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Chapter 5: Problem 262

Simplify. Give the answer as a decimal. \(5.79 \div \frac{3}{4}\)

Short Answer

Expert verified
7.72

Step by step solution

01

Write the Division as Multiplication

When dividing by a fraction, convert the operation into multiplication by the reciprocal of the fraction. The reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\). Therefore, rewrite the expression \(5.79 \div \frac{3}{4}\) as \(5.79 \times \frac{4}{3}\).
02

Multiply the Numerator

To simplify the expression \(5.79 \times \frac{4}{3}\), first multiply the numerators: \(5.79 \times 4 = 23.16\).
03

Divide by the Denominator

Divide the result from Step 2 by the denominator: \(23.16 \div 3 = 7.72\).
04

Final Answer

The simplified form of \(5.79 \div \frac{3}{4}\) is \(7.72\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reciprocal
Understanding the concept of a reciprocal is key when dealing with division by fractions. A reciprocal of a number is simply one divided by that number. For example, the reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\). This is because when you multiply a number by its reciprocal, the product is always 1.

In our specific exercise, we had to divide 5.79 by \(\frac{3}{4}\). Instead of performing direct division, we changed it to a multiplication problem using the reciprocal. Therefore, \(\frac{3}{4}\) became \(\frac{4}{3}\). This simplification step turns a complicated division problem into a much easier multiplication problem.
Multiplication of Fractions
Now that we have the reciprocal, the next step involves multiplying fractions. Multiplying fractions is straightforward: multiply the numerators together and the denominators together. In our example, once the division was converted to multiplication, we had \(5.79 \times \frac{4}{3}\).

Multiply \(5.79 \times 4\) to get 23.16. This handles the numerator.

Then, set the new product over the original denominator, 3, giving us \( \frac{23.16}{3}\). This completes the multiplication process of the fraction.
Simplifying Expressions
The last step is to simplify the expression to get the final answer. Simplifying an expression often means performing the final steps in calculations to achieve the simplest form.

In this problem, after multiplying, we got \(\frac{23.16}{3}\).

To simplify, you merely perform the division: \(23.16 \div 3 = 7.72.\) This simplification reduces the numerical expression to a tidy decimal format.

From start to finish, breaking down each step鈥攔eciprocating, multiplying, and then simplifying鈥攎akes solving complex fraction problems manageable and logical.

As a final note, always ensure your final answer is in the simplest form for accuracy and clarity.

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