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Magazine Chapter 5: Problem 46
Divide. Simplify, if possible. $$ 4 \frac{6}{7} \div \frac{1}{4} $$
Short Answer
Expert verified
\( \frac{136}{7} \) or \( 19 \frac{3}{7} \)
Step by step solution
01
Convert Mixed Number to Improper Fraction
Convert the mixed number 4 \( \frac{6}{7} \) into an improper fraction. To do this, multiply the whole number by the denominator of the fraction and add the numerator: \( 4 \times 7 + 6 = 28 + 6 = 34 \). Thus, \( 4 \frac{6}{7} \) becomes \( \frac{34}{7} \).
02
Set Up the Division
Rewrite the division problem with improper fractions: \( \frac{34}{7} \div \frac{1}{4} \).
03
Convert Division to Multiplication
To divide by a fraction, multiply by its reciprocal. The reciprocal of \( \frac{1}{4} \) is \( 4 \). So, the problem becomes \( \frac{34}{7} \times 4 \).
04
Multiply the Fractions
Multiply the numerators and multiply the denominators: \( \frac{34 \times 4}{7 \times 1} \). This simplifies to \( \frac{136}{7} \).
05
Simplify the Fraction
Check if the fraction \( \frac{136}{7} \) can be simplified. Since 136 and 7 have no common factors other than 1, the fraction is already in its simplest form.
06
Convert to Mixed Number (if needed)
Optionally, convert \( \frac{136}{7} \) back to a mixed number by dividing 136 by 7. The quotient is 19 and the remainder is 3, so \( \frac{136}{7} \) can be written as \( 19 \frac{3}{7} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
mixed number conversion
A mixed number contains both a whole number and a fraction. For instance, in the exercise, we start with the mixed number 4 \( \frac{6}{7} \). To perform operations involving mixed numbers, it's often easier to convert them into improper fractions. This makes calculations straightforward. Here鈥檚 how you convert a mixed number into an improper fraction:
- Multiply the whole number by the denominator of the fraction. In our example, multiply 4 by 7: \(4 \times 7 = 28 \).
- Add the result to the numerator of the fraction. So, \(28 + 6 = 34\).
- Place the result over the original denominator to get the improper fraction: \(\frac{34}{7}\).
improper fraction
An improper fraction has a numerator larger than or equal to its denominator. It represents a value greater than or equal to one. In our example, after converting 4 \( \frac{6}{7} \) to an improper fraction, we get \( \frac{34}{7} \). Improper fractions can be easier to work with, especially in division or multiplication problems.
If you want, you can convert an improper fraction back into a mixed number. To convert \( \frac{136}{7} \) back to a mixed number:
If you want, you can convert an improper fraction back into a mixed number. To convert \( \frac{136}{7} \) back to a mixed number:
- Divide the numerator by the denominator: \(136 \div 7 = 19\) with a remainder of 3.
- The quotient becomes the whole number part: 19.
- The remainder becomes the numerator of the fractional part: \(\frac{3}{7} \).
- The mixed number is \( 19 \frac{3}{7} \).
reciprocal multiplication
When dividing by a fraction, you can simplify the process by multiplying by its reciprocal. A reciprocal of a fraction is obtained by swapping its numerator and denominator. In our exercise, we're dividing by \( \frac{1}{4} \). Its reciprocal is \(4 \) (or \( \frac{4}{1} \)).
So, instead of dividing \( \frac{34}{7} \) by \( \frac{1}{4} \), you multiply by the reciprocal:
So, instead of dividing \( \frac{34}{7} \) by \( \frac{1}{4} \), you multiply by the reciprocal:
- Rewrite the division as multiplication: \( \frac{34}{7} \div \frac{1}{4} \) becomes \( \frac{34}{7} \times 4 \).
fraction simplification
Simplifying fractions is the process of making a fraction as simple as possible. You do this by ensuring that the numerator and denominator have no common factors other than 1. In the exercise, we arrived at \( \frac{136}{7} \). To check if this can be simplified further:
- Find the greatest common divisor (GCD) of the numerator and the denominator.
- If the GCD is 1, the fraction is already in its simplest form. For \( \frac{136}{7} \), the GCD is 1 because 7 is a prime number, and 136 is not divisible by 7.
