Learning Objectives

By the end of this section, you will be able to:

- Solve equations with fraction coefficients
- Solve equations with decimal coefficients

Note

Before you get started, take this readiness quiz.

- Multiply: (8cdot 38).

If you missed this problem, review Exercise 1.6.16. - Find the LCD of (frac{5}{6}) and (frac{1}{4}).

If you missed this problem, review Exercise 1.7.16. - Multiply 4.78 by 100.

If you missed this problem, review Exercise 1.8.22.

## Solve Equations with Fraction Coefficients

Let’s use the general strategy for solving linear equations introduced earlier to solve the equation, (frac{1}{8}x+frac{1}{2}=frac{1}{4}).

To isolate the x term, subtract (frac{1}{2}) from both sides. | |

Simplify the left side. | |

Change the constants to equivalent fractions with the LCD. | |

Subtract. | |

Multiply both sides by the reciprocal of (frac{1}{8}). | |

Simplify. |

This method worked fine, but many students do not feel very confident when they see all those fractions. So, we are going to show an alternate method to solve equations with fractions. This alternate method eliminates the fractions.

We will apply the Multiplication Property of Equality and multiply both sides of an equation by the least common denominator of all the fractions in the equation. The result of this operation will be a new equation, equivalent to the first, but without fractions. This process is called “clearing” the equation of fractions.

Let’s solve a similar equation, but this time use the method that eliminates the fractions.

Exercise (PageIndex{1}): How to Solve Equations with Fraction Coefficients

Solve: (frac{1}{6}y - frac{1}{3} = frac{5}{6})

**Answer**

Exercise (PageIndex{2})

Solve: (frac{1}{4}x + frac{1}{2} = frac{5}{8})

**Answer**(x= frac{1}{2})

Exercise (PageIndex{3})

Solve: (frac{1}{8}x + frac{1}{2} = frac{1}{4})

**Answer**(x = -2)

Notice in Exercise (PageIndex{1}), once we cleared the equation of fractions, the equation was like those we solved earlier in this chapter. We changed the problem to one we already knew how to solve! We then used the General Strategy for Solving Linear Equations.

STRATEGY TO SOLVE EQUATIONS WITH FRACTION COEFFICIENTS.

- Find the least common denominator of
*all*the fractions in the equation. - Multiply both sides of the equation by that LCD. This clears the fractions.
- Solve using the General Strategy for Solving Linear Equations.

Exercise (PageIndex{5})

Solve: (7 = frac{1}{2}x + frac{3}{4}x - frac{2}{3}x)

**Answer**(x = 12)

Exercise (PageIndex{6})

Solve: (-1 = frac{1}{2}u + frac{1}{4}u - frac{2}{3}u)

**Answer**(u = -12)

In the next example, we again have variables on both sides of the equation.

Exercise (PageIndex{8})

Solve: (x + frac{1}{3} = frac{1}{6}x - frac{1}{2})

**Answer**(x = -1)

Exercise (PageIndex{9})

Solve: (c + frac{3}{4} = frac{1}{2}c - frac{1}{4})

**Answer**(c = -2)

In the next example, we start by using the Distributive Property. This step clears the fractions right away.

Exercise (PageIndex{11})

Solve: (-11 = frac{1}{2}(6p + 2))

**Answer**(p = -4)

Exercise (PageIndex{12})

Solve: (8 = frac{1}{3}(9q + 6))

**Answer**(q = 2)

In the next example, even after distributing, we still have fractions to clear.

Exercise (PageIndex{14})

Solve: (frac{1}{5}(n + 3) = frac{1}{4}(n + 2))

**Answer**(n = 2)

Exercise (PageIndex{15})

Solve: (frac{1}{2}(m - 3) = frac{1}{4}(m - 7))

**Answer**(m = -1)

Exercise (PageIndex{17})

Solve: (frac{4y - 7}{3} = frac{y}{6})

**Answer**(y = 2)

Exercise (PageIndex{18})

Solve: (frac{-2z - 5}{4} = frac{z}{8})

**Answer**(z = -2)

Exercise (PageIndex{20})

