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Solving Proportions: Definition, Solved Examples

solve proportions

Have you ever had to split a cookie with your friend? If so, you have used the concept of proportions without even realizing it! Proportions, which are mathematical relationships, are excellent real-world examples of arithmetic. Grocery shopping, cooking, and traveling from one place to another are prominent real-life scenarios in which ratios are common and necessary for proper, cost-effective performance.

This article will help you understand how to solve proportions and the different formulas you can employ. 

What is a proportion?

A proportion can be defined in a variety of ways. According to one definition, a proportion is an equation having two equal ratios. In other terms, a percentage is when two fractions are joined in the center by an equal sign. Variables can be found in one or both of the fractions in proportions.

Apart, share, or quantity regarded in comparison to a total is generally referred to as a proportion. When two ratios are equal, they are in proportion, according to the definition of proportion. If two values increase or decrease in the same ratio, the ratios are said to be directly proportional to each other. The symbols “::” or “=” are used to represent proportions.

Example of Proportion

When two ratios are equal, they are said to be in proportion. For example, the time it takes a train to go 50 kilometers per hour is the same as the time it takes to travel 250 kilometers in 5 hours. It can be expressed as 50 km/hr = 250 km/5 hours.

Ratio and Proportion, what is the difference?

Many students often use the terms ratio and proportion interchangeably. The term proportion refers to the proportional relationship between two or more ratios. The differences between ratio and proportion given in the table may help you grasp the concept better.

RatioProportion
The ratio is used to compare the sizes of two objects that have the same unit of measurement.The proportion expresses the relationship between two ratios.
It’s written with a colon (:) or a slash (/).It’s written with a double colon (::) or the equal sign (=).

Types of Proportions

Proportions can be classified into different types depending on the type of relationship that two quantities have. There are two types of proportions.

  • Direct Proportion
  • Inverse Proportion

Direct Proportion

If there is a direct relationship between two physical quantities, then it is known as direct proportion. In other words, you can call quantities to be in direct proportion if one quantity increases, the other quantity also increases, and vice-versa. For example, if the speed of a car is increased, it covers more distance in a fixed amount of time. The direct proportion is written as y ∝ x.

Inverse Proportion

This type describes the indirect, i.e. there is no relationship between the two physical quantities. Two physical quantities are said to be indirectly proportional if one quantity increases, the other quantity decreases, and vice-versa. In notation, an inverse proportion is written as y ∝ 1/x. When a car’s speed is raised, for example, it will be able to cover a larger distance in less time.

How do you solve a proportion?

Cross-multiplication is one method to solve proportions. The Means Extremes Property is a property that you can utilize. It states that a proportion’s cross products will be equal.

It states that if ab = cd, then ad = bc. 

So, what is the Means Extremes Property, and why is it named that? The proportion can be expressed as a:b = c:d using colons. On the exterior, the extremes are the terms that are the farthest apart: the a and the d. The terms on the inside are the means: b and c.

Let us look at an example of the same: 

x:9 :: 2:3

To solve the proportion equation through cross multiplication, first, you need to multiply diagonally. Multiply the x and the 3 together, then set the result equal to the result of multiplying the 2 and the 9.

This gives us: 

(x) (3) = (2) (9) 

Now divide both sides by 3 to get x by itself

x(3)/3=18/3

 x = 6

How to Solve Proportions with Variables

What is a variable in Mathematics? A variable is a letter that denotes an unknown number, value, or quantity. In the case of algebraic expressions, variables are used, for example. x+9=4 is a linear equation in which x is a variable and 9 and 4 are constants. So how do you solve proportions with variables?

Example: 

(x+1):5 :: (x-3):3

Solve for x. 

Solution: 

Cross-multiply first. When simplifying each side, be cautious. Make sure the 5 and 3 are evenly distributed. 

3(x+ 1) = 5(x- 3)

Open the brackets and simplify the equation. 

3x+ 3 = 5x-15

Shift all the variables to one side and the constants on another. 

