Proportions are mathematical statements that indicate that two ratios or are equal. They are used in a wide range of applications, from solving in everyday life to more complex scenarios in fields such as finance and physics. When dealing with proportions that involve fractions, it is essential to understand the steps and techniques to them effectively. In this article, we will explore how to solve proportions that include fractions, providing a clear and concise guide for solving such problems.

To solve proportions with fractions, we need to follow a few simple steps. Let’s consider the following example:

4/7 = x/21

Step 1: Cross-multiply
To start solving the proportion, we cross-multiply. This means multiplying the numerator of the first fraction by the denominator of the second fraction and vice versa. In our example, we multiply 4 by 21 and 7 by x, as shown below:

4 * 21 = 7 * x

Step 2: Simplify
Now, we simplify the equation obtained from the cross-multiplication. In this case, the equation becomes:

84 = 7x

Step 3: Solve for x
To solve for x, we isolate the variable on one side of the equation. In the current equation, we divide both sides by 7:

84/7 = x

x = 12

Therefore, the value of x in the proportion 4/7 = x/21 is 12.

It is important to note that the same steps can be applied to more complex proportions that involve multiple fractions. Let’s consider another example:

2/3 = 5/x

In this case, the steps remain the same. After cross-multiplying, we have:

2x = 3 * 5

We then simplify the equation:

2x = 15

To solve for x, we divide both sides by 2:

x = 15/2

x = 7.5

Therefore, the value of x in the proportion 2/3 = 5/x is 7.5.

It is worth mentioning that proportions can involve different types of fractions, such as mixed numbers or improper fractions. The steps to solve them remain consistent. Let’s consider one final example:

1/2 = x/3/4

To avoid confusion, we can convert the second fraction, 3/4, into an improper fraction. To do this, we multiply the whole number (3) by the denominator (4) and add the numerator (3) to obtain the new numerator:

3 * 4 + 3 = 15

Now, our proportion becomes:

1/2 = x/15

We cross-multiply:

1 * 15 = 2 * x

Simplifying the equation:

15 = 2x

To isolate x, we divide both sides by 2:

15/2 = x

x = 7.5

Thus, the value of x in the proportion 1/2 = x/3/4 is 7.5.

In conclusion, solving proportions with fractions involves following a few straightforward steps. By cross-multiplying, simplifying the equation, and isolating the variable, we can find the value of the unknown fraction. These techniques apply to proportions that involve various types of fractions, whether they are proper, improper, or mixed numbers. In applying these methods, individuals can confidently and accurately solve proportions involving fractions in a wide range of real-life scenarios, as well as in more complex mathematical problems.

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