Solve: (frac{b}{10} + 2 = frac{b}{4} + 5)

**Answer**(b = -20)

Exercise (PageIndex{21})

Solve: (frac{c}{6} + 3 = frac{c}{3} + 4)

**Answer**(c= -6)

Exercise (PageIndex{23})

Solve: (frac{3r + 5}{6}+ 1 = frac{4r + 3}{3})

**Answer**(r = 1)

Exercise (PageIndex{24})

Solve: (frac{2s + 3}{2}+ 1 = frac{3s + 2}{4})

**Answer**(s = -8)

## Solve Equations with Decimal Coefficients

Some equations have decimals in them. This kind of equation will occur when we solve problems dealing with money or percentages. But decimals can also be expressed as fractions. For example, (0.3 = frac{3}{10}) and (0.17 = frac{17}{100}). So, with an equation with decimals, we can use the same method we used to clear fractions—multiply both sides of the equation by the least common denominator.

Exercise (PageIndex{25})

Solve: (0.06x + 0.02 = 0.25x - 1.5)

**Answer**Look at the decimals and think of the equivalent fractions.

(0.06 = frac { 6 } { 100 } quad 0.02 = frac { 2 } { 100 } quad 0.25 = frac { 25 } { 100 } quad 1.5 = 1 frac { 5 } { 10 })

Notice, the LCD is 100.

By multiplying by the LCD, we will clear the decimals from the equation.

Multiply both side by 100. Distribute. Multiply, and now we have no more decimals. Collect the variables to the right. Simplify. Collect the variables to the right. Simplify. Divide by 19. Simplify. Check: Let x=8

Exercise (PageIndex{26})

Solve: (0.14h + 0.12 = 0.35h - 2.4)

**Answer**(h = 12)

Exercise (PageIndex{27})

Solve: (0.65k - 0.1 = 0.4k - 0.35)

**Answer**(k = -1)

The next example uses an equation that is typical of the money applications in the next chapter. Notice that we distribute the decimal before we clear all the decimals.

Exercise (PageIndex{29})

Solve: (0.25n + 0.05(n + 5) = 2.95)

**Answer**(n = 9)

Exercise (PageIndex{30})

Solve: (0.10d + 0.05(d -5) = 2.15)

**Answer**(d = 16)

## Key Concepts

**Strategy to Solve an Equation with Fraction Coefficients**- Find the least common denominator of all the fractions in the equation.
- Multiply both sides of the equation by that LCD. This clears the fractions.
- Solve using the General Strategy for Solving Linear Equations.

## Pre-Algebra : One-Step Equations with Fractions

**Step 1:** Multiply both sides of the equation by the fraction's reciprocal to get alone on one side:

**Step 2:** Multiply:

### Example Question #3 : One Step Equations With Fractions

The goal is to isolate the variable on one side.

The opposite operation of division is multiplication, therefore , multiply each side by :

The left hand side can be reduced by recalling that anything divided by itself is equal to 1:

The identity law of multiplication takes effect and we get the solution as:

### Example Question #1 : One Step Equations With Fractions

The goal is to isolate the variable on one side.

The opposite operation of multiplication is division, therefore, we can either divide each side by or multiply each side by its reciprocal :

The left hand side can be reduced by recalling that anything multiplying a fraction by its reciprocal is equal to 1:

The identity law of multiplication takes effect and we get the solution as:

However, this solution can be reduced by dividing both the numerator and denominator by 3:

### Example Question #5 : One Step Equations With Fractions

The goal is to isolate the variable on one side.

The opposite operation of addition is subtraction so subtract from each side:

In order to complete the subtraction on the right hand side, we must first determine the common denominator, or common multiples of 3 and 6. The least common multiple of 3 and 6 is 6 itself.

Simplifying, we get the final solution:

### Example Question #1 : One Step Equations With Fractions

The goal is to isolate the variable on one side.

The opposite operation of subtraction is addition so add to each side:

In order to complete the addition on the right hand side, we must first determine the common denominator, or common multiples of 6 and 12. The least common multiple of 6 and 12 is 12 itself.

