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Chapter 3: Problem 21

Simplify. $$ \frac{\frac{1}{40}}{\frac{1}{50}} $$

Short Answer

Expert verified
The simplified form is \(\frac{5}{4}\).

Step by step solution

01

Write Down the Problem

Start with the given fraction: \(\frac{\frac{1}{40}}{\frac{1}{50}}\)
02

Understand the Division of Fractions

Remember that dividing by a fraction is equivalent to multiplying by its reciprocal.
03

Rewrite the Problem Using Reciprocal

Rewrite the division as multiplication by the reciprocal: \(\frac{1}{40} \times \frac{50}{1}\)
04

Simplify the Multiplication

Multiply the numerators together and the denominators together: \(\frac{1 \times 50}{40 \times 1} = \frac{50}{40}\)
05

Simplify the Resulting Fraction

Simplify \(\frac{50}{40}\) by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 10: \(\frac{50 \div 10}{40 \div 10} = \frac{5}{4}\)

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Key Concepts

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

simplifying fractions
Simplifying fractions means to make the fraction as simple as possible. You do this by dividing the numerator (top part) and the denominator (bottom part) by their greatest common divisor (GCD).
For example, in our solution, we found that the GCD of 50 and 40 is 10. This means we can divide both 50 and 40 by 10 to get the simplified fraction.
So, \(\frac{50}{40}\) became \(\frac{5}{4}\) after dividing both parts by 10.
Always look for the largest number that can evenly divide both the numerator and the denominator to simplify a fraction.
reciprocal
A reciprocal of a fraction is just a flipped version of that fraction. If you have \(\frac{a}{b}\), the reciprocal is \(\frac{b}{a}\).
Reciprocals are really useful when you are dividing fractions.
In our exercise, we were given the fraction \(\frac{1}{40} \/\ \frac{1}{50}\).
Instead of dividing, we multiplied by the reciprocal of \(\frac{1}{50}\), which is \(\frac{50}{1}\).
So, our division problem turned into a multiplication problem: \(\frac{1}{40} \times \frac{50}{1}\).
greatest common divisor
The greatest common divisor (GCD) is the largest number that can evenly divide both the numerator and the denominator of a fraction.
Finding the GCD helps you simplify fractions easily.
When we had \(\frac{50}{40}\), we needed to find their GCD to simplify. We checked and found that 10 is the largest number that can divide both 50 and 40.
This is why we divided both the numerator and the denominator by 10, simplifying \(\frac{50}{40}\) to \(\frac{5}{4}\).
Finding the GCD is a crucial step in working with fractions because it makes the numbers much easier to handle.

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Most popular questions from this chapter

A tile \(\frac{5}{8}\) in. thick is glued to a board \(\frac{7}{8}\) in. thick. The glue is \(\frac{3}{32}\) in. thick. How thick is the result?

A park naturalist hiked \(\frac{3}{5}\) mi to a lookout, another \(\frac{3}{10}\) mi to an osprey's nest, and finally \(\frac{3}{4}\) mi to a campsite. How far did the naturalist hike?

Simplify. $$ \frac{7}{10} \cdot \frac{4}{5} \div \frac{2}{3} $$

Find the average of \(\frac{1}{6}, \frac{1}{8},\) and \(\frac{3}{4}\).

Simplify. $$ \text { List all the factors of } 42 . \quad[2.1 \mathrm{a}] $$
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