Algebra: What Is Direct Proportion
A person went to a super market and purchased some items and calculated the bill. After that he added few more items to the list and observed that there is an increasing of the total cost in the latest bill. Here the total cost is getting increased with the increase of the total number of items. Here we say the total cost and the number of items are in the direct proportion.
Mathematically the direct proportion is defined as follows.
Two variables a and b are said to be in the direct proportion if both of them increase together such that the ratio of corresponding values remains constant. i.e. a and b are in direction proportion in only the following cases.
i) a increases when b increases
ii) b decreases when a decreases.
So, if a and b are in direct proportion their ratio is constant.
i.e., a/b = k
If a1 and a2 are any two values of a and b1 and b2are any two values of b then
a1/b1 = k and a2/b2 =k
From these two equations, we get
Learning Direct Proportion And Inverse Proportion Equations And Graphs
In math functions, we learn about direct proportion and inverse proportion. Direct proportion and inverse proportion are familiar, and many people use them in their daily lives, such as when shopping. If you cant calculate direct proportion and inverse proportion, you will be in trouble in everyday life. So, lets understand the concept of direct proportion and inverse proportion.
Direct proportion is that when the value doubles, the number you want to find also doubles. On the other hand, inverse proportion means that when the value is doubled, the number you want to find is halved.
We also learn about graphs in direct proportion and inverse proportion at the same time. So, lets understand the concept of coordinates so that we can create direct proportional and inverse proportional equations from graphs.
Distinguish whether it is direct or inverse proportion. After that, you will be able to find and calculate the constant of proportionality to get the answer.
Direct Proportion Practice Problems
Use the following direct proportion problems to test your knowledge on this topic. Select one of the options and check it to verify that you selected the correct answer. Use the solved examples above if you have a problem with these exercises.
If 10 melons cost 5 dollars, how much do 8 melons cost?
Choose an answer
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Solved Examples Inverse Proportion
Q.1. A brick wall is being cemented by 12 men. The work is completed in 18 days. If 18 men do the same work, how long will it take to complete the work?Ans: Let the number of days to complete the work \ days be \.The more men, the less will be the time taken.Therefore, the two quantities vary inversely.
|Number of men \\)|
So, \\\\Therefore, the time taken is \ days.
Q.2. If 40 workers can finish a job in 15 days, how many workers should be employed if the job is to be finished in 8 days.Ans: Let the number of workers to be employed to finish the job in \ days be \. The more the number of workers, the less will be the time taken.Therefore, the two quantities vary inversely.
|Number of workers \\)|
|Time the rice lasts \\)||\||\|
The more the people, the less the time the rice will last. Hence, the number of people and time the rice will last vary inversely.We therefore have,So, \\\\Therefore, the rice will last in \ days.
Direct Proportion Symbol And Constant Of Proportionality
The symbol for direct proportional is . Two quantities existing in direct proportion can be expressed as
T = 15 hours
Q. 3: The scale of a map is given as 1:20000000. Two cities are 4cm apart on the map. Find the actual distance between them.
Solution: Map distance is 4 cm.
Let the actual distance be x cm, then 1:20000000 = 4:x.
1/20000000 = 4/x
x = 80000000 cm = 800 km
For a detailed discussion on the concept of direct proportion and developing a relationship between two quantities based on direct proportion, inverse proportion and other topics download BYJUS The Learning App from Google Play Store and watch interactive videos. Also, take free tests to practice for exams.
Put your understanding of this concept to test by answering a few MCQs. Click Start Quiz to begin!
Select the correct answer and click on the Finish buttonCheck your score and answers at the end of the quiz
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Number Of Points Scored And Goals Made
The number of points scored and the number of goals made by a particular soccer team, both entities are directly proportional to each other. If a player scores two goals, his/her team earns two points. Similarly, if the players score four goals, the points earned by their team increases by two, and so on.
How Are Ratio And Proportion Used In Daily Life
Ratios and proportions are used on a daily basis. Ratios and proportions are used in business transactions when dealing with money, comparing quantities for the price while shopping, etc. For example, a business might have a ratio for the amount of profit earned per sale of a certain product such as $5:1, which says that the business gains $2.50 for each sale.
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What Does Direct Proportion Mean In Physics
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Then, what is direct proportion in physics?
Definition: Direct proportion is the relationship between two variables when their ratio is equal to a constant value.
Additionally, what is the example of direct proportion? Other examples of direct proportion are: the cost of apples at $5 per kilogram. total wages earned at $20 per hour. amount of flour needed to make muffins
Furthermore, what does proportion mean in physics?