Simplifying, we obtain the solution:

Reducing the fraction to its simplest terms we get the final solution:

### Example Question #1 : One Step Equations With Fractions

To get y by itself, you must divide by 6 on both sides

### Example Question #8 : One Step Equations With Fractions

To get x by itself, you must multiply both sides of the equation by 5

### Example Question #9 : One Step Equations With Fractions

Solve the equation below for x:

For this equation, isolate the variable by preforming equivalent operations on both sides of the equation.

To isolate a variable multiplied by a fraction, any fraction multiplied by it's reciprocal equals one.

Because , we isolate , and the equation becomes

multiplying the values on the right side gives us

### Example Question #10 : One Step Equations With Fractions

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## EQUATIONSWITH FRACTIONS

T O SOLVE AN EQUATION WITH fractions, we transform it into an equation without fractions -- which we know how to solve. The technique is called clearing of fractions .

Solution . Clear of fractions as follows:

Multiply both sides of the equation -- every term -- by the LCM of denominators. Each denominator will then divide into its multiple. We will then have an equation without fractions.

The LCM of 3 and 5 is 15. Therefore, multiply both sides of the equation by 15.

15 · | x 3 | + | 15 · | x &minus 2 5 | = 15 · 6 |

On the left, distribute 15 to each term. Each denominator will now divide into 15 -- that is the point -- and we have the following simple equation that has been "cleared" of fractions:

5 x + 3( x &minus 2) | = | 90. |

It is easily solved as follows: | ||

5 x + 3 x &minus 6 | = | 90 |

8 x | = | 90 + 6 |

x | = | 96 8 |

= | 12. |

We say "multiply" both sides of the equation, yet we take advantage of the fact that the order in which we multiply or divide does not matter. (Lesson 1.) Therefore we divide the LCM by each denominator first, and in that way clear of fractions.

We choose a multiple of each denominator, because each denominator will then be a divisor of it.

Example 2. Clear of fractions and solve for x :

Solution . The LCM of 2, 6, and 9 is 18. (Lesson 23 of Arithmetic.) Multiply both sides by 18 -- and cancel.

It should not be necessary to actually write 18. The student should simply look at and see that 2 will go into 18 nine (9) times. That term therefore becomes 9 x .

Next, look at , and see that 6 will to into 18 three (3) times. That term therefore becomes 3 · &minus5 x = &minus15 x .

Finally, look at , and see that 9 will to into 18 two (2) times. That term therefore becomes 2 · 1 = 2.

Here is the cleared equation, followed by its solution:

9 x &minus 15 x | = | 2 | |

&minus6 x | = | 2 | |

x | = | 2 &minus6 | |

x | = | &minus | 1 3 |

Solution . This is an equation with a fraction. Clear of fractions by mutiplying both sides by 2:

5 x &minus 2 | = | 4 x + 8 |

5 x &minus 4 x | = | 8 + 2 |

x | = | 10. |

In the following problems, clear of fractions and solve for x :

To see each answer, pass your mouse over the colored area.

To cover the answer again, click "Refresh" ("Reload").

Do the problem yourself first!

Problem 1. | x 2 | &minus | x 5 | = | 3 |

The LCM is 10 . Here is the cleared equation and its solution: | |||||

5 x | &minus | 2 x | = | 30 | |

3 x | = | 30 | |||

x | = | 10. |

On solving any equation with fractions, the very next line you write --

Problem 2. | x 6 | = | 1 12 | + | x 8 |

The LCM is 24 . Here is the cleared equation and its solution: | |||||

4 x | = | 2 + 3 x | |||

4 x &minus 3 x | = | 2 | |||

x | = | 2 |

Problem 3. | x &minus 2 5 | + | x 3 | = | x 2 |

The LCM is 30 . Here is the cleared equation and its solution: | |||||

6 (x &minus 2) + 10 x | = | 15 x | |||

6 x &minus 12 + 10 x | = | 15 x | |||

16 x &minus 15 x | = | 12 | |||

x | = | 12. |

Problem 4. A fraction equal to a fraction.

x &minus 1 4 | = | x 7 |

The LCM is 28 . Here is the cleared equation and its solution: | ||

7( x &minus 1) | = | 4 x |

7 x &minus 7 | = | 4 x |

7 x &minus 4 x | = | 7 |

3 x | = | 7 |

x | = | 7 3 |

We see that when a single fraction is equal to a single fraction, then the equation can be cleared by "cross-multiplying."

