highly beneficial in practical situations. As you encounter more problems and apply these techniques, your problem-solving skills will improve.

### Ratio and Proportion short methods with examples

**1. Reducing Ratios to Simplest Form:**

- When given a ratio, simplify it to its simplest form by dividing both parts by their greatest common divisor (GCD). This makes calculations easier.
- Example: Simplify the ratio 12:18.
- GCD of 12 and 18 is 6.
- Divide both parts by 6: (12/6):(18/6) = 2:3.

**2. Solving Direct Proportion Problems:**

- In direct proportion problems, if one quantity is doubled, the other is also doubled.
- Example: If 4 pens cost $8, how much do 8 pens cost?
- Set up the proportion: 4/8 = 8/x.
- Cross-multiply: 4x = 8 * 8.
- Solve for x: x = (8 * 8)/4 = $16.

**3. Solving Inverse Proportion Problems:**

- In inverse proportion, when one quantity doubles, the other halves.
- Example: If 5 workers can finish a job in 10 hours, how many workers are needed to finish it in 5 hours?
- Set up the proportion: 5 * 10 = x * 5.
- Simplify: 50 = 5x.
- Solve for x: x = 50/5 = 10 workers.

**4. Unitary Method:**

- The unitary method involves finding the value of one unit and then using it to find the value of the whole quantity.
- Example: If 5 apples cost $10, how much does 1 apple cost?
- Set up the proportion: 5/10 = 1/x.
- Cross-multiply: 5x = 10 * 1.
- Solve for x: x = 10/5 = $2 per apple.

**5. Scaling and Proportionality:**

- When scaling up or down, use the scale factor to find the new values.
- Example: A map has a scale of 1:25,000. If a river is 4 cm on the map, what is its actual length?
- Set up the proportion: 1/25,000 = 4/x.
- Cross-multiply: 1x = 25,000 * 4.
- Solve for x: x = (25,000 * 4)/1 = 100,000 cm = 1,000 meters.

**6. Averages and Ratios:**

- To find an average based on a ratio of parts, use the concept that the average is the middle value when the values are arranged in ascending order.
- Example: The average of three numbers is 5. If the ratio of the first to the second number is 2:3, what is the third number?
- Set up the proportion: 2/3 = x/5 (since 5 is the average).
- Cross-multiply: 2 * 5 = 3x.
- Solve for x: x = (2 * 5)/3 = 10/3.

These short methods and examples should help you quickly and effectively solve various ratio and proportion problems by simplifying the process and using proportional reasoning.

## Ratio and Proportion math class 6 standard question with solution

Here’s a class 6 standard math question involving ratios and proportions, along with its solution:

**Question:** Sarah and Emma are sharing some candies in the ratio of 3:5. If Sarah has 12 candies, how many candies does Emma have?

**Solution:** To solve this problem, you can use the concept of proportions. The given ratio is 3:5, which means for every 3 candies Sarah has, Emma has 5 candies.

- Set up a proportion with Sarah’s candies (12) and an unknown quantity for Emma’s candies (let’s call it “x”):3/5 = 12/x
- Cross-multiply to solve for x:3x = 5 * 12
- Solve for x by dividing both sides by 3:x = (5 * 12) / 3
- Calculate the value of x:x = 60 / 3
x = 20

So, Emma has 20 candies.

**Question:** In a school, the ratio of boys to girls is 2:3. If there are 50 boys, how many girls are there in the school?

**Solution:** To solve this problem, you can use the concept of ratios and proportions. The given ratio is 2:3, which means for every 2 boys, there are 3 girls.

- Set up a proportion with the number of boys (50) and an unknown quantity for the number of girls (let’s call it “x”):2/3 = 50/x
- Cross-multiply to solve for x:2x = 3 * 50
- Solve for x by dividing both sides by 2:x = (3 * 50) / 2
- Calculate the value of x:x = 150 / 2
x = 75

So, there are 75 girls in the school.

**Question 1:** The ratio of the number of red marbles to blue marbles in a bag is 4:7. If there are 28 blue marbles, how many red marbles are there?

**Solution 1:** To solve this problem, use the given ratio and the number of blue marbles to find the number of red marbles.

- Set up a proportion with the given ratio (4:7) and the number of blue marbles (28) and an unknown quantity for the number of red marbles (let’s call it “x”):4/7 = x/28
- Cross-multiply to solve for x:4 * 28 = 7 * x
- Solve for x by dividing both sides by 7:x = (4 * 28) / 7
- Calculate the value of x:x = 112 / 7
x = 16

So, there are 16 red marbles.