Proportionality. In physics, we often talk about proportionality. This is a relationship between two quantities where they increase or decrease at the same rate. In other words, when quantity A changes by a certain factor, quantity B will change by the same factor.
What is the difference between direct and indirect proportion?
In a direct proportion, the ratio between matching quantities stays the same if they are divided. In an indirect proportion, as one quantity increases, the other decreases.
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How To Solve Direct Proportion Problems
To solve direct proportion word problems, follow the steps given below:
- Identify the two quantities which vary in the given problem.
- Make sure that the variation is directly proportional.
- Form an equation in terms of y = kx and find the value of k base on the given values of x and y.
- Find the unknown value by putting the values of x and the known variable.
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Direct Proportion: Definition Symbol Examples & More
A proportion tells that two ratios are equal. Four numbers are said to be in proportion if the ratio of the first two numbers is equal to the ratio of the last two numbers. There are two types of proportion, Direct Proportion and indirect/inverse proportion.
Two quantities are said to be in direct proportion if they increase or decrease together so that the ratio of their corresponding values remains constant. When the value of one quantity increases concerning the decrease in other or vice-versa, we call it inversely proportional. The symbol to represent the proportionality is \.
In this article, we have provided complete details about direct proportion, symbols, examples, etc. Continue reading to know more.
Age And Height Of A Person
The age and height of a person tend to maintain a direct proportionality for the first few years of his/her life. With an increase in age, a significant and proportionate increase in the height of a person can be observed easily however, the reverse is not possible as the age or height of a person cannot be reversed.
Goods Manufactured At A Factory Per Hour
Let us say that a goods manufacturing industry produces 25 articles in an hour, then the number of goods that can be manufactured in 2 hours is definitely equal to 50. Similarly, 100 articles are produced in 4 hours and so on. This example clearly establishes a directly proportional relationship between the number of articles manufactured and the time taken by the industry for production.
See The Word Linear Think Line
The relationship between a and F in the previous example is linear. When you see the word linear in physics, think line. A graph of two variables which are directly proportional to one another will always be a straight line through the origin.
In this case, we can see that a plot of a against F is indeed a straight line through the origin, and that a doubles when F doubles . This is further confirmation of the fact that a F.
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The Concept Of Coordinates: $x$ Axis $y$ Axis And Origin
After learning the definition of functions, to understand direct proportion and inverse proportion, we must understand the concept of coordinates. A graph represents the position of a point and the shape of a line.
In a graph, the horizontal axis is called the $x$ axis. On the other hand, the vertical axis is called the $y$ axis. The intersection point of the $x$ and $y$ axes is called the origin.
When we use graphs in mathematics, we write points. It is the coordinates that indicate where the point is located in the graph. For example, in the following graph, the coordinate of the point P is $$.
In the coordinates of $P$, 3 is called the $x$ coordinate, and -2 is called the $y$ coordinate. In any case, the points that exist in the graph are called coordinates.
Number Of Visitors And Earnings Of A Restaurant
The total money earned by a particular restaurant tends to vary directly with respect to the number of customers or visitors. If the number of customers visiting a restaurant increases, the sales of the restaurant tend to shoot, thereby increasing the money earned by it. Similarly, when the customer count at the same restaurant reduces, the sales drop low and comparatively less money is earned. The ratio between the two entities remains constant as the change in values of the number of visitors and the money earned is proportional to each other.
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What Is Proportion In Math
Proportion is a mathematical comparison between two numbers. According to proportion, if two sets of given numbers are increasing or decreasing in the same ratio, then the ratios are said to be directly proportional to each other. Proportions are denoted using the symbol :: or =. For example, 2:5 :: 4:8 or 2/5 = 4/8. Here, 2 and 8 are the extremes, while 5 and 4 are the means.
Earning Of A Worker Per Day
Suppose, a worker is paid 500 rupees for one day work. This means that the wages earned in two days are equal to 1000 rupees. Similarly, the worker tends to earn 2000 rupees for four days of work and so on. One can easily observe the pattern and relationship between the number of days and the amount of money earned. With an increase in the number of days, the amount of money earned increases. This verifies the application of direct proportion in real life.
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Direct Variation Example Problems
1. In one of the real situations of direct proportion examples, a bus travels 150 km in 5 hours. What is the time taken by the bus to travel 700 km?
Distance travelled and time taken are directly proportional to each other.
In the given question, the distance travelled in case 1 is x1 = 150 km
The distance travelled in case 2 is x2 = 700 km
The time taken in case 1 is y1 = 5 hours
Time taken in case 2 is y2 =?