Problem 5. | x &minus 3 3 | = | x &minus 5 2 |

Here is the cleared equation and its solution: | |||

2( x &minus 3) | = | 3( x &minus 5) | |

2 x &minus 6 | = | 3 x &minus 15 | |

2 x &minus 3 x | = | &minus 15 + 6 | |

&minus x | = | &minus9 | |

x | = | 9 |

Problem 6. | x &minus 3 x &minus 1 | = | x + 1 x + 2 |

Here is the cleared equation and its solution: | |||

( x &minus 3)( x + 2) | = | ( x &minus 1)( x + 1) | |

x ² &minus x &minus 6 | = | x ² &minus 1 | |

&minus x | = | &minus1 + 6 | |

&minus x | = | 5 | |

x | = | &minus5. |

Problem 7. | 2 x &minus 3 9 | + | x + 1 2 | = | x &minus 4 |

The LCM is 18 . Here is the cleared equation and its solution: | |||||

4 x &minus 6 + 9 x + 9 | = | 18 x &minus 72 | |||

13 x + 3 | = | 18 x &minus 72 | |||

13 x &minus 18 x | = | &minus 72 &minus 3 | |||

&minus5 x | = | &minus75 | |||

x | = | 15. |

Problem 8. | 2 x | &minus | 3 8 x | = | 1 4 |

The LCM is 8 x . Here is the cleared equation and its solution: | |||||

16 &minus 3 | = | 2 x | |||

2 x | = | 13 | |||

x | = | 13 2 |

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## Fraction Solver

Solving fractions is hard when done mentally. You need pen and paper to solve it manually if you want to get and efficient result. And even if you know how to solve it, sometimes you still make mistake in your computation. You cannot trust your computed result by doing it only one time. You need to review it atleast one time to make sure you got the correct answer. This process takes some effort, consumes more time, and drains more energy from you. In short, this process of solving fraction manually is not efficient and accurate at all. You need a sure-way process. You need a bullet-proof method to get the exact computation. And the solution for this issue is to use a fraction solver. This tool is very efficient and presented below. You can use it anytime you need a fraction calculation.

Fraction solver is a tool designed to solve all kinds of fraction problems. It deals all arithmetic operations related to fractions. All kinds of fractions such as proper, improper, mixed numbers and even their combination are dealt with. Adding, subtracting, multiplying, and dividing fractions and their combinations are done with this tool without making any error. It has been tested and used for over five years now. The user will just input the operation they need by pressing the corresponding buttons.

### Fraction Solver Parts Description And Their Usage:

Simplify - This button is used to simplify a given fraction. The user will input the fraction he wants to be simplified and press the Simplify button. To input a fraction, he just needs to press the whole number button if it has a whole number followed by the numerator button and then lastly the denominator button. If the fraction is already at its simplified state, it will remain the same even if simplify button has been hit several times.

Whole Numbers - These buttons are darker gray in color and are bigger and bulkier than numerators and denominators button. These buttons are located at the left side of the fraction solver and must be pressed if ever the operation involves whole numbers otherwise there is no need to press these buttons.

Numerators – These buttons are located at the upper right side of the fraction solver and are smaller than whole numbers button but same in size with denominators button. They are white in colors. The user may not need to press these buttons if the numerator they want to enter is one (1) because the numerator will be automatically set to one (1) if the user will directly press the denominator buttons.

Denominators – These buttons are light gray in color and located the lower right side of the fraction solver. The user may directly press these buttons and bypass the numerator buttons if the fraction’s numerator is one (1).

Backspace – This button is used when you want to delete a digit or operation. This is done one press at a time until all the operations or digits you want to delete are removed.

A/C – This button is pressed if you want to clear the screen and want to start over.

÷ - This button is used to divide a fraction.