**Question 2:** A recipe for making fruit punch calls for 2 cups of orange juice and 3 cups of apple juice. If you want to make 10 cups of fruit punch, how many cups of each juice do you need?

**Solution 2:** To determine how many cups of each juice you need to make 10 cups of fruit punch, you can use the concept of ratios.

- Calculate the ratio of orange juice to apple juice, which is 2:3.
- Set up a proportion to find how many cups of orange juice (x) and apple juice (y) are needed to make 10 cups of fruit punch:2/3 = x/10
- Cross-multiply to solve for x:2 * 10 = 3 * x
- Solve for x by dividing both sides by 3:x = (2 * 10) / 3
- Calculate the value of x:x = 20 / 3
x ≈ 6.67 cups (rounded to two decimal places)

So, you need approximately 6.67 cups of orange juice and 10 – 6.67 = 3.33 cups of apple juice to make 10 cups of fruit punch.

**Question 1:** Tom and Jerry are sharing a sum of money in the ratio 3:5. If Tom receives $24, how much does Jerry receive?

**Solution 1:** To find how much Jerry receives, use the given ratio and the amount Tom receives:

- Set up a proportion with Tom’s share (3) and Jerry’s share (5), and the known amount for Tom ($24):3/5 = 24/x
- Cross-multiply to solve for x:3x = 5 * 24
- Solve for x by dividing both sides by 3:x = (5 * 24) / 3
- Calculate the value of x:x = 120 / 3
x = $40

Jerry receives $40.

**Question 2:** The length and width of a rectangle are in the ratio 4:7. If the length is 28 cm, what is the width of the rectangle?

**Solution 2:** To find the width of the rectangle, use the given ratio and the length of the rectangle:

- Set up a proportion with the length (4) and width (7) and the known length (28 cm):4/7 = 28/x
- Cross-multiply to solve for x:4x = 7 * 28
- Solve for x by dividing both sides by 4:x = (7 * 28) / 4
- Calculate the value of x:x = 196 / 4
x = 49 cm

The width of the rectangle is 49 cm.

**Question 3:** A recipe calls for 3 cups of sugar and 5 cups of flour. If you want to make a larger batch using 9 cups of flour, how many cups of sugar do you need?

**Solution 3:** To determine how much sugar you need for 9 cups of flour, you can use the given ratio from the original recipe:

- Calculate the ratio of sugar to flour, which is 3:5.
- Set up a proportion to find how many cups of sugar (x) are needed for 9 cups of flour:3/5 = x/9
- Cross-multiply to solve for x:3 * 9 = 5 * x
- Solve for x by dividing both sides by 5:x = (3 * 9) / 5
- Calculate the value of x:x = 27 / 5
x = 5.4 cups

You need 5.4 cups of sugar for 9 cups of flour.

### Some More from ncert book of class 6

**Question 1:** In a class of 30 students, the ratio of boys to girls is 2:3. How many girls are there in the class?

**Solution 1:** To find the number of girls, set up a proportion:

2/3 = x/30

Cross-multiply:

2 * 30 = 3 * x

60 = 3x

Divide by 3:

x = 60 / 3

x = 20

So, there are 20 girls in the class.

**Question 2:** A recipe for making pancakes requires 1 cup of flour and 2 eggs. If you want to make 12 pancakes, how many cups of flour and eggs do you need?

**Solution 2:** To find the quantities needed, set up two proportions, one for flour and one for eggs:

For flour: 1/2 = x/12

Cross-multiply:

1 * 12 = 2 * x

12 = 2x

Divide by 2:

x = 12 / 2

x = 6 cups of flour

For eggs: 1/2 = y/12

Cross-multiply:

1 * 12 = 2 * y

12 = 2y

Divide by 2:

y = 12 / 2

y = 6 eggs

So, you need 6 cups of flour and 6 eggs to make 12 pancakes.

### Ratio and Proportion math Exercise

**Exercise 1: Ratios**

- The ratio of apples to oranges in a basket is 3:4. If there are 15 apples, how many oranges are there?
- A mixture of red and blue marbles is in the ratio 2:5. If there are 28 red marbles, how many blue marbles are there?
- In a garden, the ratio of roses to tulips is 5:3. If there are 24 tulips, how many roses are there?

**Exercise 2: Proportions**

- A car travels 180 miles in 4 hours. How far will it travel in 6 hours at the same speed?
- If 8 books cost $64, how much do 12 books cost at the same price per book?
- The ratio of men to women in a club is 3:5. If there are 48 women, how many men are there?

**Exercise 3: Mixing and Averaging**

- To make orange juice, you need to mix 3 cups of orange concentrate with 5 cups of water. If you want to make 15 cups of juice, how much concentrate and water do you need?
- The average score of a student in two exams is 85. If the scores in the first exam and second exam are in the ratio 3:4, what did the student score in each exam?
- A fruit vendor mixes apples and pears in the ratio 2:3. If there are 24 apples, how many pears are there in the mixture?

**Exercise 4: Scaling**

- On a map, 1 inch represents 10 miles. If two cities are 3 inches apart on the map, what is the actual distance between them?
- A model car is 1/24th the size of a real car. If a real car is 18 feet long, how long is the model car?
- A drawing is enlarged so that its sides are 5 times longer. If the original area of the drawing is 30 square inches, what is the area of the enlarged drawing?

Feel free to attempt these exercises and then check your answers. If you have any questions or need solutions to specific problems, please let me know!

### 50 Ratio and Proportion fill in the blanks Question and Answer

**Questions:**

- The __________ of boys to girls in a class is 2:3.
- If the ratio of sugar to flour in a recipe is 1:2, and you need 3 cups of flour, you’ll need __________ cups of sugar.
- In a triangle, the ratio of the lengths of the sides is 3:4:5, so the sides could be 3x, 4x, and __________.
- If the ratio of red balls to blue balls is 5:3, and there are 15 red balls, there are __________ blue balls.
- In a mixture of juice and water, the ratio is 2:7. If there are 45 liters of the mixture, there are __________ liters of water.
- The ratio of adults to children on a bus is 4:6. If there are 12 adults, there are __________ children.
- In a fruit salad, the ratio of apples to bananas is 3:2. If there are 12 bananas, there are __________ apples.
- The ratio of the lengths of two sides of a rectangle is 3:4. If one side is 9 inches long, the other side is __________ inches.
- If the ratio of students to teachers is 25:1, and there are 100 students, there are __________ teachers.
- The __________ of 8 to 12 can be simplified to 2:3.
- In a recipe, the ratio of tablespoons of sugar to tablespoons of flour is 1:4. If you use 2 tablespoons of sugar, you’ll need __________ tablespoons of flour.
- The ratio of nickels to dimes in a piggy bank is 3:2. If there are 15 dimes, there are __________ nickels.
- If you have 4 red marbles and 6 green marbles, the ratio of red to green marbles is __________.
- A recipe for pancakes calls for 1 cup of milk and 2 cups of flour. If you want to make 8 pancakes, you’ll need __________ cups of flour.
- The ratio of the area of a small square to a large square is 1:4. If the area of the small square is 16 square inches, the area of the large square is __________ square inches.
- The ratio of boys to girls in a class is 2:5. If there are 14 boys, there are __________ girls.
- The __________ of 6 to 18 can be simplified to 1:3.
- In a school, the ratio of teachers to students is 1:30. If there are 60 students, there are __________ teachers.
- If the ratio of lemons to oranges is 4:7, and there are 28 lemons, there are __________ oranges.
- The ratio of the ages of two sisters is 4:7. If the older sister is 28 years old, the younger sister is __________ years old.
- In a bag, the ratio of red marbles to blue marbles is 3:2. If there are 15 red marbles, there are __________ blue marbles.
- The ratio of pennies to nickels in a jar is 1:6. If there are 24 nickels, there are __________ pennies.
- A rectangle has sides in the ratio 3:5. If the length of the rectangle is 15 cm, the width is __________ cm.
- The ratio of men to women at a party is 2:3. If there are 30 men, there are __________ women.
- A jar contains 60 marbles, and the ratio of red marbles to blue marbles is 5:7. There are __________ red marbles.
- If the ratio of salt to water in a solution is 1:4, and you have 15 cups of water, you need __________ cups of salt.
- The ratio of the ages of two brothers is 2:3. If the younger brother is 12 years old, the older brother is __________ years old.
- In a recipe, the ratio of eggs to cups of flour is 3:2. If you use 6 cups of flour, you’ll need __________ eggs.
- The ratio of the lengths of the sides of a triangle is 2:3:4. If the shortest side is 6 inches long, the longest side is __________ inches.
- In a park, the ratio of ducks to swans is 3:2. If there are 12 swans, there are __________ ducks.
- The ratio of dimes to quarters in a piggy bank is 5:4. If there are 20 dimes, there are __________ quarters.
- A map has a scale of 1 inch to 10 miles. If two cities are 2.5 inches apart on the map, the actual distance between them is __________ miles.
- In a store, the ratio of the prices of shirts to pants is 3:5. If a shirt costs $18, a pair of pants costs __________.
- The ratio of cats to dogs in a neighborhood is 2:7. If there are 14 cats, there are __________ dogs.
- A recipe calls for 2 cups of sugar and 5 cups of flour. If you want to make 20 muffins, you’ll need __________ cups of sugar.
- The ratio of triangles to circles in a drawing is 2:3. If there are 15 circles, there are __________ triangles.
- If the ratio of lemons to limes is 3:4, and there are 9 limes, there are __________ lemons.
- The ratio of apples to oranges in a fruit basket is 5:3. If there are 30 apples, there are __________ oranges.
- A bag of candies contains 48 candies, and the ratio of red candies to green candies is 4:3. There are __________ green candies.
- The ratio of the ages of two cousins is 3:5. If one cousin is 15 years old, the other cousin is __________ years old.
- In a classroom, the ratio of boys to girls is 4:6. If there are 18 girls, there are __________ boys.
- A car travels 200 miles in 4 hours. How far will it travel in 6 hours at the same speed? __________ miles.
- The ratio of adults to children in a park is 3:7. If there are 42 adults, there are __________ children.
- If the ratio of students to teachers is 20:1, and there are 80 students, there are __________ teachers.
- A recipe for making cookies requires 1 cup of butter and 2 cups of chocolate chips. If you want to make 24 cookies, you’ll need __________ cups of butter.
- The ratio of oranges to apples in a fruit basket is 3:4. If there are 24 oranges, there are __________ apples.
- In a bag, the ratio of green marbles to blue marbles is 5:2. If there are 10 green marbles, there are __________ blue marbles.
- The ratio of nickels to quarters in a piggy bank is 1:4. If there are 16 quarters, there are __________ nickels.
- A rectangle has sides in the ratio 2:7. If the width of the rectangle is 14 cm, the length is __________ cm.
- The ratio of men to women at a party is 3:5. If there are 45 men, there are __________ women.

**Answers:**

- ratio
- 6
- 5x
- 9
- 14 liters
- 18 children
- 18 apples
- 12 inches
- 750 teachers
- ratio
- 8 tablespoons
- 10 nickels
- 2:6
- 4 cups
- 64 square inches
- 35 girls
- ratio
- 2 teachers
- 49 oranges
- 49 years old
- 10 blue marbles
- 4 pennies
- 25 cm
- 45 women
- 25 red marbles
- 3 cups
- 18 years old
- 9 eggs
- 12 inches
- 18 ducks
- 16 quarters
- 25 miles
- $30
- 49 dogs
- 8 cups
- 10 triangles
- 9 lemons
- 50 oranges
- 36 green candies
- 25 years old
- 12 boys
- 300 miles
- 98 children
- 4 teachers
- 12 cups
- 32 apples
- 4 blue marbles
- 64 nickels
- 49 cm
- 75 women

**Here are 10 more fill-in-the-blank questions related to ratio and proportion, along with their answers:**

**Questions:**

- The __________ of students to textbooks is 3:4. If there are 24 students, there are __________ textbooks.
- In a recipe for making lemonade, the ratio of lemon juice to water is 1:8. If you use 2 cups of lemon juice, you’ll need __________ cups of water.
- A bag of marbles contains 60 marbles, and the ratio of blue marbles to green marbles is 2:5. There are __________ blue marbles.
- The ratio of the number of employees to computers in an office is 5:8. If there are 40 computers, there are __________ employees.
- In a drawing, the ratio of squares to circles is 3:4. If there are 21 circles, there are __________ squares.
- If the ratio of cats to dogs in a neighborhood is 4:3, and there are 12 cats, there are __________ dogs.
- The ratio of girls to boys in a class is 5:7. If there are 35 girls, there are __________ boys.
- A recipe for making cookies requires 2 cups of sugar and 3 cups of flour. If you want to make 36 cookies, you’ll need __________ cups of sugar.
- The ratio of nickels to dimes in a piggy bank is 2:3. If there are 24 dimes, there are __________ nickels.
- In a school, the ratio of students to teachers is 30:1. If there are 90 students, there are __________ teachers.

**Answers:**

- ratio, 32 textbooks
- 16 cups
- 20 blue marbles
- 25 employees
- 16 squares
- 9 dogs
- 49 boys
- 24 cups
- 16 nickels
- 3 teachers

** Here are 10 more fill-in-the-blank questions related to ratio and proportion, along with their answers:**

**Questions:**

- The __________ of boys to girls in a school is 3:5. If there are 60 boys, there are __________ girls.
- In a fruit basket, the ratio of oranges to apples is 2:3. If there are 18 apples, there are __________ oranges.
- A recipe for making muffins calls for 1 cup of milk and 2 cups of flour. If you want to make 12 muffins, you’ll need __________ cups of milk.
- The ratio of red marbles to yellow marbles in a bag is 3:4. If there are 24 red marbles, there are __________ yellow marbles.
- In a garden, the ratio of roses to daisies is 5:2. If there are 10 daisies, there are __________ roses.
- If the ratio of men to women in a club is 4:3, and there are 12 men, there are __________ women.
- The ratio of pennies to quarters in a jar is 2:5. If there are 35 pennies, there are __________ quarters.
- A map has a scale of 1 inch to 5 miles. If two cities are 2.5 inches apart on the map, the actual distance between them is __________ miles.
- The ratio of the ages of two brothers is 5:7. If the younger brother is 14 years old, the older brother is __________ years old.
- In a jar of marbles, the ratio of green marbles to blue marbles is 4:5. If there are 36 blue marbles, there are __________ green marbles.

**Answers:**

- ratio, 100 girls
- 27 oranges
- 6 cups
- 32 yellow marbles
- 25 roses
- 9 women
- 70 quarters
- 12.5 miles
- 21 years old
- 32 green marbles

## Aptitude ratio and proportion

Ratio and proportion are essential concepts in aptitude tests and mathematical aptitude. They often appear in various aptitude and competitive exams to assess your ability to understand and solve problems involving relative quantities and relationships. Here’s a breakdown of how to approach ratio and proportion questions in aptitude tests:

**1. Understand the Basics:**

- Ensure you have a solid understanding of the basic concepts of ratios and proportions. A ratio is a comparison of two or more quantities, while a proportion is an equation that states two ratios are equal.

**2. Set Up Ratios:**

- Read the problem carefully and identify the quantities involved. Create ratios based on the information given in the question.

**3. Simplify Ratios:**

- If possible, simplify ratios by dividing all the terms by their greatest common divisor (GCD) to make calculations easier.

**4. Solve Direct Proportion Problems:**

- In direct proportion problems, as one quantity increases, the other increases proportionally.
- If the problem involves direct proportion, use the equation: $a/b=c/d$, where $a$ and $b$ are two quantities, and $c$ and $d$ are the other two quantities in proportion.

**5. Solve Inverse Proportion Problems:**

- In inverse proportion problems, as one quantity increases, the other decreases.
- If the problem involves inverse proportion, use the equation: $a/b=d/c$, where $a$ and $b$ are two quantities, and $c$ and $d$ are the other two quantities in inverse proportion.

**6. Use the Unitary Method:**

- The unitary method involves finding the value of one unit and using it to find the value of the whole quantity. This can be especially useful in solving ratio and proportion problems.

**7. Cross-Multiply and Solve:**

- In proportion problems, cross-multiply to solve for the unknown variable. This helps you establish an equation and solve for the missing value.

**8. Check for Consistency:**

- After solving a problem, check if the answer is consistent with the problem statement. Ensure the units and quantities make sense.

**9. Practice a Variety of Problems:**

- Practice solving different types of ratio and proportion problems to improve your aptitude skills. Work on problems involving money, distance, time, mixing solutions, and more.

**10. Work Efficiently:**

- Aptitude tests are often timed, so practice solving problems efficiently and quickly. Focus on eliminating answer choices that are clearly incorrect.

**11. Understand the Context:**

- Sometimes, understanding the context of a problem is crucial. Ensure you know what the problem is asking for, whether it’s a missing value, a comparison, or a change in proportion.

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