Solve the Mixed Number Equation: 70 = 4½b

Q: Why convert to an improper fraction instead of using decimals?

Converting to improper fractions keeps your work exact! Using decimals like 4.5 can lead to rounding errors. Fractions give you the precise answer every time.

Q: How do I convert a mixed number to an improper fraction?

Use this formula: $ a\frac{b}{c} = \frac{(a \times c) + b}{c} $For $ 4\frac{1}{2} $: multiply 4 × 2 = 8, add 1 = 9, so $ 4\frac{1}{2} = \frac{9}{2} $

Q: What's the reciprocal and why do I multiply by it?

The reciprocal flips the fraction: reciprocal of $ \frac{9}{2} $ is $ \frac{2}{9} $. Multiplying by the reciprocal is the same as dividing, which isolates the variable!

Q: How do I convert my final improper fraction back to a mixed number?

Divide the numerator by denominator:$ \frac{140}{9} $: 140 ÷ 9 = 15 remainder 5So $ \frac{140}{9} = 15\frac{5}{9} $

Q: Can I solve this equation a different way?

Yes! You could divide both sides by $ 4\frac{1}{2} $, but converting to improper fractions first makes the arithmetic much easier and less error-prone.

Linear Equations with Mixed Number Coefficients

$70=4\frac{1}{2}b$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:03 We want to isolate variable B

00:07 Let's isolate variable B

00:12 Convert from number and fraction to fraction

00:19 Replace division with multiplication by reciprocal

00:27 Break down 140 to 135 plus 5

00:30 Break down one fraction into whole number and remainder

00:35 This is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$70=4\frac{1}{2}b$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Convert the mixed number to an improper fraction
Step 2: Isolate $b$ using multiplication
Step 3: Simplify to find the value of $b$

Now, let's work through each step:

Step 1: Convert $4\frac{1}{2}$ to an improper fraction:

$4\frac{1}{2} = \frac{9}{2}$

Step 2: Isolate $b$ on one side of the equation:

The equation becomes $70 = \frac{9}{2}b$

To isolate $b$ , multiply both sides by the reciprocal of $\frac{9}{2}$ :

$b = 70 \times \frac{2}{9}$

Step 3: Perform the multiplication:

$b = \frac{70 \times 2}{9}$

$b = \frac{140}{9}$

The improper fraction $\frac{140}{9}$ converts to a mixed number:

$b = 15 \frac{5}{9}$

Therefore, the solution to the problem is $b = 15\frac{5}{9}$ .

Final Answer

$b=15\frac{5}{9}$

Key Points to Remember

Essential concepts to master this topic

Conversion Rule: Change mixed numbers to improper fractions first
Technique: Convert $4\frac{1}{2}$ to $\frac{9}{2}$ then multiply by reciprocal
Check: Substitute back: $4\frac{1}{2} \times 15\frac{5}{9} = 70$ ✓

Common Mistakes

Avoid these frequent errors

Adding instead of multiplying when converting mixed numbers
Don't convert $4\frac{1}{2}$ as 4 + 1/2 = 4.5 and work with decimals! This leads to rounding errors and wrong final answers. Always convert to improper fractions: $4\frac{1}{2} = \frac{4 \times 2 + 1}{2} = \frac{9}{2}$ .

Practice Quiz

Test your knowledge with interactive questions

Solve for X:

$$ 5x=25 $$

FAQ

Everything you need to know about this question

Why convert to an improper fraction instead of using decimals?

Converting to improper fractions keeps your work exact! Using decimals like 4.5 can lead to rounding errors. Fractions give you the precise answer every time.

How do I convert a mixed number to an improper fraction?

Use this formula: $a\frac{b}{c} = \frac{(a \times c) + b}{c}$

For $4\frac{1}{2}$ : multiply 4 × 2 = 8, add 1 = 9, so $4\frac{1}{2} = \frac{9}{2}$

What's the reciprocal and why do I multiply by it?

The reciprocal flips the fraction: reciprocal of $\frac{9}{2}$ is $\frac{2}{9}$ . Multiplying by the reciprocal is the same as dividing, which isolates the variable!

How do I convert my final improper fraction back to a mixed number?

Divide the numerator by denominator:

$\frac{140}{9}$ : 140 ÷ 9 = 15 remainder 5
So $\frac{140}{9} = 15\frac{5}{9}$

Can I solve this equation a different way?

Yes! You could divide both sides by $4\frac{1}{2}$ , but converting to improper fractions first makes the arithmetic much easier and less error-prone.

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