× - This button is used to multiply.

+ - This is for addition of fractions.

- - This is for subtraction of fractions.

= - When the desired fraction operations are set, the equal button is pressed to make the fraction solver start the process of computing the problems and display the result.

### Why Fraction Solver Has Been Created?

The creator has once felt the weight of the consequence when someone makes a mistake of computing a very simple fraction. There was a wealthy man who as five legal and two illegitimate children. According to the law of the country they live in, to divide the inheritance, the illegitimate child must have half the value of what the legitimate child inherit. Fortunately there are only two illegitimate children. So their inheritance value must be equal to that one legitimate child. So the computation must be each legal child must get 1/6 of inheritance while the illegitimate children must get 1/12 each. Computing these fractions must be cumbersome when done manually. That is why the idea was born to create a tool that will compute a fraction automatically.

### Scenarios Suitable For This Tool:

A math teacher is creating a questionnaire for the examination of his 60 students. To make a reference answer as well as the solutions for the items related to fractions, he can use the fraction solver tool. This will save him time that can be used for preparing other math topics as well as give him some time to relax.

A student is preparing his math exam and practicing the topic about fraction. He has no means of confirming whether his answer is right or wrong. He can only check and review it several times until he is sure that his answer is correct by having the same answer after several checking. This process is time consuming and he might not cover the whole topic to study. By using fraction solver as a guide whether his manual calculation is correct might help him learn fast and make him confident about the topic.

Another scenario is a businessman calculating his profits in business. Normal physical calculator has a limited feature in computing about fraction. There are some which has fraction computation functionality but have to set a certain settings. There is need to have a tool specific only in calculating fractions. So this fraction solver tool would really help a lot.

A cook might need to sum up all the ingredients of a certain dish but unable to do so because some of the ingredients quantity are in fractions form. The need to sum it up might be because of the need to have the correct proportion between the soup and the dish or whatever factors that needs to have a good proportion. Computing it manually is not an ideal solution because a cook is a cook and not a mathematician. Computing it with exact results might need the service of the fraction solver tool.

### What People Say About Fraction Solver?

"This tool is very cool. Eversince I found this tool, my job as a cashier in a small convenience store in our town has been easier. Normally I have to compute money and there were times when an amount is related to fractions and I have to do it manually using pen and paper. This would lead me to render some unpaid overtime because since I am dealing with cash amount, I have to make sure that every computation is accurate. Sure we are using POS system and cash register but I cannot escape some situation where the fraction solver is the right tool to use." From Neddah Kramel – Arkansas.

"Hi, I love this tool. It helps me a lot in my assignments related to fraction. I am a secondary student and I have difficulty in dealing with fractions. Sometimes I feel that fraction is a beast. Now that I found this very cool tool, I am now confident in dealing with fraction." From Jessie M. – Chicago.

## Decomposing and Composing Fractions Calculator

An online decomposing fractions calculator to decompose fraction into a unit fraction. Decomposing fractions is breaking up of fractions into several parts that can be added together. Composing fractions is the opposite of decomposing, where all part fractions will be composed as one. This online Decomposing and Composing Fractions Calculator helps you to compose and decompose fractions. Choose the option and enter the values in the calculator to find the result.

## 2.5: Solve Equations with Fractions or Decimals

Help Buffy multiply fractions by whole numbers using the standard algorithm in addition to visual fraction models in this bakery-themed, interactive tutorial. This is part4 of a 4-part series. Click below to open other tutorials in the series. Part 2: Multiplying Fractions Part 3Using Models to Multiply a Fraction by a Whole Number

Subject Area(s): Mathematics

Primary Resource Type: Original Tutorial

Help Buffy the Baker multiply a fraction by a whole using models in this sweet interactive tutorial. This is part 3 of a 4-part series. Click below to open other tutorials in the series. Part 2: Multiplying Fractions Part 4:Multiplying a Fraction by a Whole Number - Standard Algorithm

Subject Area(s): Mathematics

Primary Resource Type: Original Tutorial

Find the total amounts of repeated fraction quantities by multiplying a fraction by a whole number using visual models that represent real-world problems and cookies in this interactive tutorial.

