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Chapter 3: Problem 24

Divide. Write a mixed numeral for the answer, where appropriate. $$ 12 \frac{1}{2} \div 50 $$

Short Answer

Expert verified
\( \frac{1}{4} \)

Step by step solution

01

Convert Mixed Number to Improper Fraction

First, convert the mixed number 12 \( \frac{1}{2} \) into an improper fraction. To do this, multiply the whole number part by the denominator of the fractional part and add the numerator. \[ 12 \frac{1}{2} = \frac{12 \times 2 + 1}{2} = \frac{24 + 1}{2} = \frac{25}{2} \]
02

Write the Division as a Multiplication

Division by a number is the same as multiplying by its reciprocal. So, change the division problem to a multiplication problem by multiplying by the reciprocal of 50, which is \( \frac{1}{50} \). \[ \frac{25}{2} \times \frac{1}{50} \]
03

Multiply the Fractions

Multiply the numerators and the denominators of the fractions: \[ \frac{25 \times 1}{2 \times 50} = \frac{25}{100} \]
04

Simplify the Fraction

Simplify \( \frac{25}{100} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 25: \[ \frac{25 \rightarrow 1}{100 \rightarrow 4} = \frac{1}{4} \]

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Key Concepts

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

Improper Fraction
An improper fraction is a fraction where the numerator is larger than or equal to the denominator. To convert a mixed number to an improper fraction, use the following steps:
  • Multiply the whole number by the denominator.
  • Add the numerator to the result from step 1.
  • Write the result from step 2 over the original denominator.

For example, converting the mixed number 12 \( \frac{1}{2} \) to an improper fraction:
\[ 12 \times 2 + 1 = 24 + 1 = 25 \]
So, 12 \( \frac{1}{2} \) becomes \( \frac{25}{2} \).
Reciprocal
The reciprocal of a number is simply the number flipped upside down. For a fraction, this means swapping the numerator and the denominator. This concept is crucial for dividing fractions because dividing by a fraction is the same as multiplying by its reciprocal.

To find the reciprocal of 50 (which can be written as \( \frac{50}{1} \)), simply invert it:
\[ \frac{1}{50} \]
So, the reciprocal of 50 is \( \frac{1}{50} \).
Fraction Simplification
Simplifying a fraction means reducing it to its lowest terms.

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, let's simplify \( \frac{25}{100} \)
  • The GCD of 25 and 100 is 25.
  • Divide 25 by 25 to get 1.
  • Divide 100 by 25 to get 4.

So, \( \frac{25}{100} = \frac{1}{4} \).
Mixed Number
A mixed number is a combination of a whole number and a fraction. Mixed numbers are often easier to understand than improper fractions because they clearly show a whole part and a fractional part.

To convert an improper fraction to a mixed number:
  • Divide the numerator by the denominator to get the whole number part.
  • The remainder becomes the numerator of the fractional part.
  • The denominator remains the same.

For example, let's convert the improper fraction \( \frac{25}{2} \) back to a mixed number:
  • Divide 25 by 2 (25 梅 2 = 12 R1).
  • The whole number part is 12.
  • The remainder is 1, so the fractional part is \( \frac{1}{2} \).

So, \( \frac{25}{2} \) as a mixed number is 12 \( \frac{1}{2} \).

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Most popular questions from this chapter

Simplify. $$ \left(\frac{2}{3}\right)^{2}-\frac{1}{3} \cdot 1 \frac{1}{4} $$

Add and simplify. $$ \frac{1}{2}+\frac{3}{8}+\frac{1}{4} $$

Find the average of \(10 \frac{2}{3}\) and \(24 \frac{5}{6}\).

Estimate each of the following by estimating each mixed numeral as a whole number or as a mixed numeral where the fraction part is \(\frac{1}{2}\) and by estimating each fraction as \(0, \frac{1}{2},\) or 1 . $$ 24 \div 7 \frac{8}{9} $$

Add and simplify. $$ \frac{3}{16}+\frac{5}{16}+\frac{4}{16} $$
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