What is the difference between the ratios $\frac{1}{4}$ and $\frac{1}{5}$?
Answer (1)
Find a common denominator: Rewrite 4 1 and 5 1 as 20 5 and 20 4 .
Increase the denominator: Multiply both fractions by 4 to get 80 20 and 80 16 .
Identify three rational numbers between the fractions: 80 17 , 80 18 , and 80 19 .
Simplify the fractions: The three rational numbers are 80 17 , 40 9 , and 80 19 . 80 17 , 40 9 , 80 19
Explanation
Problem Analysis We are asked to find three rational numbers between 4 1 and 5 1 .
Finding Common Denominator First, we need to find a common denominator for the two fractions. The least common multiple of 4 and 5 is 20. So, we rewrite the fractions as 4 1 = 20 5 and 5 1 = 20 4 .
Increasing the Denominator Since we need to find three rational numbers between 20 5 and 20 4 , we need to increase the denominator. Multiply both fractions by 4 to get 20 5 × 4 4 = 80 20 and 20 4 × 4 4 = 80 16 .
Finding Rational Numbers Now we can easily find three rational numbers between 80 16 and 80 20 . For example, 80 17 , 80 18 , and 80 19 .
Simplifying Fractions We can simplify the fractions if possible. 80 18 simplifies to 40 9 .
Final Answer Therefore, three rational numbers between 4 1 and 5 1 are 80 17 , 40 9 , and 80 19 .
Examples
Imagine you're baking a cake and the recipe calls for a certain amount of flour, but you want to add a little more to make it extra fluffy. If the recipe says to use between 4 1 and 5 1 cups of flour, finding rational numbers between these two fractions helps you determine exactly how much more flour you can add. For instance, you could choose to add 80 17 , 40 9 , or 80 19 cups of flour, ensuring your cake turns out just right!