## Improper Fractions

Now when aiming to locate an improper fraction on a fraction number line. It's usually easier to convert to mixed fraction form first.

Details of how to do so can be seen __ here__.

Let's consider the improper fractions oldsymbol

In mixed fraction form these improper fractions are:

oldsymbol_{1} oldsymbol_{3} oldsymbol

Using the same approach as before for locating mixed fractions, can now be used to plot the improper fractions.

## Solving Equations with Multiplication/Division

In order **to move a number that is multiplied or divided by the variable, you must do the inverse to both sides.** That means, if the number is multiplied, you must divide it by both sides. If it's divided, you must multiply it by both side.

the 3's will cancel out and disappear (3/3 = 1 and 1x = x).

We have the variable all by itself, so this is our answer.

3x = 10 We can check the answer by putting 10/3 back in the

true original equation in place of x.

Final answer: This is true, so we know we have the right answer.

2) Even if you can do this in your head, you need to learn the

Now we will work with each side. On the left hand side,

y = -10 the 5's will cancel out and disappear. On the right,

-2(5) = -10

original equation in place of y.

true

Final answer: y = -10 This is true, so we know we have the right answer.

3) The fraction 2/3 is on the same side as the x, so that is

the 3/2 and 2/3 will cancel out and disappear. On the right,

We have the variable all by itself, so this is our answer.

We can check the answer by putting -9 back in the

original equation in place of x.

Final answer: x = -9 This is true, so we know we have the right answer.

**Practice:** Solve the equation and check your answer

2)

3)

5)

## Having fun with unit fractions

Dividing both sides by 28 results in a decomposition of 1 into a sum of unit fractions with five terms:

There are many other perfect numbers (496, 8128, 33550336, and 8589869056, for example). While we could similarly decompose these numbers as above, the condition that each denominator must be 99 or less limits which ones we can use. For example 496 is out because two of its divisors have three digits:

We can generalise our methods for decomposition and composition into five rules.

### Rule 1: Decomposition into two terms

Doing this is relatively simple by hand, but you can also use spreadsheet software to speed things along.

### Rule 2: Decomposition into n terms

### Rule 3: Decomposition into two terms by another method

### Rule 4: Decomposition into three terms

### Rule 5: Composition of two terms into one

Occasionally in the process of constructing a sum of unit fractions it might be useful to replace two existing unit fractions by one which then gives us more possibilities. For example, given the decomposition into two termsUsing these five rules I have found a decomposition of 1 into 42 distinct unit fraction terms. You can see it below.

Repeated decomposition and composition of two and three terms to find increasingly long sums equal to 1 is an excellent way to kill time. But just how far can we take it? A bit of integral calculus allows one to predict that the maximum value must be 62. (If you're not familiar with calculus, you can skip this section.)

### A maximum of 62 terms

Figure 2 shows the same curve, this time with rectangles whose areas sum to less than the area under the curve.

Putting the two inequalities together we have

( 1 ) |

( 2 ) |

A similar manipulation as above then gives the analogue of inequality (2)

Working out the integral gives

The inequality on the right says that in order for the sum of the unit fractions to not exceed 1 we must have

### A tree diagram of my 42 term solution

My current solution with 42 distinct unit fraction terms looks like this

will give you an alternative 42-term answer, which I leave for you to work out (here is the answer).

Figure 4 shows the solution as a tree diagram, in which the trunk is our original value of 1 and the branches show the process of decomposition into two and three terms. Each of the 42 leaves is a unique unit fraction term in the solution.

Perhaps a bit more searching would repair those places by removing the need for the compositions.

### About the author

"boomerang professor". He is interested in the mathematics of daily life and has written eight books about the subject. The most recent one,*The mystery of five in nature*, investigates, amongst other things, why many flowers have five petals.

## How to solve a proportion?

Students have to solve proportions by hand in their exams and classrooms. This tool can be very helpful to quickly complete the assignments and homework. Moreover, it can be used to learn the calculations of proportion. Here, we will illustrate how to calculate proportion by yourself.