3 + 15 = 5x – 3x

18 = 2x

Divide both sides by 2 to get x by itself. 

18/2 = 2x/2

x = 9

Practice Questions for Solving Proportions

Q1. Solve for x. 

x:9 :: 4:x

Answer: When you multiply one x with another, the result is x-squared. 

(x) (x) = (9) (4) 

x 2 = 36

To remove the square, we need to take the square root of both sides. 

√x 2 = √36

x = 6

Q2. Solve for x. 

(x+3):8 :: 5:(x+9)

Answer: When (x+3) is multiplied with (x+9), it is called binomial multiplication. You need to use the FOIL method. (First outside, inside last). 

(x+3)(x+9) = (8) (5)

Use FOIL to simplify the left side. 

x 2 +9x +3x + 27 = 40

x 2 +12x + 27 = 40

Now we need to set this quadratic equation equal to zero. 

x 2 +12x + 27 = 40

                   -40  =  -40

 x 2 +12x -13 = 0

There are various methods for solving quadratic equations. You can factor, complete the square, or use the Quadratic Formula to answer the problem. In this case, we will use the factoring method to solve. 

 x 2 +12x -13 = 0

(x + 13) (x – 1) = 0

Split up the factors and set each equal to zero. 

x + 13 = 0 or x – 1 = 0 

Thus, the two values of x are -13 and x = + 1

Q3. A ten-foot-long metal bar weighs 128 pounds. What is the weight of a two-foot-four-inch-long comparable bar?

Answer:

We should start by converting “two feet four inches” to a feet-only measurement. Four inches is one-third of afoot because one foot has twelve inches. So, in feet only, the length is as follows:

2 feet + 0.333 feet 

= 2.333feet

The ratio of length to the weight of the bar is given as length (ft)weight (lbs)

The proportion of the two comparable bars can be expressed as 10:128 :: 2.333:w

w is the unknown weight that we are expected to find. 

(10) (w) = (128) (2.333)

w = 128 * 2.333 / 10

    = 29.8624 pounds

  

Q4. One piece of pipe 21 meters long is to be cut into two pieces, with the lengths of the pieces being in a 2: 5 ratio. What are the lengths of the pieces?

Answer: 

The length of the short piece will be denoted by the letter “s.” The longer piece, after chopping offs meters, must therefore be 21 – s long. Then, the ratio would be expressed as:

short piece long piece: 25 = s21 – s

Solving the proportion by cross multiplying, 

2 (21 – s) = (5) (s) 

42 – 2s = 5s

42 = 7s 

s = 6 

From the original statement, we can see that the length of the long piece is 

21 – s = 21 – 6 = 15 

So, the two pieces of the pipe are 6 cm and 15 cm. 

Q5. Calculate how many centimeters are there in thirty inches if twelve inches equals 30.48 centimeters.

Answer: 

We can start our ratios with “inches” at the top and then use the letter “c” to represent the requested centimeter value. The equation would look like this:

inches centimeters : 12:30.48 :: 30:c

Solving the proportion by cross multiplying, 

12 (c) = (30) (30.48)

c = 914.412

c = 76.2 

Therefore, in thirty inches there are 76.2 centimeters. 

Q6. You’re having rain gutters installed across the back of your house. The gutters should drop 14 inches for every four feet of lateral flow, according to the instructions. Thirty-seven feet will be covered with gutters. How sloped should the gutters’ low end be than the beginning point?

Answer: Rain gutters must be sloped downwards to direct rainwater toward the end. The gutters should decline by 14 inches for every four-foot length as you move from the high end to the low end. So, over a thirty-seven-foot span, how much must the guttering deteriorate? In the proportion, we will take “d” as the distance we need to discover.

declination (inches)Length (feet) : 1/44 = d37

d = (37) )(1/4)4

   = (37/4)4= 9.254

d = 2.3125

The lower end should be 2.3125 or 2 615 inches lower than the high end.

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