The proportionality relationship can be stated as:
\ = \
\ = \
y2 = \ × 5
y2 = 23.33
So, the time taken by the bus to travel 700 km is 23.33 hrs
2. Given that a and b are directly proportional to each other, complete the table given below.
From the table x1 = 4, y1 = 6, x2 = 5, x3 = 12, x4 = 6
y2 = ? y3 = ? y4 = ?
Case 1: To find y2
\ = \
\ = \
y2 = \ × 6
y2 = 7.5
Case 3: To find y4
\ = \
\ = \
y4 = \ × 6
Y4 = 9
So, the completed table is as below:
3. Sumanth has Rs. 400/- with him. If he can purchase 5 kgs of ghee for 2180, how much ghee can he purchase with the amount he has?
Total amount for 5 kg ghee is x1 = Rs. 2180/-
Ghee purchased with Rs. 2180/- is y1 = 5 kg
Amount with Sumanth is x2 = Rs. 400/-
Ghee purchased with Rs. 400/- is y2 = ?
The money and amount of ghee purchased are directly proportional to each other.
The direct proportionality relationship can be written as:
\ = \
\ = \
y2 = \ × 5
y2 = 0.917
Sumanth can purchase 0.917 kgs of ghee with Rs. 400.
Applications Of Inverse Proportion
Inversely proportionality is a term that is commonly utilized in everyday life. Inverse proportion helps to solve numerous problems in science, statistics, and other fields. In physics, the concept of inverse proportionality is used to create several formulas. Ohms law, the speed and time relationship, the wavelength and frequency of sound are only a few examples.
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Shadow And Height Of Objects
At any particular time of the day, the height of an object is directly proportional to the length of the shadow cast by it on the ground. For instance, suppose that two poles stand across the opposite corners of a playground. One of the poles is 3m high, while the height of the second pole is unknown. The pole with a height equal to 3m casts a shadow that is 6.3m long. At the same time, the other pole casts a shadow that is 8.4m long. Now, with the help of the unitary method, the height of the second pole can be calculated easily. The height of the second pole comes out to be 4m. If you compare the heights of the poles and the lengths of the shadow cast by them, you can easily observe that they are directly proportional to each other. This means that with an increase in the height of the pole, the length of the shadow increases accordingly.
Number Of Students And Number Of Benches In A Class
The number of benches installed in a classroom is always maintained proportional to the strength of the class. Let us say that a class consists of 40 students, then the number of two-seater benches required to accommodate all of the students of the class is equal to 20. If you increase the number of students, then the number of benches required should also increase accordingly.
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Direct Proportion Examples In Real Life
In our day-to-day life, we observe that the variations in the values of various quantities depending upon the variation in values of some other quantities.
For example: if the number of individuals visiting a restaurant increases, earning of the restaurant also increases and vice versa. If more number of people are employed for the same job, the time taken to accomplish the job decreases.
Sometimes, we observe that the variation in the value of one quantity is similar to the variation in the value of another quantity that is when the value of one quantity increases then the value of other quantity also increases in the same proportion and vice versa. In such situations, two quantities are termed to exist in direct proportion.
Some more examples are:
- Speed is directly proportional to distance.
- The cost of the fruits or vegetable increases as the weight for the same increases.
Cycling Speed And Force Applied
The relationship between the magnitude of mechanical force applied by a cyclist and the speed with which the cycle moves is a prominent example of direct proportionality. This is because on increasing the magnitude of applied force, the speed of the cycle increases proportionally. Similarly, when the rider slows down the pedalling speed, the overall speed of the cycle reduces proportionally.
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The Concept Of Proportion And Constant Of Proportionality
After you understand the definitions and concepts about functions and coordinates, you can learn about proportionality. Proportionality refers to a function that as the value of $x$ increases, the number of $y$ increases at the same rate.
Lets consider the following example, which is the same as the previous one.
- Walking for 3 hours at $x$ km per hour, the distance walked is $y$ km.
This function is $y = 3x$. Therefore, as the value of $x$ increases by 1, $y$ is increased by 3. Also, if the value of $x$ decreases, the value of $y$ decreases by the same ratio. The table looks like the following.
A change in the value of $x$ causes $y$ to increase by 3. This is direct proportionality. In this equation, $y=3x$, so the value of $y$ changes by 3. On the other hand, for example, if $y=5x$, the value of $y$ will change by 5 as the value of $x$ changes.
Therefore, in direct proportional coordinates, we use the following formula.
$a$ is called a constant of proportionality. The value of $a$ is different for each. In the previous equation, walking at $x$ km/h for 3 hours, resulting in $y=3x$. On the other hand, if you walk at $x$ km/h for 4 hours, $y=4x$. The constant of proportionality, $a$, changes depending on the problem.