### Cross multiplication method

To calculate a missing value or unknown variable in a proportion, follow the below steps:

- Write the given values in fraction form. Use any variable to represent the unknown value.
- Join both fractions using an equal sign.
- Multiply both fractions diagonally. I.e. multiply the numerator of the first fraction with the denominator of the second fraction and multiply the numerator of the second fraction with the denominator of the first fraction.
- Write both numbers after multiplication and put an equal sign between them.
- Find the value of the variable by isolating it on either side of the equation.

### Example &ndash Direct proportion

A train from Seoul to Busan travels 300 km distance in 4 hours. How much time will it take to reach a distance of 500 km?

** Step 1:** Write the given values in fraction form. Use any variable to represent the unknown value.

It travels 300 km in 4 hours, the ratio will be: 300 / 4

The ratio for the distance of 500 km will be: 500 / **X**

** Step 2:** Join both fractions using an equal sign.

300 / 4 = 500 / **X**

300 : 4 = 500 : **X**

** Step 3:** Multiply both fractions diagonally. I.e. multiply the numerator of the first fraction with the denominator of the second fraction and multiply the numerator of the second fraction with the denominator of the first fraction.

** Step 4:** Write both numbers after multiplication and put an equal sign between them.

**300 X = 500 × 4**

**Step 5:** Find the value of the variable by isolating it on either side of the equation.

**X = 2000 / 300**

**X = 6.6** hours approx.

So, the train will take approximately 6 and a half hours to covers a distance of 500 km.

### Example &ndash Inverse proportion

In a toy manufacturing factory, 3 workers make a box of toys in 8 days. The company has hired 2 more workers to increase the production of the unit. There are a total of 5 workers in the factory now. How long will it take to complete that same task by 5 workers?

** Step 1:** Write the given values in fraction form. Use any variable to represent the unknown value.

Ratio of workers before and after = 3 / 5

Ratio of days of completion = **X** / 8

** Step 2:** Join both fractions using an equal sign.

3 / 5 = **X** / 8

** Step 3:** Multiply both fractions diagonally. I.e. multiply the numerator of the first fraction with the denominator of the second fraction and multiply the numerator of the second fraction with the denominator of the first fraction.

** Step 4:** Write both numbers after multiplication and put an equal sign between them.

** Step 5:** Find the value of the variable by isolating it on either side of the equation.

So, 5 workers will manufacture the toys in approx. **4.8** days.

### How do you calculate proportion?

Proportion can be calculated by using a cross multiplication method. In the cross multiplication method, we diagonally multiply the numerator and denominator of both fractions and calculate the value of an unknown variable by isolating it on one side of the equation.

For example, we have two fractions as:

By cross multiplication, we get:

2**x** = 12 è **x** = 6

### What are examples of proportions?

The followings are some examples of proportions:

- We use proportions of ingredients for cooking a specific amount of food.
- We compare the prices of various shopping centers by using proportions.
- Builders mix sand and cement to the gravel to make a solution by using the specific proportions.
- Chemists make several chemical formulas and medicines by using proportions of various chemicals and drugs.
- A rope of specific weight and length. The length and weight of the rope are proportional.
- The sizes of the shapes of any objects can be proportional to each other.

### What is the ratio of 4 to 3?

The ratio of 4 to 3 can be written as **4 : 3.** It means that the second quantity is 1/3 of the first quantity.

A few equivalent ratios of **4 : 3** are:

4 : 3 | 8 : 6 | 12 : 9 | 16 : 12 | 20 : 15 |

24 : 18 | 28 : 21 | 32 : 24 | 36 : 27 | 40 : 30 |

44 : 33 | 48 : 36 | 52 : 39 | 56 : 42 | 60 : 45 |

64 : 48 | 68 : 51 | 72 : 54 | 76 : 57 | 80 : 60 |

84 : 63 | 88 : 66 | 92 : 69 | 96 : 72 | 100 : 75 |

### What is the ratio of 1 to 5?

A ratio of 1 to 5 can be written as **1 : 5.** It shows that the quantity in second place is five-time the quantity in the first place. Some equivalent ratios of **1 : 5** are: