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Chapter 4: Problem 46

Write the mixed number in proper form (that is, as a whole number with a proper fraction that is simplified to lowest terms). $$4 \frac{8}{7}$$

Short Answer

Expert verified
\(5 \frac{1}{7}\)

Step by step solution

01

Identify the Mixed Number

The given mixed number is \(4 \frac{8}{7}\).
02

Simplify the Fraction

Identify that \(\frac{8}{7}\) is an improper fraction because the numerator is larger than the denominator. Convert it into a proper fraction.
03

Convert the Improper Fraction

Divide 8 by 7 to get \(1 \frac{1}{7}\). This means \(\frac{8}{7} = 1 \frac{1}{7}\).
04

Add the Whole Numbers

Since \(4 \frac{8}{7}\) is the original mixed number, add the whole numbers together. \[4 + 1 = 5.\] Now we have \(5 \frac{1}{7}\).
05

Simplify the Mixed Number

The final simplified mixed number is \(5 \frac{1}{7}\).

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Key Concepts

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

Improper Fractions
An improper fraction is a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). For example, in the fraction \(\frac{8}{7}\), the numerator 8 is greater than the denominator 7. Improper fractions are quite common and can be easily converted into mixed numbers.
To convert an improper fraction to a mixed number:
  • Divide the numerator by the denominator to find the whole number part.
  • The remainder from this division is the numerator of the proper fraction part.
  • Keep the same denominator for the fraction part.
For example, with the improper fraction \(\frac{8}{7}\):
  • Divide 8 by 7 to get 1 with a remainder of 1.
  • The whole number part is 1, and the fraction part is \(\frac{1}{7}\).
Therefore, \(\frac{8}{7}\) converts to the mixed number \(\text{1}\frac{1}{7}\). This process helps in understanding and simplifying fractions and making calculations easier.
Fraction Simplification
Simplifying fractions means reducing them to their simplest form. A fraction is simplified when the numerator and the denominator have no common factors other than 1. For instance, \(\frac{8}{7}\) is already in its simplest form because 8 and 7 have no common factors.
Steps to simplify a fraction:
  • Find the greatest common divisor (GCD) of the numerator and the denominator.
  • Divide both the numerator and the denominator by the GCD.
For example, to simplify \(\frac{10}{15}\):
  • The GCD of 10 and 15 is 5.
  • Divide both 10 and 15 by 5 to get \(\frac{2}{3}\).
Simplifying helps in making the fraction easier to understand and work with in different arithmetic operations. For our exercise, since \(\frac{8}{7}\) is already simplified, no further reduction is necessary.
Arithmetical Operations with Mixed Numbers
Adding, subtracting, multiplying, and dividing mixed numbers involves a few extra steps compared to regular fractions. Mixed numbers have both a whole number and a fraction part, making operations slightly more complex.
Let's break it down:
  • Adding/Subtracting Mixed Numbers: Convert to improper fractions first, perform the operation, then convert back if necessary.
  • Multiplying Mixed Numbers: Convert to improper fractions, multiply, then simplify and convert back if needed.
  • Dividing Mixed Numbers: Convert to improper fractions, multiply by the reciprocal, simplify, and convert back if needed.
For example, to add \(\text{2}\frac{1}{3}\) and \(\text{1}\frac{2}{5}\):
  • Convert both to improper fractions: \(\text{2}\frac{1}{3} = \text{}\frac{7}{3}\) and \(\text{1}\frac{2}{5}=\text{}\frac{7}{5}\).
  • Find a common denominator and perform the addition: \(\frac{7}{3} + \frac{7}{5} = \frac{35}{15} + \frac{21}{15} = \frac{56}{15}\).
  • Convert back to a mixed number: \(\frac{56}{15} = \text{3}\frac{11}{15}\).
Understanding and mastering these operations makes it easier to handle real-world problems involving mixed numbers and fractions.

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Most popular questions from this chapter

Simplify the expression. $$\frac{16 a b}{10 a}$$

Simplify to lowest terms by first reducing the powers of 10. $$-\frac{2000}{1500}$$

Mr. Bishop and Ms Waymire both gave exams today. By mid-afternoon, Mr. Bishop had finished grading 16 out of 36 exams, and Ms. Waymire had finished grading 15 out of 27 exams. a. What fractional part of her total has Ms. Waymire completed? b. What fractional part of his total has Mr. Bishop completed?

True or false? Suppose \(m\) and \(n\) are nonzero numbers, where \(m>n .\) Then \(\frac{n}{m}\) is a proper fraction.

Convert the improper fraction to a mixed number. $$-\frac{27}{10}$$
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