Table of Contents

## Direct Variation

Direct Variation is an essential concept in Ratios and Proportions. We are providing the important formulas, explanations, and definitions here. Direct Proportion is one type of Ratios and Proportions. Go through the below sections to know the various details like Formulas Types, Definitions, Solved Questions, etc.

## Importance of Direct Variation

Direct Proportion or Variation is the relationship between two different variables in which one variable is the constant of another variable.

If the variable is directly proportional to another variable, then we define that one of the variables changes with the same ratio as the other increases.

Also, if one variable decreases, then the ratio of the other variable decreases.

**For example:** If you save a huge amount of money every month, then you will increase your savings by a definite amount. This is called the constant of variation.

Because there was a constant rate of increase. The constant rate of increase or decrease is called the “constant of variation”.

Also, you will know more details regarding the direct variation in the upcoming sections. We also provide some tips, tricks, shortcuts, books, and solved questions.

### Direct Variation Definition

Two quantities or equations are said to be variant if there is a consistent increase or decrease in quantity causes an increase or decrease in other quantity. In simple terms, Direct Variation is a relation between two numbers such that one number should be a constant multiple for another number.

Mathematics generally deal with constant quantities or variable quantities.

If a value changes with different situations, it is called a variable and if a value does not change with different situations, it is called a constant. Consider an example. 22/7, 4, etc., are examples of constants. The population of a city/ town, speed of a car, etc., are examples of variables.

When a value of relative variable changes, there will be a change in the value of the variable, this is called variation.

The Direct Variation is like a simple relation between two variables. Consider an equation that says y varies directly with x if y=kx.

k is a constant called constant of proportionality or constant of variation.

It means that x is directly proportional to y, it implies if x increases, y increases, and if x decreases, y decreases. The ratio also will be the same.

So considering the above statements, the graph of the above direct variation equation is a straight line.

### Books for Direct Variation

- New Mathsahead: Book 7 (Rev. Edn.)
- New Learning Composite Mathematics 8 by S.K. Gupta & Anubhuti Gangal
- Maths Wiz Book by S.K. Gupta & Anubhuti Gangal
- Direct Methods in the Calculus of Variations by Enrico Giusti
- CALCULUS OF VARIATIONS WITH APPLICATIONS by A. S. GUPTA
- Algebra: A Step-by-Step Guide by Jennifer Dagley
- The Calculus of Variations by N.I. Akhiezer
- CliffsNotes Algebra I Quick Review, 2nd Edition by Jerry Bobrow

### How to find the Direct Variation?

Here are a few steps you need to follow in order to solve a direct variation problem

Step 1: Note down the formula for direct variation.

Step 2: In order to get variables, substitute the given values.

Step 3: Now, solve to get the constant of variation.

Step 4: Write the equation which satisfies x and y.

### Solved Questions on Direct Variation

**Question 1:** A wooden box is made which is directly proportional to the no of wooden blocks. 120 wooden blocks are needed to make 30 boxes. How many wooden blocks are needed to prepare a box?

**Solution:**

In the above-given problem,

No of wooden blocks needed for 30 boxes = a= 120

Number of boxes = b = 30

No of wooden blocks needed for a box = y

The direct variation formula is

a=y*b

120=y*30

y=120/30

y=4

No of wooden blocks needed for a box = 4

**Question 2:** Given that a varies directly as b, with x constant of variation y=1/3, find a when b=12

**Solution:**

According to the given equation,

a=1/3b

Substitute the given b value,

a=1/3.12

a=4

**Question 3:** Suppose a varies directly as b and a=30 when b=6. What is the value of a when b=100?

**Solution:**

From the direct variation equation

b=ka

Substitute the given a and b values in the equation, and solve them for “k”

30=k*6

k=5

The equation is a=5b. Now substitute b=100 and find a

a=5.100

a=500

**Question 4:** Suppose that a car runs at a speed constantly and takes 3 hours to cover a distance of 180 km. How much time does the car take to cover a distance of 100km?

**Solution:**

Let T be the time taken to run the total distance.

Let S be the distance.

Suppose V is the speed of the car.

As per the Direct Variation equation S=kT where k is the constant

From the given question

S=180, T=3

Therefore, 180 = k*3 = 180/3 = 60

So, the constant speed of the car = 60km/hr

For 100km distance

S=kT

100=60*T

T=100/60=5/3hours=1 hour 40 mins

Therefore, the car takes 1 hour 40 mins to cover a distance of 100km.

**Question 5:** If X varies directly as Y and the value of X is 60 and Y is 40, find the equation that determines the direct variation of X and Y?

**Solution:**

As X varies directly with Y, the ratio of X and Y is constant for any value of X and Y.

So, constant V=X/Y=60/40=3/2

Therefore, the equation that determines the direct proportion of X and Y is X=3/2Y.

### Preparation Tips

- Attend as many as challenging questions you can.
- Solve most of the previously asked questions to become perfect.
- Know your strengths and weaknesses.
- Improve your problem-solving techniques.
- Know the mistakes you make beforehand and rectify them.
- Have good sleep and rest before the exam.
- Build self-confidence and ease of solving the problems.

The given Direct Variation will help you for your better preparation. Stay in touch with our website Learcbse.in to check the preparation tips, syllabus, various mathematical concepts, etc.

Candidates have to note that we are providing various information for your reference. Click on the bookmark button to get the latest updates.