Solving Mixed Number Equations: A Step-by-Step Guide

Nov 17, 2025 by Alex Johnson 53 views

Mathematics can sometimes feel like a puzzle, and one common type of puzzle involves solving equations. Today, we're going to tackle a specific kind of math problem: solving an equation with mixed numbers. Specifically, we'll be looking at how to solve the equation $7 rac{1}{8}=4 rac{3}{4}+n$ . This type of problem is fundamental in understanding algebraic concepts and how to manipulate fractions, which are essential building blocks in higher mathematics. Whether you're a student looking to master your homework or just someone curious about how these problems are solved, you've come to the right place. We'll break down the process into easy-to-understand steps, making sure you feel confident by the end.

Understanding Mixed Numbers and Equations

Before we dive into solving the equation $7 rac{1}{8}=4 rac{3}{4}+n$ , let's take a moment to understand what we're dealing with. A mixed number is a whole number and a proper fraction combined. For example, $7 rac{1}{8}$ means 7 whole units and one-eighth of another unit. An equation is a mathematical statement that asserts the equality of two expressions. In our case, the equation $7 rac{1}{8}=4 rac{3}{4}+n$ tells us that the value on the left side ( $7 rac{1}{8}$ ) is exactly equal to the value on the right side ( $4 rac{3}{4}+n$ ). Our goal is to find the value of 'n', which is the unknown variable in this equation. To do this, we need to isolate 'n' on one side of the equation. This involves using inverse operations, much like solving any other algebraic equation. The added complexity here comes from working with mixed numbers, which often require conversion to improper fractions for easier manipulation.

Converting Mixed Numbers to Improper Fractions

One of the most effective strategies for solving equations involving mixed numbers is to convert them into improper fractions. An improper fraction is a fraction where the numerator is greater than or equal to the denominator. This conversion makes it much simpler to perform addition, subtraction, multiplication, and division. Let's convert our mixed numbers: $7 rac{1}{8}$ and $4 rac{3}{4}$ .

To convert a mixed number like $a rac{b}{c}$ into an improper fraction, you use the formula: $rac{(a imes c) + b}{c}$ .

For $7 rac{1}{8}$ :
- Multiply the whole number (7) by the denominator (8): $7 imes 8 = 56$ .
- Add the numerator (1) to the result: $56 + 1 = 57$ .
- Keep the same denominator (8).
- So, $7 rac{1}{8}$ becomes the improper fraction $rac{57}{8}$ .
For $4 rac{3}{4}$ :
- Multiply the whole number (4) by the denominator (4): $4 imes 4 = 16$ .
- Add the numerator (3) to the result: $16 + 3 = 19$ .
- Keep the same denominator (4).
- So, $4 rac{3}{4}$ becomes the improper fraction $rac{19}{4}$ .

Now, our original equation $7 rac{1}{8}=4 rac{3}{4}+n$ can be rewritten using these improper fractions: $rac{57}{8} = rac{19}{4} + n$ . This transformed equation is much easier to work with algebraically.

Isolating the Variable 'n'

With our equation now in terms of improper fractions, $rac{57}{8} = rac{19}{4} + n$ , our next step is to isolate the variable 'n'. To do this, we need to get rid of the $rac{19}{4}$ that is being added to 'n' on the right side of the equation. The inverse operation of addition is subtraction. Therefore, we will subtract $rac{19}{4}$ from both sides of the equation to maintain the equality.

$rac{57}{8} - rac{19}{4} = ( rac{19}{4} + n) - rac{19}{4}$

This simplifies to:

$rac{57}{8} - rac{19}{4} = n$

Now we have 'n' isolated, but we still need to perform the subtraction of the two fractions. To subtract fractions, they must have a common denominator. Our current denominators are 8 and 4. The least common multiple of 8 and 4 is 8. So, we need to convert $rac{19}{4}$ into an equivalent fraction with a denominator of 8.

To do this, we multiply both the numerator and the denominator of $rac{19}{4}$ by 2 (since $4 imes 2 = 8$ ):

$rac{19}{4} imes rac{2}{2} = rac{19 imes 2}{4 imes 2} = rac{38}{8}$

Now we can substitute this back into our equation:

$rac{57}{8} - rac{38}{8} = n$

Performing the Fraction Subtraction

We are now at the stage where we can perform the fraction subtraction: $rac{57}{8} - rac{38}{8} = n$ . Since both fractions have the same denominator (8), we can simply subtract the numerators and keep the common denominator.

$n = rac{57 - 38}{8}$

$n = rac{19}{8}$

So, the value of 'n' is $rac{19}{8}$ . This is an improper fraction. Often, answers are preferred in their simplest form, which can be either an improper fraction or a mixed number. Let's check if $rac{19}{8}$ can be simplified further. The greatest common divisor of 19 and 8 is 1, so the fraction is already in its simplest form.

Converting the Answer Back to a Mixed Number (Optional but often preferred)

While $rac{19}{8}$ is a correct answer, sometimes it's more intuitive to express it as a mixed number. To convert an improper fraction like $rac{19}{8}$ back to a mixed number, we divide the numerator (19) by the denominator (8).

How many times does 8 go into 19? It goes in 2 times ( $8 imes 2 = 16$ ).
What is the remainder? $19 - 16 = 3$ .
The quotient (2) becomes the whole number part.
The remainder (3) becomes the numerator of the fractional part.
The denominator (8) stays the same.

So, $rac{19}{8}$ converts to the mixed number $2 rac{3}{8}$ .

Now, let's look at the options provided in the original question: a) $1 rac{9}{4}$ b) $rac{19}{4}$ c) $rac{19}{8}$ d) $1 rac{9}{8}$ .

Our calculated value for 'n' is $rac{19}{8}$ , which matches option (c). It's worth noting that option (d) $1 rac{9}{8}$ is not a proper mixed number because the fractional part $rac{9}{8}$ is an improper fraction itself. If we were to convert $1 rac{9}{8}$ to an improper fraction, it would be $rac{(1 imes 8) + 9}{8} = rac{17}{8}$ , which is not our answer.

Verifying the Solution

To ensure our answer is correct, we can substitute $n = rac{19}{8}$ back into the original equation: $7 rac{1}{8}=4 rac{3}{4}+n$ . We already converted the mixed numbers to improper fractions: $rac{57}{8} = rac{19}{4} + rac{19}{8}$ .

To check if the equality holds, we need to add the fractions on the right side. We need a common denominator, which is 8.

$rac{19}{4} = rac{19 imes 2}{4 imes 2} = rac{38}{8}$

So, the right side becomes:

$rac{38}{8} + rac{19}{8} = rac{38 + 19}{8} = rac{57}{8}$

The left side of the equation is $rac{57}{8}$ . Since the left side equals the right side ( $rac{57}{8} = rac{57}{8}$ ), our solution $n = rac{19}{8}$ is correct.

Conclusion

Solving equations involving mixed numbers might seem daunting at first, but by following a systematic approach, it becomes quite manageable. The key steps involve converting mixed numbers to improper fractions, using inverse operations to isolate the variable, finding common denominators for fraction arithmetic, and finally, verifying your solution. We successfully solved the equation $7 rac{1}{8}=4 rac{3}{4}+n$ and found that $n = rac{19}{8}$ , which corresponds to option (c). Remember, practice is crucial for mastering any mathematical concept. Keep working through similar problems, and you'll find your confidence growing!

For further exploration into algebraic concepts and fraction manipulation, you might find the resources at Khan Academy very helpful. Their comprehensive lessons and practice exercises cover a wide range of mathematical topics.