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Chapter 5: Problem 25

For Exercises 25鈥�32, write the improper fraction as a mixed number. $$\frac{30}{7}$$

Short Answer

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Question: Convert the improper fraction $$\frac{30}{7}$$ into a mixed number. Answer: The improper fraction $$\frac{30}{7}$$ can be converted into the mixed number $$4\,\frac{2}{7}$$.

Step by step solution

01

Identify the given fraction

The given improper fraction is: $$\frac{30}{7}$$
02

Divide the numerator by the denominator

To convert the improper fraction into a mixed number, divide the numerator (30) by the denominator (7): $$30 \div 7 = 4\,\text{remainder}\, 2$$
03

Write the result as a mixed number

From the division in Step 2, we know that the quotient is 4 and the remainder is 2. Use this information to write the result as a mixed number: $$4\,\frac{2}{7}$$ So, the improper fraction $$\frac{30}{7}$$ can be written as the mixed number $$4\,\frac{2}{7}$$.

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Key Concepts

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

Improper Fractions
Understanding improper fractions is key when you are dealing with fractions that seem a bit out of balance. An improper fraction has a numerator (the top number) that is larger than its denominator (the bottom number). This can create more than a 'whole' number, which is why they're termed 'improper.'Here鈥檚 how you can spot them and what they mean:
  • Look at the fraction. If the top number is bigger than the bottom one, it's improper.
  • They represent a quantity greater than one.
  • They often need conversion for easy interpretation, especially in practical scenarios like measurements.
Think of an improper fraction like \[ \frac{30}{7} \] which means 30 pieces of something that each fit into 7 at a time. Essentially, it shows that you have several whole parts plus some extra, unruly, leftover pieces! Converting these to mixed numbers helps simplify and make more sense of them.
Mixed Numbers
Mixed numbers are an elegant way to express improper fractions. In essence, a mixed number combines an integer and a proper fraction. For example, when we converted \[ \frac{30}{7} \] into the mixed number \[ 4 \frac{2}{7} \], we discovered:
  • The whole number 4 represents the full parts you could divide from the total.
  • The fraction \( \frac{2}{7} \) captures what is left over after filling these whole parts.
To make a mixed number, you divide the numerator by the denominator:1. The quotient turns into the whole number part.2. The remainder becomes the new numerator of the fractional part.3. The denominator stays the same as the one in the original improper fraction.Mixed numbers are useful as they turn unwieldy improper fractions into more intuitive numbers, making them easier to understand in daily contexts.
Division
The concept of division is at the heart of converting improper fractions to mixed numbers. Division in this context can be thought of as sharing equally and finding out how many whole parts and leftovers you have. Here's how it relates to fractions:
  • Consider the division shown in our example: \(30 \div 7 = 4 \) remainder \, \(2\).
  • This operation splits your numerator, 30, into parts of 7, which gives you 4 whole parts.
  • The remainder here (2) indicates that beyond the 4 whole parts, there are 2 more pieces yet to fill another whole part of 7.
So when you divide while working with fractions, you're essentially distributing the total into even sections, finding out how many full sections you have, and seeing what鈥檚 left over. This division process is fundamental in converting improper fractions to their mixed number counterparts.

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Most popular questions from this chapter

Solve. Amanda wants to hang a picture frame so that the bottom of the frame is 54 , \(\frac{1}{2}\) inches from the floor. The hanger on the back of the picture is \(11 \frac{5}{8}\) inches from the bottom of the frame. How high should she place the nail?

Solve. Write all answers in lowest terms. \(\frac{5}{6}\) of a company's employees live within a 15 -mile radius of the company. Of these employees, \(\frac{3}{4}\) live within a 10 -mile radius. What fraction of all employees live within a 10 -mile radius?

Use \(<,>,\) or \(=\) to make a true statement. $$\frac{9}{16} ? \frac{12}{18}$$

Use \(d=r t\). If a family drove \(4 \frac{3}{4}\) hours at an average rate of 65 miles per hour, how far did they travel?

Solve. If Lonnie has a test average (mean) of 94 or higher, she will be exempt from the final exam. She has taken three out of the four tests for the course. Her scores on the three tests are \(88,92,\) and \(90 .\) What is the lowest she can score on the last test to be exempt from the final exam?
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